COMPUTATIONAL METHODS FOR NON-IDEAL PLASMAS By Lucas J. Stanek A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Computational Mathematics, Science and Engineering – Doctor of Philosophy 2022 ABSTRACT Plasmas are many-body systems of interacting charged particles that exist naturally and can be created experimentally. For example, plasmas are found in many astrophysical systems like the corona of the sun, the Earth’s ionosphere, and in the interior of white-dwarf stars. In engineering and medicine, plasmas are used during the design process of semi-conductors or for inactivating viruses like COVID-19. Plasmas also occur in nuclear fusion experiments which promise a nearly infinite supply of clean, renewable energy. Quantifying the behaviors of plasmas experimentally can be challenging due to the short time- scales and small length-scales that interactions between the plasma particles occur. In many cases, computational approaches are used to simulate the dynamics of plasmas to supplement the dearth of experimental data. The accuracy of these computational methods is largely unknown across the entire parameter regimes plasmas occupy limiting their predictive capabilities. This dissertation is composed of four distinct projects all with the common goal of developing numerical methods for rapidly and accurately computing properties of non-ideal plasmas. First, we focus on the data-driven discovery of pair interaction potentials for molecular dynamics simulations of dense plasmas across a wide range of temperatures and elements. We find that our pair interaction potentials simulate the ionic interactions in a plasma with accuracy comparable to Kohn-Sham molecular dynamics but with orders of magnitude less computation cost. Second, we develop theoretical models that avoid the need for numerical simulations of plasma mixtures altogether. Our theoretical models show reasonable agreement with molecular dynamics data across the both the weak and strong coupling regimes. Third, we use techniques in machine learning to interpolate plasma properties data with multiple sources of data. We find that our machine learning method accurately predicts trends in data even in the absence of high-fidelity calculations. Lastly, we implement a numerical scheme for solving kinetic equations with applications to ultracold neutral plasma mixtures and high-energy-density plasmas. With our simulation results, we suggest plasma conditions for future experiments and we discuss natural extensions of our numerical method that will be the basis of future work. Copyright by LUCAS J. STANEK 2022 To my mother, father, and sister. iv ACKNOWLEDGEMENTS There have been numerous individuals that have helped me throughout the pursuit of completing my Ph.D. These individuals exist in the form of mentors, colleagues, friends, and family who have tirelessly supported me in a variety of ways for which I am eternally grateful; without them, I would not be where I am at today. In particular, I have had the pleasure of working with and learning from patient, thoughtful, and talented mentors who have guided me in my academic endeavors and have pushed me to become a better student and researcher. I would like to thank my research advisor Michael Murillo for guiding me along this journey. You have always extended a helping hand and a critical ear whenever needed. You have mentored me since my first days in CMSE and have made learning about all things plasma physics fun and exciting. Your mentoring spans far beyond physics in that you have made deliberate efforts to ensure that I have the necessary skills in presenting and technical writing. You have instilled in me your affinity for exploring alternative avenues in research and love for data visualization. I think that you’ll find the plots in this dissertation are, “just like downtown." Thank you to Brian O’Shea for his constant support throughout graduate school. Brian, you were an excellent graduate advisor, and have always provided me with guidance. I’d like to also thank Shaunak Bopardikar for introducing me to the field of multi-fidelity modeling and offering a unique and valuable perspective on research. Additionally, I want to thank Chandre Dharma- wardana for his continued involvement in furthering my understanding of physics and technical writing. A big shoutout to my colleagues Jeff Haack and Liam Stanton. Jeff, you have always been insightful and helpful in “everything numerics" and have made yourself available whenever I have needed to phone-a-friend. Liam, you have offered me advice in both life and math. I deeply value our many conversations about and am grateful that you have pushed me to be precise in my understanding of physics. Thank you to Lisa Murillo for her careful editing of many of my manuscripts. Lisa, because of you, I write much more gooder now. I want to extend my gratitude to John Luginsland and Frank Graziani for being great mentors and crucial members of my professional network. On numerous occasions, both of you have allowed v me access to doors that would otherwise be shut. I’d also like to thank all of my mentors at Sandia National Laboratories for their involvement in my Ph.D. work over the years. In particular, I’d like to thank Kris Beckwith, Ray Clay, Pat Knapp, and Stephanie Hansen for making my internship experience at Sandia exciting and fruitful. I am indebted to my friends who have cheered me on and supported me for many years. Adam, you have supported me in numerous ways and have always offered me encouragement. Tyler, I value our conversations about life and grad school. Dustin, thank you for your random phone calls to check-in and offering me advice whenever needed. I would also like to acknowledge my friends in CMSE. Nat, you have been a constant fount of support and I am so grateful for you. I appreciate our video game session, our mutual ridiculousness, and your help throughout this journey. Sarah, I look back fondly on when we’d study together for our qual classes and end up distracting each other by playing battleship. Our friendship has offered a much needed respite from the hectic life of graduate school and I could not have done it without you. Bill, without you, I would have not learned about CMSE. Thank you for being a proponent of me over the years; I am glad we were able to experience grad school together. Lastly, my family has consistently provided unwavering support. Mom, you have taught me to be compassionate and caring, Dad, you have always offered unbiased advice to me when I’ve needed it, Mike, you have provided me with endless guidance, and Lindsey, you have been a role model for me since we were kids and have instilled your work ethic and tenacity in me. vi TABLE OF CONTENTS CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Plasma Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Computational Modeling of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHAPTER 2 DATA-DRIVEN FORCE LAWS FOR MOLECULAR DYNAMICS SIMULATIONS OF DENSE PLASMAS . . . . . . . . . . . . . . . . . . . 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Models for the Interaction Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 CHAPTER 3 ANALYTIC MODELS FOR INTERDIFFUSION IN DENSE PLASMA MIXTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Interdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Thermodynamic Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Interdiffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 CHAPTER 4 MULTI-FIDELITY REGRESSION FOR PLASMA PROPERTIES DATA . 74 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Dataset and Regression Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Multi-fidelity Regression of Plasma Transport-Coefficient Data . . . . . . . . . . . 90 4.4 Regression of Sparse Disparate Data . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 CHAPTER 5 KINETIC MODELING OF STRONGLY COUPLED PLASMA MIXTURES 106 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3 The Vlasov Equation in Conservative Form . . . . . . . . . . . . . . . . . . . . . 116 5.4 Plasma Expansion Into a Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.5 Numerical Methods for Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . 134 5.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 CHAPTER 6 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 171 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 APPENDIX A ANALYTIC SOLUTION TO POISSON’S EQUATION IN SPHERICAL COORDINATES . . . . . . . . . . . . . . . . . . . . . . . 199 vii APPENDIX B DYNAMIC STRUCTURE FACTOR IN THE RANDOM PHASE APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 APPENDIX C SECOND-ORDER UPWINDING STENCIL IN SPHERICAL COORDINATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 viii CHAPTER 1 INTRODUCTION 1.1 Background Plasmas are charged particle systems that can be commonly found in nature. Plasmas respond strongly to electric and magnetic fields – requiring free electrons much like a metal. Free electrons in a system are created by a process called ionization. When an element is ionized, bound electrons are removed from an atom by sources of extreme pressure, temperature, and/or radiation. For metals, the free electrons result from pressure ionization due to the density of the metal. In ultracold neutral plasmas (UNPs), ionization occurs due to radiation. In other states of matter, like hot dense matter, ionization can result from combinations of pressure, temperature, and radiation sources. As shown in Fig. 1.1, plasmas occupy a wide temperature and density range and can be classified as hot, magnetized, non-ideal, ultracold, dusty, non-neutral, etc. Many astrophysical systems like the Sun’s corona, the interiors of gas giants, and Earth’s ionosphere are considered plasmas. Plasmas have many applications in medicine, technology, and energy production. For example, plasmas can be used for inactivating viruses like COVID-19 [1], during the manufacturing process of semiconductors [2], or in nuclear fusion experiments for generating large amounts of clean, renewable energy. Nuclear fusion experiments typically require intense lasers or high-energy x-rays that can only be produced at a select number of facilities in the world. Examples of these facilities include the Z Machine at Sandia National Laboratories, The National Ignition Facility at Lawrence Livermore National Laboratory, and the OMEGA laser facility at The Laboratory for Laser Energetics at The University of Rochester. While nuclear fusion is currently not feasible for energy production, data collected during these experiments are used to optimize their set-up to maximize the likelihood of sustained thermonuclear burn. Such data include equations of state, plasma transport coefficients, neutron yield, plasma temperature, and data on collective effects like plasma waves or hydrodynamic 1 Figure 1.1: Common plasmas at different temperatures and densities. This figure was originally presented in Ref. [3]. The red curve corresponds to Γ = 1 [see Eq. (1.1)], for mean-ionization state hZi = 1. Below the curve, Γ > 1 and above the curve Γ < 1. instabilities. The cost associated with carrying out these experiments prohibits the availability of wide-ranging experimental data which is further limited by the accessible parameter regime of the experiments. Computational approaches are used to generate large amounts of plasma properties data to support the few high-fidelity experimental calculations. Often, these computational approaches are used in physical regimes absent of experimental data for “extrapolation." By coupling data from experimental platforms to results from numerical simulations, burning plasmas have recently been created in inertial confinement fusion (ICF) experiments at The National Ignition Facility [4–6] resulting in the largest amount of fusion energy output to date. This progress has highlighted the importance of accurate and rapid calculations of plasma properties using computational methods. A major challenge in computational plasma physics is quantifying the accuracy of the numerical methods used for simulating plasmas. The aim of this dissertation is focused on the development of numerical methods for rapidly computing accurate plasma properties of non-ideal plasmas. 2 1.2 Plasma Preliminaries Fundamentally, plasmas are many-body systems of interacting charged particles. To estimate the nature of these interactions, dimensionless parameters such as the Coulomb coupling parameter, degeneracy parameter, and Knudsen number are used. These dimensionless parameters estimate the significance of correlations between particles, if quantum statistics are necessary, and if the plasma behaves as a fluid. The Coulomb coupling parameter is often described as the ratio of the average potential energy of particles in a plasma to their average kinetic energy and is defined as [7–11] hZi 2 e2 Γ= , (1.1) ai T where hZi is the effective charge (or mean-ionization state) of an ion [12], e is the elementary charge, ai = (4πni /3)−1/3 is the Wigner-Seitz radius for the ion number density ni , and T is the temperature of the plasma in energy units. Equation (1.1) quantifies the importance of many-body interactions to the plasma’s properties. A plasma is said to be “weakly coupled" when Γ < 1 and “strongly coupled" when Γ > 1. For Γ  1, the average kinetic energy of the particles in the plasma is far greater than their average potential energy. In this limit, the plasma particles behave like an ideal gas and the plasma is referred to as an “ideal plasma." In the opposite limit, when the Γ  1, the average potential energy of the particles dominates the average kinetic energy and the particles in the plasma are strongly correlated; a plasma in this state is referred to as a “non-ideal" plasma. Note that there are a variety of plasma conditions that correspond to a Γ > 1. For example, when a plasma is dense (e.g., white dwarf stars), cold (e.g., ultracold neutral plasmas), or if the mean-ionization state is large which is true for dusty plasmas where the plasma macroparticles (referred to as “dust grains") can have hZi ∼ 1000 [13, 14]. We stress that multiple types of plasma may share the same value of Γ; this fact allows for connections between disparate plasmas such as high-energy-density (HED) plasmas and UNPs [15]. Note that Eq. (1.1) is for a single, unscreened1 species. If electron screening is significant which is often the case in strongly coupled plasmas [11], 1 “Screening" or “shielding" refers the reduction of a plasma particle’s force range due to the accumulation of oppositely charged particles. 3 we can modify Eq. (1.1) to include electron screening with a screening function. The resulting coupling parameter is referred to as an “effective coupling parameter" as shown in Refs. [16–18]. Moreover, if we wish to extend Eq. (1.1) to binary plasma mixtures of ion species i and j, (see Chapter 3) we have hZi ihZ j ie2 Γi j = , (1.2) atotTi j where atot = (4πntot /3)−1/3 with ntot = ni + n j , and Ti j = (mi T j + m j Ti )/(mi + m j ) [19]. In addition to the coupling of the ions, we must also consider the nature of the electrons. That is, if we can treat the electrons as classical point particles or if we need to include quantum statistics to describe their interactions. If quantum statistics are needed to describe the electrons, the plasma is said to be "degenerate" or "partially degenerate." If we can treat the electrons as classical point particles, the plasma is said to be "non-degenerate." The magnitude of the degeneracy of a plasma is given by the degeneracy parameter which we define as [20] Te θ= , (1.3) EF where EF = ~2 (3π 2 ne )2/3 /2me is the Fermi energy for the electron number density ne , electron mass me , and the electron temperature Te is in energy units. A system is non-degenerate when θ > 1, and degenerate or partially degenerate when θ < 1. For dense or cold plasmas we have that θ < 1 and the wave packet describing the electrons begin to overlap. In this case, it is no longer appropriate to approximate the electrons by classical statistical mechanics and quantum statistics are needed. However, for high temperatures and/or low densities, the overlap of wave packets does not occur and the electrons may be treated as classical point particles. The final dimensionless quantity we introduce is the Knudsen number which helps determine if a system is in a hydrodynamic state i.e., if the plasma may be treated with continuum mechanics instead of statistical mechanics. The Knudsen number is defined as [21] λ Kn = , (1.4) L where λ is the mean-free-path, the average distance that a particle travels before collision, and L is some reference length scale. As highlighted by Fig. 1.2, the magnitude of Kn suggests appropriate 4 Figure 1.2: Different theoretical regimes for different Knusden numbers. This figure was originally presented in Ref. [21]. As Kn becomes small hydrodynamic approximations such as the Euler or Navier-Stokes-Fourier (NSF) equations are valid. For larger Kn , kinetic theory becomes an accurate description of the system. plasma element ni (cm−3 ) Ti (eV) Te (eV) hZi Γ θ Kn type UNP Ca 1×109 8.6×10−5 8.6×10−3 1 2.3 2.4×106 1.1 × 10−3 CDM H 2×1022 0.01 0.01 0.34 71 3.7×10−3 6.1×10−8 WDM H 6×1024 10 10 0.84 3 0.083 5×10−6 HDM H 6×1025 1000 1000 0.97 0.086 1.8 8.4×10−4 Table 1.1: Comparison of non-dimensional parameters for different plasmas. Here we abbreviate cold-, warm-, and hot-dense matter as CDM, WDM, and HDM, respectively. We have chosen experimentally relevant conditions for each plasma type with a focus on fusion fuel in ICF ex- periments for the CDM, WDM, and HDM cases. To compute the Knudsen numbers, a reference length-scale of LUNP = 1 mm and LICF = 1 µm, were used for the UNP and ICF plasmas. A Thomas-Fermi model was used to compute the mean-ionization state hZi for the ICF plasmas; for UNPs, hZi is known exactly and is often unity. models for the plasma. For example, in the highly collisional limit (Kn  1), the plasma is close to equilibrium and Euler hydrodynamics is an applicable model. In contrast, in the transition/kinetic regimes, particles collide and interact but are out of equilibrium so non-equilibrium statistical mechanics (i.e., kinetic theory) must be employed. In each chapter of this dissertation, we reference each of these three dimensionless parameters to 1) support the choice of models/numerical approaches we implement, and 2) make connections to relevant experimental platforms that benefit from this work. As an example, Table 1.1 displays all three dimensionless parameters for different plasma types at various number densities and temperatures. Note that warm dense matter (WDM) and UNPs share a similar Coulomb coupling parameters suggesting that UNPs can be used as a proxy for WDM. 5 1.3 Computational Modeling of Plasmas The numerical techniques used to simulate plasmas are just as diverse as the parameter space they span. These techniques range from various forms of molecular dynamics (MD) [22–38], where explicit interactions between particles are calculated, to coarse-grained methods such as the particle- in-cell technique [39–43], models based on kinetic theory [44–60], and hydrodynamics [61–78]. Usually, the choice of which numerical is appropriate for a given plasma depends on certain limiting physical regimes e.g., high density and low temperatures or low density and high temperatures. At low densities and temperatures, Γ > 1 and θ < 1 resulting in a strongly coupled, degenerate plasma. To numerically treat strong coupling and quantum statistics, computational methods such as Kohn-Sham MD [79–87], and orbital-free MD [32, 83, 88–92] are employed. In Kohn- Sham MD [93], the quantum statistics are treated by solving Schrödinger’s equation for the one- particle wave function referred to as “Kohn-Sham orbitals" or simply “orbitals." Once the orbitals are obtained, the electron density is constructed and the force acting on the ions due to the electronic structure (i.e., the placement of the electrons) is computed at every simulation time step. Determining the electronic structure in the Kohn-Sham MD formalism is a computationally expensive procedure that scales numerically as O(Ne3 ) where Ne is the number of free electrons in the system. To mitigate the computational cost associated with Ne , pseudopotentials are implemented to represent the non-interacting core of an atom which reduces the number of orbitals that need to be numerically obtained reducing computation cost. At higher temperatures (T ∼ 15 eV), pseudopotentials must be used with caution because an ion may become fully ionized and an “all-electron" calculation may be needed. Orbital-free MD [94] is the generalized, original formalism of which Kohn-Sham MD is based. In orbital-free MD, the orbitals are not calculated and algorithms such as the fast Fourier transform can be implemented which results in a numerical scaling of O(Ne log Ne ) [94–96]. Orbital-free MD methods allow for simulations with orders of magnitude more particles and are especially effective at high system temperature. As shown in Ref. [97], orbital-free MD numerically scales as ∼ O(1) with system temperature T in contrast to Kohn-Sham MD which numerically scales as 6 Figure 1.3: Time to update one particle in Kohn-Sham MD and orbital-free MD versus system temperature (see Ref. [97]). The orbital-free MD data does not fit a power-law form unlike the Kohn-Sham MD data which scales with temperature as ∼ O(T 2.6 ). ∼ O(T 2.6 ). Figure 1.3 shows data of simulation time versus temperature using both Kohn-Sham MD and orbital-free MD obtained from Ref. [97]. The points correspond to the time to update one particle using both methods for a given system temperature. A fit has been obtained from the data showing the aforementioned temperature scaling. The best-fit parameters to the orbital-free MD data indicate that a power-law does not describe its numerical scaling in contrast to Kohn-Sham MD. The approximations of orbital-free MD (and thus Kohn-Sham MD) lie in the determination of various functionals that account for the kinetic energy and the exchange and correlation of the system. Figure 1.4 displays data obtained from the literature that showcases simulations carried out using Kohn-Sham MD and orbital-free MD for different systems in a space-time diagram. We see that Kohn-Sham MD is sequestered to physical systems on the length- and time-scales of angstroms (Å) and pico-seconds (ps); this clustering is a direct consequence of the computational complexity associated with Kohn-Sham MD. In contrast, orbital-free MD can access much larger spatial scales, even on the order of micrometers (µm). Importantly, there is a third axis of temperature that is not displayed here. We emphasize that orbital-free MD is able to simulate plasmas at temperatures of hundreds of eV making orbital-free MD an ideal method for simulating high-energy-density 7 plasmas. In contrast, Kohn-Sham MD is more accurate than orbital-free MD at lower temperatures (e.g., on the order of fractions of an eV) making Kohn-Sham MD an ideal candidate for simulating warm-dense-matter, when Γ ≈ θ ≈ 1. In addition to Kohn-Sham and orbital-free MD, Figure 1.4 also displays data from MD sim- ulations which span a much larger time and space scales. The difference in viable simulation scales is due to the fact that MD makes use of pre-computed potentials that can be derived from Kohn-Sham/orbital-free methods or obtained empirically. Some examples of pre-computed poten- tials include the Coulomb potential, the Yukawa potential, or the Lennard-Jones potential. These potentials eliminate the need for an electronic structure calculation as it is accounted for in the potentials functional form. Molecular dynamics typically scales as O(N log N) where N is the number of particles in the MD simulation by utilizing fast-neighbor-list algorithms to compute particle interactions. In Chapter 2 we make comparisons of Kohn-Sham MD, orbital-free MD, and MD by carrying out simulations of dense plasmas ranging in temperature, density, and nuclear charge. The last remaining methods in Figure 1.4 are kinetics and hydrodynamics. Instead of simulating explicit particle interactions, these methods simulate statistical distributions of particles, greatly reducing computation cost and allows these methods to access even larger space and time scales than MD. The computational complexity associated with kinetics and hydrodynamics varies based on the numerical methods implemented, but the key distinction between these methods and MD is that they are often grid-based methods, instead of particle-based methods. Chapter 5 is dedicated to the development of a grid-based numerical method for solving kinetic equations with applications to UNPs. In recent years, machine learning has become a widely used tool for both obtaining plasma prop- erties data as well as utilizing and analyzing existing datasets. Commonly used machine learning methods for plasma properties data range from deep neural networks, which are favored as they can learn non-linear relationships between features in a dataset, to dimensionality reduction techniques like principal component analysis. In reference to Fig. 1.4, the influence of machine learning 8 Figure 1.4: Computational methods employed at various time- and length-scales for physical systems. Each data point represents published data from a variety of disciplines starting in the year 2012; ellipses computed based on the principal components of the data are also plotted. Note that Kohn-Sham molecular dynamics [79–87] and orbital-free molecular dynamics [32, 83, 88–92] simulations are typically done at the scale of angstroms and picoseconds; this is a direct result of their computational complexity which limits these methods to small system sizes. Coarse-grained methods such as kinetics [44–60] and hydrodynamics [61–78] are able to access larger spatio- temporal scales because instead of computing interactions between explicit particles as is done in MD [23–38], a statistical average of particles is computed which greatly reduces computation cost. would be seen across the entire spatio-temporal domain making machine learning the most widely applicable method for determining plasma properties. Some concrete examples of machine learn- ing include its use in Kohn-Sham MD to determine the necessary exchange correlation functional or in hydrodynamics for obtaining the necessary constituent data (i.e., transport coefficients and equations of state). In Chapter 4 we present a machine learning method for regressing multi-modal datasets to rapidly generate accurate predictions of plasma transport data. 9 1.4 Dissertation Overview This dissertation consists of multiple objectives which are directed toward the common goal of creating more accurate, computationally efficient, methods for simulating plasmas across disparate temperature and density regimes. The computational methods focused on herein are those of MD, machine learning, and kinetic theory. Each chapter is dedicated to one of these methods which are applied to numerical calculations of non-ideal plasmas. The focus of Chapter 2 is to elucidate the efficacy of force laws used for MD simulations of dense plasmas. A review of commonly used force laws is presented along with a discussion of the current state-of-the-art models that stem from data-driven machine learning approaches. We present a non-parametric, data-driven approach for obtaining force laws for a wide range of elements at various temperatures and densities allowing for an increase in simulation time by orders of magnitude while also reducing statistical errors. The scope of the work presented in Chapter 2 was limited to plasmas comprised of a single ionic species. However, in many scenarios (e.g., ICF experiments or UNP mixtures), an under- standing of the interactions between particles in plasma mixtures is vital. Chapter 3 addresses the theoretical challenges associated with dense plasma mixtures where we derive analytic models for the thermodynamic factor and interdiffusion transport coefficient. Our analytical formulae agree with molecular dynamics data in certain limiting regimes. Thus, we are able to avoid the need for molecular dynamics simulation, and as a result, eliminates the associated computation cost and reduces statistical error. While Chapters 2 and 3 focused on generating accurate plasma properties data, Chapter 4 focuses on interpolating multi-modal datasets to improve predictions in regions were high-fidelity measurements do not exist. We compare our multi-fidelity machine learning approach to its single- fidelity counterpart and show that by using data from multi-modal datasets, the predictions across regions absent of high-fidelity data are more accurate with reduced uncertainty. In Chapter 5 we model the interactions of UNP mixtures using kinetic theory to investigate their expansion into a vacuum as well as their entropy production. We begin by providing an overview 10 of kinetic theory, as well as a derivation of the Vlasov equation in spherical coordinates which is scarce in the literature and appears seemingly nowhere. Using a multi-species kinetic equation in Cartesian coordinates, we simulate UNP mixtures and find that the initial plasma conditions can be chosen such that the dynamics are time-reversible. Additionally we apply our kinetic simulations to HED plasmas where we analyze the dominant drivers of diffusive mixing. The findings of this dissertation are summarized in Chapter 6 along with a discussion on how the methods presented herein can be improved. 11 CHAPTER 2 DATA-DRIVEN FORCE LAWS FOR MOLECULAR DYNAMICS SIMULATIONS OF DENSE PLASMAS 2.1 Introduction ln strongly coupled plasmas, N-body simulations are required to correctly account for collective effects (e.g., plasma waves) or instabilities (e.g., the two-stream or bump-on-tail instability [98]). As discussed in Sec. 1.3, MD1 is a simulation method that encodes N-body effects by integrating Newton’s second law for a system of interacting particles. The critical input of MD is the force-law or interaction potential that quantifies the particle interactions. For example, the Lennard-Jones interaction potential is used for MD simulation of noble gasses, and the Yukawa interaction potential is used to simulate a system of charged particles (i.e., plasmas). As most plasmas are made up of ions and electrons where electron screening is present, the Yukawa potential, which approximates electron screening, is commonly used for MD simulations of plasmas and is favored for its short- range interactions and pre-computed form. However, the range of validity of the Yukawa potential is unknown for different plasma conditions. While MD “stands alone" as its own simulation method, it may be unable to simulate large time and length scales that may be required for some applications (see Fig. 1.4). In these situations coarse-grained methods based on hydrodynamics or kinetic theory are employed. However, coarse- grained methods still require detailed microscopic information, often from MD simulations, as “closures." The quality of the microscopic information directly influences the results of the coarse- grained methods and therefore quantifying the sensitivity of the choice of interaction potential for MD simulations is crucial for obtaining high-fidelity macroscopic simulations of plasmas. This chapter2 focuses on the delineation of the accuracy boundary between microscale simulation methods by comparing a variety of force laws for plasma simulation across a wide range of elements, 1 See Ref. [23] for an overview of MD. 2 The content described in this chapter has been reproduced from Lucas J. Stanek, Raymond C. Clay III, M. W. C. Dharma-wardana, Mitchell A. Wood, Kristian R. C. Beckwith, and Michael S. Murillo , "Efficacy of the radial pair potential approximation for molecular dynamics simulations of dense plasmas", Physics of Plasmas 28, 032706 (2021) https://doi.org/10.1063/5.0040062," with the permission of AIP Publishing and has been modified to address the requirements of this dissertation; see Ref. [87] for the full published article. 12 Figure 2.1: Different plasma conditions studied in this chapter; each color corresponds to a different element. The cases studied here are strongly coupled and span the non-degenerate and degenerate regimes. For all cases, the Knudsen number 10−6 < Kn < 10−2 where we have chosen a reference length scale of L = 5 Å which is approximately the size of the simulation cell for all cases. The diamonds () denote the MD simulations for which we have both RPP-MD and KS-MD data. The circles (•) denote MD simulations for which we only have RPP-MD data. temperature, and density; these conditions are are shown in Fig. 2.1. As a baseline, we employ Kohn-Sham density functional theory molecular dynamics (KS-MD) and compare results obtained from radial pair potential molecular dynamics (RPP-MD). By extracting the optimal RPP from KS- MD data using force matching, we constrain its functional form and dismiss classes of potentials that assume a constant power law for small interparticle distances. Our results show excellent agreement between RPP-MD and KS-MD for multiple metrics of accuracy at temperatures of only a few electron volts. The use of RPPs offers orders of magnitude decrease in computational cost and indicates that three-body potentials are not required beyond temperatures of a few eV. Due to its efficiency, the validated RPP-MD provides an avenue for reducing errors due to finite-size effects that can be on the order of ∼ 20%. A wide variety of RPPs have been developed for modeling dense plasmas. In some cases the 13 accuracy of the model can be inferred from its theoretical underpinnings; in other cases, comparison to higher-fidelity approaches or experiments is needed. Limitations of the RPP approximation are generally unknown unless compared to an N-body potential simulation result such as KS-MD. Both KS-MD simulations and this comparison are time-consuming processes that are limited to the temperature regime in which the pseudopotentials necessary for KS-MD are valid [99, 100]. Moreover, comparisons between RPP-MD and KS-MD are limited in the literature, have not been carried out for a range of elements and temperatures, and are often validated with integrated quantities where individual particle dynamics have been averaged and results are subject to cancellation of errors. We carry out KS-MD simulations for a range of elements, temperatures, and densities, allowing for a systematic comparison of three RPP models. While multiple RPP models can be selected [101–105], we choose to compare the widely used Yukawa potential, which accounts for screening by linearly perturbing around a uniform density in the long-wavelength (Thomas-Fermi) limit, a potential constructed from a neutral pseudo-atom (NPA) approach [106–109], and the optimal force-matched RPP that is constructed directly from KS-MD simulation data. Each of the models we chose impacts our physics understanding and has clear computational consequences. For example, success of the Yukawa model reveals the insensitivity to choices in the pseudopotential and screening function and allows for the largest-scale simulations. Large improvements are expected from the NPA model, which makes many fewer assumptions with a modest cost of pre-computing and tabulating forces. The force-matched RPP requires KS-MD data, and is therefore the most expensive to produce, but it reveals the limitations of RPPs themselves, since they are by definition the optimal RPP. Using multiple metrics of comparison between RPP-MD and KS-MD including the relative force error, ion-ion equilibrium radial distribution function g(r), Einstein frequency, power spectrum, and the self-diffusion transport coefficient, the accuracy of each RPP model is analyzed. By simulating disparate elements, namely an alkali metal, multiple transition metals, a halogen, a non-metal, and a noble gas, we see that force-matched RPPs are valid for simulating dense plasmas at temperatures 14 above fractions of an eV and beyond. We find that for all cases except for low temperature carbon, force-matched RPPs accurately describe the results obtained from KS-MD to within a few percent. By contrast, the Yukawa model appears to systematically fail at describing results from KS-MD at low temperatures for the conditions studied here validating the need for alternate models such as force-matching and NPA approaches at these conditions. In Sec. 2.2 we discuss how RPPs arise from second order perturbation theory and how their representation influences the shape of g(r) due to particle crowding and/or attraction. Comparisons between RPPs and KS-MD are done in Sec. 2.3, where we begin by comparing interparticle forces illustrating how an increase in temperature indicates an increase in accuracy. In addition, the microfield distribution of forces, Einstein frequency, power spectrum, self-diffusion coefficient, and g(r) are compared, highlighting how an approximately accurate g(r) does not ensure similar accuracy in time correlation functions and transport coefficients. A description of how we accurately compute the self-diffusion coefficient and its uncertainty when finite-size errors are non-negligible is given in Sec. 2.3.3. This further emphasizes the need for RPPs, as we minimize finite-size errors in KS-MD simulations by making the necessary corrections as shown in Sec. 2.3.5. We conclude by comparing fully converged (in particle number and simulation time) self-diffusion coefficients to an analytic transport theory; benchmarking its accuracy and providing an effective interaction correction to extend the range of applicability. 2.2 Models for the Interaction Potential The theoretical foundations of the models we will compare are described in this section; their connections are shown in Fig. 2.2. We compare three classes of interactions that are based on the ionic N-body energy, shown in the top box, pair interactions that are pre-computed and are analytic or tabulated, shown in the lower-left box, and optimal pair interactions extracted from the N-body results, shown in the lower-right box. By comparing these three approaches we aim to answer several specific questions. First, given the nuclear charge Z, ionic number density ni , and temperature T, what ranges in {Z, ni, T } space are the fast, pre-computed interactions valid and therefore allow for large-scale heterogeneous simulations? Second, how accurate is the “optimal" 15 Figure 2.2: Connections between different portions of this work. N-body potentials, shown in the top box, are used to validate pair potential models (lower left) and produce optimal tabulated potentials (lower right). Both pre-computed RPPs and tabulated force-matched RPPs provide finite- size corrections to KS-MD data; assuming they accurately reproduce the Kohn-Sham potential energy surface. The tabulated force-matched RPPs highlight the appropriate RRP representation (e.g., oscillations). The pre-computed RPPs give physical intuition to the representation determined by the KS-MD data. pair interaction, and what do its limitations reveal about the need for three-body interactions (and perhaps beyond)? Can these interactions be used to test and correct for finite-size errors? Third, can the optimal interactions guide the development of pre-computed interactions? To simplify the discussion we will consider single species matter with a range of Z, each species at its normal solid ionic mass density ρi , or in some cases half of that, and in thermodynamic equilibrium at temperature T. While we do not consider mixtures in here, the framework is general and can be straightforwardly applied to them. Assuming the Born-Oppenheimer approximation holds, we define a potential energy surface for the ions as Utot = UN (r1, r2, . . . , rN ). (2.1) Physically, the ionic potential energy surface is determined by the electronic charge distribution arising from ions at a particular set of coordinates; in general, Eq. (2.1) does not simplify into sums over pairwise terms. There are two major approaches to obtaining Eq. (2.1) in practice. The 16 approach represented by the top box in Fig. 2.2 computes the electronic charge distribution for each ionic distribution. This is achieved computationally in Kohn-Sham approaches by reducing the electron many-body problem to a single-electron problem in which the Kohn-Sham electron moves in the external field of N-ionic centers. The dominant computational cost comes from solving an No × No set of eigenvalue equations, where No is the number of single-particle orbitals. Even though the electron many-body problem has been simplified to a one-body problem, matrix diagonalization incurs a cost of O(No3 ), and at high temperatures the smearing of the Fermi- Dirac distribution requires an increasing number of orbitals leading to significant increases in computational cost. The complexity of the electron charge distribution also demands the use of an advanced “Jacob’s ladder" of exchange-correlation functions to address the electron many body problem. This approach yields an intrinsically ionic N-body potential energy surface; the electronic density is computed using a description appropriate to the choice of {Z, ni, T }. The second approach to calculating the potential energy surface is to use a cluster-type expansion, which takes the form ÕN ÕN Õ N Utot = U1 (ri ) + U2 (ri, r j ) + U3 (ri, r j , r k ) + · · · . (2.2) i i, j i, j,k When this expansion can be truncated with only a few terms, interactions can be pre-computed, and fast neighbor algorithms allow for a very rapid evaluation of forces, typically many orders of magnitude faster than through use of Eq. (2.1). This allows, for example, for simulations with trillions of particles [110–112]. However, the disadvantages are that the computational cost increases rapidly as more terms are included, and the accuracy of a specific truncation and choice of functional forms with that truncation are not usually known; part of our goal is to assess how accurate the potential energy surface in Eq. (2.1) can be represented by the first two terms of Eq. (2.2). 17 2.2.1 N-body Interaction Potentials The most accurate forces are obtained from the gradient of the total energy in Eq. (2.1), which requires the entire ionic configuration. Although machine learning approaches are enabling the ability to pre-learn that relationship [113–115], it remains more common to compute the forces for each ionic configuration during the simulation (“on-the-fly"). We obtain the electronic number density for each ionic configuration in the Kohn-Sham-Mermin formulation of the density Õ ne (r) = fi (T)|φi (r)| 2, (2.3) i where T is the temperature of the system in energy units, the Fermi occupations are given by  −1 fi (T) = 1 + e β(Ei −µ) , and the Kohn-Sham-Mermin orbitals φi (r) satisfy    1 2 − ∇ + ve f f (r) φi (r) = i φi (r), (2.4) 2 where ne (r0) δE xc [ne ] ∫   ve f f (r) = Vext (r) + dr 0 + , (2.5) |r − r0 | δne (r) is a sum of the external (N ion-electron), Hartree, and exchange-correlation energies. Our KS-MD simulations were done using the Vienna Ab-initio Simulation Package (VASP) [116–119]. The finite temperature electronic structure was treated with the Mermin free-energy functional, and we used the Perdew–Burke-Ernzerhof functional for the exchange correlation energy [120]. To improve computational efficiency, we eliminated the chemically inactive core electrons with the projector augmented-wave [121] pseudopotential. Due to the anticipated high temperatures and small interionic separations, we used the smallest core “GW" pseudopotentials available in VASP. Here, “GW" denotes that the “GW approximation" has been made; the details of the approximation can be found in Ref. [122]. Sixty-four atoms (N = 64) were used in these simulations, with an energy cutoff of 800 eV and at the Baldereschi mean-value k-point [123] for all temperatures ranging from T = 0.5 to 15 eV. A simulation time step of 0.1 fs was used, and the total simulation lengths for each case vary and are on the order of a few picoseconds. All KS-MD simulations were first equilibrated in the NVT ensemble and then carried out in the NVE ensemble where data was collected. 18 2.2.2 Force Matching After the Kohn-Sham potential energy surface has been computed, we aim to construct a compact representation of Eq. (2.1) with Eq. (2.2). By assuming a parameterized functional form for Eq. (2.2), the force-matching procedure [124–129] was used to generate the optimal RPP model based on the KS-MD force data. From each KS-MD simulation, a dataset of K ≡ 3N M forces (3 force components, N atoms, and M atomic configurations) is obtained. Atomic simulation data at nearby time points is highly correlated; thus, a stride between atomic configurations was used to generate 100-200 independent configurations. With each KS-MD dataset, we determine the optimal RPP for that system by minimizing the loss function ÕK L(ζ) = w k [Fk (ζ) − Fk0 ]2 . (2.6) k=1 Here, ζ is a set of optimizable parameters, Fk (ζ) is the k-th force for the parameterized model with parameters ζ, Fk0 is the k-th force from KS-MD reference dataset, and w k is a weight factor. The weight factor w k = 1/(Fk0 + ε)2 ensures that both large, and small forces contribute equally to Eq. (2.6). The parameter ε should be varied for each temperature and element but in most cases here, ε ≈ 1. The choice of parameterization can either have a pre-computed functional form, such as Eq. (2.8), or be determined completely from the data as is the case for a tabulated potential [130] with spline interpolation – the choice here. For each system, we begin by sampling a Thomas-Fermi Yukawa (see Sec. 2.2.3) RPP at 15 locations in r and use that as the initial condition for the force- matching procedure. The Thomas-Fermi Yukawa RPP is sampled such that rmin < r < 8 Å where rmin is the minimum ionic separation in the KS-MD dataset. To ensure that the core repulsion and/or attractive oscillation regions are sampled sufficiently, 10 points are placed in the region where rmin < r < 4 Å, leaving the 5 remaining points to be placed where r > 4 Å. To test for convergence of the optimal force-matched RPP, two optimization methods were used (specifically simulated annealing, and differential evolution.) By choosing a tabulated potential form for the RPP, the explicit form of the model is entirely determined from the KS-MD force data and not 19 limited to a fixed functional form. While the force-matched RPP yields the best RPP to reproduce the KS-MD force data, it could be the case 3-body and higher interactions are non-negligible. To check this, we selectively employ the Spectral Neighborhood Analysis Potential (SNAP) which constructs a potential energy surface from a set of 4-body descriptors (bispectrum components),where each descriptor is independently weighted, and these weights are determined by regressing against KS-MD data of energies and forces. A descriptor captures the strength of density correlations between neighboring atoms and the central atom within a given cutoff distance, details can be found in Refs. [131, 132]. The parameterization of the SNAP uses descriptors of the local atomic environment capturing up to 4-body interactions when represented in the form of Eq. (2.2), so lower errors associated with SNAP compared to an optimal RPP are entirely due to 3- and 4-body interactions. While higher bodied inter-atomic potentials exist in the literature [133], it can be expected there are diminishing accuracy returns with higher interaction moments, thus SNAP offers a leading order check on the RPP compared here. SNAP potentials utilizing 56 bispectrum component descriptors were trained on 10% of the KS-MD dataset, and additionally tested against an additional 10% to ensure regression errors were properly minimized and avoided over-fitting of the KS-MD data. 2.2.3 Radial Pair Potentials As the computational cost of using on-the-fly N-body interactions is often prohibitive, the least expensive approach utilizes pre-computed RPPs ignoring most of the terms in Eq. (2.2). Many functional forms for the RPP have been proposed for application to warm dense matter often using the second-order perturbation-theory interaction energy u(k) = hZi 2 uC (k) + |uei (k)| 2 χ(k), (2.7) which is the standard Fourier-space result [134] written in terms of the mean ionization state hZi, the bare Coulomb potential uC (k) = 4πe2 /k 2 , the electron-ion pseudopotential uei (k) and the susceptibility χ(k). 20 In practice, pair interactions are constructed using nearly the same steps as for the N-body interactions, with the primary difference being that each ion is replaced with a single “average atom" (AA), which is an all-electron, non-linear, finite-temperature density functional theory calculation [12]; such calculations can also be relativistic [135,136]. From the AA, a pseudopotential uei (k) and an accurate free/valence electron response function χ(k) are constructed and Eq. (2.7) is formed. This approach has three strengths: (1) typical AA models are not limited to low temperatures, (2) the interaction Eq. (2.7) can be pre-computed for use in MD, and (3) pair interactions with a fast nearest neighbor algorithm are very computationally efficient. As we alluded to above, the accuracy loss attendant to these strengths is what we wish to determine. The AA itself is aware of the ionic number density ni , which sets the ion-sphere radius ai = (3/4πni )1/3 , and includes the fact that there is only one ion in the ion sphere, which implies a g(r); this indirect inclusion of higher-order terms in Eq. (2.2) is true for all AA-based interactions. Among the simplest variants of Eq. (2.7), one approximates the pseudopotential as uei (k) ≈ −4πhZie2 /k 2 , where the mean ionization state hZi results from a AA calculation [12], and χ(k) in its long-wavelength (Thomas-Fermi) limit χT F (k); this is known as the “Yukawa" interaction [104, 137]. Here, we employ a Yukawa interaction with inputs from a Thomas-Fermi AA [88], which we will refer to as “TFY." This procedure yields an analytic potential in real space of the form hZi 2 e2 uT FY (r) = exp (−r/λT F ) , (2.8) r where the electron screening is approximated by the Thomas-Fermi screening length √ 8T λT F = −2 F−1/2 (βµe ), (2.9) π where F−1/2 is the Fermi-Dirac integral of order −1/2, β = 1/T , and µe is the electron chemical potential. Padé fits of Fermi-Dirac integrals and their inverses are carried out in [138, 139]. An approximation to these fits [140] yields 4πne e2 λT−2F ≈q , (2.10) T 2 + ( 32 EF )2 21 where the Fermi energy EF = ~2 (3π 2 ne )2/3 /2me . Note that the TFY interaction is monotonically decreasing (purely repulsive). Computationally, the TFY model is highly desirable because of its radial, pair, analytic form with an exponentially-damped short range. Its weaknesses are the relatively approximate treatments of uei (k) and χ(k). The TFY model can be extended by including the gradient corrections to χT F (k), but otherwise retaining the other approximations. This improvement yields the Stanton-Murillo potential [104]; the gradient correction to χT F (k) introduces oscillations in the potential in some plasma regimes that are absent in the monotonic TFY model. Moreover, gradient corrections add improvements to the cusp at the origin and the large-r asymptotic behavior. Here, however, we will only employ the simpler TFY model. A great deal of accuracy can be gained by abandoning analytic inputs to Eq. (2.7). In this case, self-consistent numerical calculations of each of the terms can be carried out, still allow- ing for pre-computed interactions; there is essentially no computational overhead for tabulated interactions [130]. Here, we employ a NPA model that yields both the mean ionization state and its pseudopotential using a Kohn-Sham-Mermin approach, as described above, but with a finite-temperature exchange-correlation potential; the susceptibility is determined by the Lind- hard function with local field corrections [107]. Note that the electron-ion pseudopotential uei (k) introduces additional oscillations on length scales different from χ(k), although the Friedel os- cillations in χ(k) contribute much more to the pair interaction. Note that the name “NPA" has been used by many authors to several different average-atom models, and many of them involve approximations that limit those models to higher temperatures, e.g., T > EF ; however, here we use the one-center density functional theory model developed by Dharma-wardana and Perrot as this model has been tested at high temperatures as well as at very low temperatures, and found to agree closely with more detailed N-center density functional theory simulations and path-integral quantum calculations where available. It is worth comparing predictions based on Eq. (2.7) with other forms suggested previously. A popular RPP for warm dense matter studies is the short-range repulsion interaction, which adds a long-range, power-law correction to the TFY model of the form A/r 4 [32,103,141–145]; for A > 0, 22 200 f(r) = u(r) 175 f(r) = r2u(r) f(r) = r4u(r) 150 f(r) = r6u(r) 125 6 f(r) 100 2 4 75 50 25 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r (Å) Figure 2.3: NPA RPP for Al at 2.7 g/cm3 and T = 1 eV. Various power laws are valid at different values of r. The appropriate power law for a given range of r is shaded and denoted with a “2," “4," or “6." this is also a monotonic interaction, with the goal of increasing the strength of the TFY model, which underestimates the peak height of g(r). In Fig. 2.3, we examine this ansatz by computing a NPA interaction for Al at solid density and T = 1 eV. To find the “best" power law, we multiply the NPA interaction by various powers r a to find regions where the interaction is flat; a flat region with a = 4 would recover the short-range repulsion interaction. It is clear that the A/r 4 is only valid over a very small range of r values; importantly, the NPA interaction shows that the exponent a increases as r becomes large, which is a true short-ranged interaction - the empirical correction the short-range repulsion model adds greatly overestimates the strength of the interaction at large interparticle separations [106]. Worse, the short-range repulsion model potentially gets an accurate answer for the wrong reason, as we explore in Fig. 2.4. Because the form of Eq. (2.7) generally has oscillations, the enhanced peak height of g(r) from the NPA model over the TFY model occurs for two, independent reasons. Attractive regions of the interaction, as shown in the top panel of Fig. 2.4, can produce very strong peaks in g(r). Conversely, stronger overall repulsion at intermediate r can lead to a similar g(r) behavior, as shown in the bottom panel of Fig. 2.4, but with rapid decay of the interaction at larger r. The functional form of Eq. (2.7) naturally contains both the “crowding" and “attraction" behaviors as special cases. Fig. 2.5 23 5 4 (a) 4 3 g(r) 2 3 1 u(r)/T 2 0 0 2 4 r (Å) 1 0 1 5 2.0 (b) 4 1.5 g(r) 1.0 3 0.5 u(r)/T 2 0.0 0 2 4 r (Å) 1 0 TFY NPA 1 1 2 3 4 5 6 r (Å) Figure 2.4: Comparison of TFY and NPA RPPs for C and Al with corresponding g(r) computed from MD simulation: (a) C at 2.267 g/cm3 and T = 0.5 eV. The increase in magnitude of the first g(r) peak results, in this case, from particle attraction. (b) Al at 2.7 g/cm3 and T = 1 eV. In this case, it is particle crowding increases the magnitude of the first g(r) peak. shows a comparison of the RPPs for C, Al, V, and Au at T = 0.5 and 5 eV. The TFY model is purely monotonic whereas the force-matched and NPA RPPs have attractive and repulsive regions in their oscillations. Below, we will explore the consequences of these features of the interaction on ionic transport. Once the RPPs have been constructed, MD simulations were carried out using in the Large- scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [146]. For the tabulated RPPs (force-matched and NPA) a linear interpolation was needed to determine the force value between tabulation points. To make a direct comparison between the RPP-MD and KS-MD results, all simulations were carried out in a 3 dimensional periodic box with 64 atoms and a time step of 0.1 fs. The length of each simulation is identical to the corresponding simulation performed with KS-MD. Keeping these conditions identical avoids the unintentional reduction in statistical errors 24 4 C (2.267 g/cm3) Al (2.7 g/cm3) V (6.11 g/cm3) Au (19.30 g/cm3) 3 T = 0.5 eV T = 0.5 eV T = 0.5 eV T = 0.5 eV 2 TFY u(r)/T FM NPA 1 0 4 C (2.267 g/cm3) Al (2.7 g/cm3) V (6.11 g/cm3) Au (19.30 g/cm3) 3 T = 5 eV T = 5 eV T = 5 eV T = 5 eV 2 u(r)/T 1 0 0 2 4 60 2 4 60 2 4 60 2 4 6 r (Å) r (Å) r (Å) r (Å) Figure 2.5: The RPP models normalized by temperature versus distance for C, Al, V, and Au. Top row, T = 0.5 eV: the representation of the RPP is element dependent with strong agreement for aluminum. Bottom row, T = 5 eV: The agreement between models improves significantly. The differences in the representation can be connected back to Eq. (2.7) where the treatment of the mean ionization, electron-ion pseudopotential, and susceptibility define the RPP. between KS-MD and RPP-MD. All simulations were first equilibrated in the NVT ensemble so that the average temperature for each simulation during the data collection phase is within 1% of the reported temperature in Table 2.1. The data collection phase was carried out in the NVE ensemble. In Sec. 2.3.5, a finite-size effect study was done for the cases of C at 2.267 g/cm3 and V at 6.11 g/cm3 where the total simulation length was increased by 10 times and the number of atoms N increases from 64 to 256, 3375, and 8000. 2.3 Numerical Results 2.3.1 Force Error Analysis One metric for establishing the accuracy of approximations to the Kohn-Sham potential energy surface is to compute relative force errors between Kohn-Sham force data and a parameterized model (RPP or many-body potential) for M particle coordinate configurations. For this, we compute the mean-absolute force error 1 Õ (P AR) (KS−M D) M AE = |Fα,i,m − Fα,i,m |, (2.11) 3M N α,i,m 25 0.5 C (2.267 g/cm3, T = 0.5 eV) Al (2.7 g/cm3, T = 0.5 eV) V (6.11 g/cm3, T = 0.5 eV) Au (19.30 g/cm3, T = 0.5 eV) 0.4 P(|F|) (arb. units) TFY FM 0.3 NPA 0.2 SNAP KS-MD 0.1 0.0 0.20 C (2.267 g/cm , T = 5 eV) 3 Al (2.7 g/cm3, T = 5 eV) V (6.11 g/cm3, T = 5 eV) Au (19.30 g/cm3, T = 5 eV) 0.15 P(|F|) (arb. units) 0.10 0.05 0.000 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 |F| (eV/Å) |F| (eV/Å) |F| (eV/Å) |F| (eV/Å) Figure 2.6: Microfield distributions for C, Al, V, and Au. The observed trends of the microfields agree with the trends of the self-diffusion coefficients in Table 2.1. In general, when the microfields are similar to that of KS-MD, the agreement between the self-diffusion coefficient increases. To assess the importance of 3-body or higher interactions, SNAP results are reported for C and V at both T = 0.5 and 5 eV. C (2.267 g/cm3) TFY Al (2.7 g/cm3) V (6.11 g/cm3) Au (19.30 g/cm3) FM NPA Relative Force Error 100 SNAP 10 1 0.1 0.05 10 2 100 101 100 101 100 101 100 101 T (eV) T (eV) T (eV) T (eV) Figure 2.7: Relative force error versus temperature computed from Eq. (2.13) for C, Al, V, and Au. The red shaded region indicates force accuracy of ≤ 0.1 and the blue shaded region indicates force accuracy of ≤ 0.05. SNAP and force-matched RPP yields the lowest relative force error and decreases or remains constant as temperature increases. This indicates an increases in accuracy of the RPP models as temperature increases. 26 (P AR) (KS−M D) where Fα,i,m and Fα,i,m are the α-th force components (x, y, or z) on the i-th atom in particle coordinate configuration number m for the parameterized model and the KS-MD force data respectively. Note that a direct comparison of the mean absolute error between different elements, tempera- tures, and densities cannot be done as the distributions of forces associated with systems of different elements at different thermodynamic conditions are in general quite different. This can be observed in Fig. 2.6 where a microfield distribution of the force magnitudes is shown. In all cases but C at 2.267 g/cm3 and T = 5 eV, the TFY model peaks at a smaller field value than KS-MD. In contrast, for C, V, and Au at T = 0.5 eV the NPA RPP peaks at a higher field value than KS-MD. These trends can be connected back to Eq. (2.7) where the choice of hZi, uei (k), and χ(k) all contribute to the construction of a RPP model and hence the force magnitudes. More work needs to be done to determine how each term influences the RPP model, the predicted forces, and observables. As the microfield force distributions vary for different elements and temperatures, the mean absolute error will also vary. To this end, we seek a scale factor for Eq. (2.11) to normalize the results across the different elements, temperatures, and densities studied here. Such a scale factor is the “mean absolute force" defined as 1 Õ (KS−M D) M AF = |F |. (2.12) 3M N α,i,m α,i,m Using Eq. (2.11) and Eq. (2.12), we define the relative force error as Í (P AR) (KS−M D) i,α,m |Fα,i,m − Fα,i,m | RFE = Í (KS−M D) . (2.13) α,i,m |Fα,i,m | The metric, Eq. (2.13), has the following desirable property: if the mean absolute error changes with density or temperature in the same way as the underlying force distribution, the relative force error will maintain roughly the same value. Therefore, as we change the thermodynamic conditions for a given element, Eq. (2.13) provides a temperature independent metric as measured with respect to a KS-MD force data “baseline." Intuitively, when Eq. (2.13) evaluates to 1, the mean absolute 27 error is the same order of magnitude as the mean absolute force and when Eq. (2.13) is zero, the parameterized model is exactly reproducing the per-component KS-MD force data. Fig. 2.7 displays Eq. (2.13) as a function of temperature for C, Al, V, and Au where general trends can be observed. One trend is that for most RPPs, the relative force error decrease towards higher temperatures, which confirms an intuition long held for the validity of the NPA and TFY models. However, for all systems pictured except C, force matching drastically reduces the relative force error compared to the NPA and TFY results. Specifically, the force-matched RPPs routinely achieve a relative force error of roughly 0.05 above T = 5 eV. Except for the case of the NPA RPP for Al, the NPA and TFY RPPs maintain an error of around 0.2 across the entire the temperature range. The second major observation from Fig. 2.7 is that while force-matched RPPs drastically lower the observed relative force errors across temperatures compared against other RPPs, we immediately see where a RPP approximation is likely invalid. For example, the relative force error for C using the force-matched RPP is uncharacteristically high (roughly 0.6) until T = 5 eV. A similar situation appears for the case of V at T = 0.5 eV where the relative force error for the force-matched RPP is roughly 0.25. We can demonstrate explicitly that these discrepancies come from the neglect of 3-body and higher interactions by showing relative force errors using a SNAP model. For C, the relative force error drops from roughly 0.6 using a force-matched RPP to 0.2 using a SNAP model at T = 0.5 eV. Likewise for V, the relative force error drops from roughly 0.25 using a force-matched RPP to 0.07 using a SNAP model at the same temperature. Ultimately, it is not the component-wise force or the interaction potential we care about gen- erating, but rather observables such as g(r) and the self-diffusion coefficient. To address this connection, we examine correlations between the force error and the self-diffusion coefficient error, as shown in Fig. 2.8. While there is a general trend with increasing errors in both quantities (shown with a linear fit), there are also some clear outliers. For the case of C at 2.267 g/cm3 and T = 0.5 eV, we find that the NPA and TFY RPPs produce a self-diffusion coefficient that differs from the KS-MD result by many factors. However, C under these conditions exists in several charge states 28 350 TFY C (2.267 g/cm3, T = 0.5 eV) FM Relative Self-Diffusion Error (%) 300 NPA 250 200 150 100 50 0 V (6.11 g/cm3, T = 0.5 eV) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Relative Force Error Figure 2.8: Relative (to KS-MD) self-diffusion error versus the relative force error for C, Al, V, and Au. The size of each point corresponds to the atomic number. The grey dashed line is a linear fit to the points showing a positive correlation between self-diffusion error and force error. with transient bonding; only the NPA accounts for this. This case is marked with arrows in Fig. 2.8. Conversely, for V at T = 1 eV, the relative self-diffusion percent error is low, yet the relative force error is high. The imperfect mapping of relative self-diffusion error versus relative force error suggests that physics beyond a RPP is needed, possibly at least a three-body angular dependence, but further work is needed. 2.3.2 Radial Distribution Function and The Einstein Frequency The radial distribution function [125] is a measure of spatial correlations normalized by the ideal gas. It has been shown that in general, there always exists a RPP that can reproduce g(r) from a N-body simulation [147], and the force-matching procedure provides an avenue for obtaining this RPP. Fig. 2.9 compares g(r) computed from MD simulations for all RPP models for C, Al, V, and Au. Each row corresponds to a different temperature, and clear trends can be observed, such as the improvement in agreement between models as the temperature increases. We note that the force-matched RPP always obtains the correct g(r), and the NPA model generally predicts the location of the first peak but sometimes over-predicts the magnitude or misses the location of the first peak altogether as observed in the case of V at 6.11 g/cm3 for T = 0.5 eV. The TFY model always underestimates the magnitude of the first peak height, and the location is usually shifted. 29 4 C (2.267 g/cm , T = 0.5 eV)KS-MD Al (2.7 g/cm3, T = 0.5 eV) V (6.11 g/cm3, T = 0.5 eV) Au (19.30 g/cm3, T = 0.5 eV) 3 3 TFY FM g(r) 2 NPA 1 0 2 C (2.267 g/cm3, T = 2 eV) Al (2.7 g/cm3, T = 2 eV) V (6.11 g/cm3, T = 2 eV) Au (19.30 g/cm3, T = 2 eV) 1.5 g(r) 1 0.5 0 C (2.267 g/cm3, T = 5 eV) Al (2.7 g/cm3 T = 5 eV) V (6.11 g/cm3 T = 5 eV) Au (19.30 g/cm3, T = 5 eV) 1.5 1 g(r) 0.5 00 1 2 3 4 0 1 2 3 4 50 1 2 3 4 0 1 2 3 4 5 r (Å) r (Å) r (Å) r (Å) Figure 2.9: The radial distribution functions for C, Al, V, and Au are shown. The top row corresponds to T = 0.5 eV, the middle row T = 2 eV, and the bottom row T = 5 eV. The force-matched RPP always reproduces the g(r) obtained from KS-MD. Insight into the connection between the g(r) peak height and the the self-diffusion coefficient can be obtained from the normalized velocity autocorrelation function [148] hv(t) · v(0)i Z(t) = , (2.14) hv(0) · v(0)i where v(t) is the velocity of a particle at time t and h·i is an ensemble average over particles and time. A short time expansion of Eq. (2.14) yields t2 Z(t) = 1 − Ω20 +··· , (2.15) 2! where Ω0 is the Einstein frequency ∫ ∞ 4πρi Ω20 = dr r 2 g(r)∇2 u(r), (2.16) 3mi 0 where mi is the ion mass in grams. The Einstein frequency gives insight into the relationship between u(r) and g(r), highlighting how different regions are weighted more or less depending 30 200 TFY 150 ×1/3 FM ×1/3 ×1/2 100 NPA r2g(r) 2u(r) ×1/3 50 0 50 C (2.267 g/cm3, T = 0.5 eV) Al (2.7 g/cm3, T = 0.5 eV) V (6.11 g/cm3, T = 0.5 eV) Au (19.30 g/cm3, T = 0.5 eV) 200 150 100 r2g(r) 2u(r) ×1/2 ×1/2 50 0 50 C (2.267 g/cm3, T = 5 eV) Al (2.7 g/cm3, T = 5 eV) V (6.11 g/cm3, T = 5 eV) Au (19.30 g/cm3, T = 5 eV) 0 1 2 3 4 0 2 4 0 1 2 3 4 0 2 4 r (Å) r (Å) r (Å) r (Å) Figure 2.10: The integrand of the Einstein frequency Eq. (2.16). All integrands are consistent with values reported in Table 2.1 as the self-diffusion coefficient decrease as the integral of the Einstein frequency increases. This allows for a “by eye" comparison of the self-diffusion coefficient from different RPP models. on the curvature of u(r). In Fig. 2.10, the integrand of Eq. (2.16) is shown. For the TFY model, the integrand is always smaller than those predicted by force-matched and NPA RPPs. The area under each curve in Fig. 2.10 can be directly connected to the self-diffusion coefficient through the Green-Kubo relation (in 3 dimensions) ∫ ∞ T D= dt Z(t), (2.17) m 0 by substituting, Eq. (2.15) into Eq. (2.17). Doing so shows that the TFY model will always predict a larger self-diffusion coefficient than the force-matched or NPA model as the area under these curves is larger. This is confirmed later when the self-diffusion coefficients are explicitly calculated as discussed in Sec. 2.3.3. 2.3.3 Self Diffusion Another approach to compute the self-diffusion coefficient is via the the slope of the mean- squared displacement from the Einstein relation h|r(t) − r(0)| 2 i D = lim . (2.18) t→∞ 6t 31 100 C (2.267 g/cm3) Al (2.7 g/cm3) Ar (1.3954 g/cm3) V (6.11 g/cm3) 10 1 D (Å2/fs) 10 2 KS-MD 10 3 TFY FM NPA 101004 V (3.055 g/cm3) Fe (7.874 g/cm3) Fe (3.937 g/cm3) Au (19.30 g/cm3) 10 1 D (Å2/fs) 10 2 10 3 10 4 10 1 100 101 10 1 100 101 10 1 100 101 10 1 100 101 T (eV) T (eV) T (eV) T (eV) Figure 2.11: Self-diffusion coefficients for different elements and densities versus temperature. The numerical values are reported in Table 2.1. For all cases all models predict values that have roughly the same order of magnitude. The only case where the force-matched RPP fails to reproduce the KS-MD self-diffusion coefficient is for C at 2.267 g/cm3 and T = 0.5 eV. The TFY RPP model generally predicts larger self-diffusion coefficients, which is consistent with the Einstein frequency in Fig. 2.10. Note that the NPA RPP model, in contrast, agrees with results obtained from the force-matched RPP and KS-MD models very well. Both Eqs. (2.17) and (2.18) can be used to compute the self-diffusion coefficient and have been shown to be equivalent [149]. Here, the self-diffusion coefficient has been calculated from a linear fit to the mean-squared displacement, h|r(t) − r(0)| 2 i. Due to finite-size effects, two problems arise when computing the slope and uncertainty of the linear fit. First, we must ensure that the linear fit is carried out in the late-time linear regime of the mean-squared displacement. Second, we dismiss statistically unconverged late-time behavior of the mean-squared displacement where the ensemble average contains sparse amounts of data. To remedy both of these concerns, we uniformly randomly sub-sample the mean-squared displacement 100 times with 10 points along each sub-sample. Next, a linear fit is determined for each sub-sample, and the standard deviation of the sub-sample slopes is computed. Once the standard deviation is known, a cutoff time is calculated by determining the point in time that the standard deviation of the sub-sample fits is less than half of the standard deviation computed from sub-sample fits to the 32 entire mean-squared displacement. The simulation data for the mean-squared displacement after the cutoff time is discarded, and the fitting procedure described above is repeated. The average, and standard deviation of the fits to the reduced dataset yield self-diffusion coefficient and the uncertainty in the fit respectively and are reported in Table 2.1. Given the values for the self-diffusion coefficient reported in Table 2.1, we can answer the following question: at what temperature are computationally inexpensive models adequate? To do this, we compute the relative self-diffusion coefficients DN P A/DKS−M D and DT FY /DN P A. For example, the top panel in Fig. 2.12 suggests that NPA models may be accurate from T = 1 eV and above if the target error tolerance is 50% of the self-diffusion coefficient computed from KS-MD. Similarly in the bottom figure, the TFY model is generally accurate to within 50% of the NPA model from T = 5 eV and beyond. Two important observations can be made from the trends in Fig. 2.12. The top panel illustrates temperatures at which an N-body potential is needed and when NPA is adequate. The bottom panel shows a comparison with TFY, which has the simplest uei (k) and χ(k), and we see temperatures at which TFY becomes comparable to NPA, suggesting when we can exploit simpler approximations for those inputs. 2.3.4 Power Spectrum The self-diffusion coefficient is useful for comparing and quantifying the accuracy of RPP models and transport theories, but in order to assess how accurately the particle dynamics are reproduced, we look at the power spectrum of the velocity autocorrelation function Z(t) ∫ ∞ Z̃(ν) = dt cos (2πνt) Z(t). (2.19) 0 In Fig. 2.13, we compare Z̃(ν) calculated using TFY, force-matched, and NPA RPPs against results obtained from KS-MD. We find that with the exception of low temperature C and V, force-matched RPPs agree with the KS-MD results across the entire frequency range. This, combined with the low relative force errors and accurate reproduction of static properties discussed previously, indicates that the the force-matched RPPs accurately approximate the Kohn-Sham potential energy surface. 33 Element ρi (g/cm3 ) T (eV) DKS−M D DF M DT FY DN P A Li 0.513 0.054 1.4 ± 0.13 1.27 ± 0.054 5.6 ± 0.39 1.26 ± 0.077 C 2.267 0.47 2.4 ± 0.12 9.3 ± 0.20 11.0 ± 0.60 1.69 ± 0.060 1.0 18.6 ± 0.7 27.2 ± 0.77 25 ± 1.64 11.0 ± 0.45 2.0 46 ± 1.68 49 ± 1.0 42 ± 3.70 43 ± 3.86 4.9 85 ± 5 94 ± 5.82 106 ± 5.37 92 ± 3.45 10.0 – – 215 ± 4.49 151 ± 4.63 15 – – 266 ± 3.46 198 ± 5.10 20 – – 349 ± 7.60 249 ± 2.35 28 – – 423 ± 13.28 324 ± 12.17 Al 2.7 0.1 – – 1.6 ± 0.14 0.35 ± 0.0217 0.50 3.8 ± 0.16 4.1 ± 0.13 9.17 ± 0.099 3.9 ± 0.11 1.1 9.8 ± 0.30 9.4 ± 0.11 17.5 ± 0.70 8.5 ± 0.44 2.0 18.7 ± 0.50 18.8 ± 0.68 34.8 ± 0.52 18.8 ± 0.40 4.9 48 ± 3.56 49 ± 3.17 72 ± 3.03 54 ± 2.7 9.2 83 ± 1.63 84 ± 5.67 122 ± 5.83 94 ± 8.80 10.0 – – 131 ± 6.77 105 ± 13.53 15.4 134 ± 3.68 129 ± 3.37 169 ± 5.26 142 ± 6.17 20.0 – – 197 ± 5.21 151 ± 2.55 30.0 – – 252 ± 4.74 203 ± 6.44 Ar 1.395 0.48 10.7 ± 0.43 12 ± 1.03 19 ± 1.10 – 1.0 20.1 ± 0.89 26 ± 3.0 39 ± 2.22 – 2.0 48 ± 1.75 45 ± 2.84 85 ± 8.75 – 5 – – 143 ± 6.55 171 ± 4.09 10.0 – – 210 ± 14.53 179 ± 6.21 15.0 – – 235 ± 13.34 193 ± 11.95 20.0 – – 255 ± 6.73 209 ± 8.91 30.0 – – 268 ± 2.73 228 ± 8.26 V 6.11 0.49 2.25 ± 0.050 2.86 ± 0.079 3.9 ± 0.18 0.91 ± 0.027 1.0 5.5 ± 0.21 6.5 ± 0.16 7.9 ± 0.36 6.6 ± 0.15 2.1 11.6 ± 0.78 12.5 ± 0.68 17.8 ± 0.74 14.8 ± 0.50 4.8 24.2 ± 0.63 24.7 ± 0.88 41 ± 2.76 27.7 ± 0.90 9.5 46 ± 3.41 42 ± 2.65 68 ± 2.10 47.6 ± 0.93 14.6 53 ± 1.81 57 ± 3.25 84 ± 4.83 63 ± 1.19 20.0 – – 103 ± 6.10 82.7 ± 0.78 30.0 – – 134 ± 8.57 96 ± 1.86 Table 2.1: The self-diffusion coefficient for all systems. For each RPP model, the number of particles, time step, and simulation length were kept identical for each element, density, and temperature. Finite-size corrections are carried out in Sec. 2.3.5. 34 Table 2.1 (cont’d) Element ρi (g/cm3 ) T (eV) DKS−M D DF M DT FY DN P A V 3.055 0.5 9.0 ± 0.81 11.3 ± 0.29 8.7 ± 0.23 – 0.97 14.7 ± 0.47 15.4 ± 0.43 19 ± 1.39 – 2.0 23 ± 1.13 24 ± 1.84 31 ± 1.26 27 ± 1.84 4.9 47 ± 4.38 44 ± 2.52 66 ± 7.30 48 ± 2.02 Fe 7.874 0.51 2.13 ± 0.047 2.34 ± 0.042 2.84 ± 0.030 – 1.1 5.27 ± 0.098 5.5 ± 0.16 5.9± 0.39 – 2.1 10.4 ± 0.72 10.4 ± 0.73 14.8 ± 0.46 9.2 ± 0.47 5.0 20.4 ± 0.61 22.0 ± 0.97 32 ± 1.41 27.1 ± 0.19 10.4 35 ± 1.14 38 ± 1.40 54 ± 1.25 49 ± 2.90 15.0 – – 83 ± 5.18 60 ± 2.60 20.0 – – 97 ± 2.93 70 ± 1.04 30.0 – – 103 ± 4.69 83.0 ± 0.94 Fe 3.937 0.51 6.0 ± 0.39 8.5 ± 0.94 6.2 ± 0.21 – 1.1 15.8 ± 0.70 15.6 ± 0.67 14.4 ± 0.33 – 2.1 20 ± 1.18 22 ± 2.07 27 ± 1.28 – Au 19.30 0.52 0.92 ± 0.028 0.71 ± 0.084 1.67 ± 0.12 0.51 ± 0.042 1.1 2.0 ± 0.11 1.92 ± 0.088 3.9 ± 0.42 1.66 ± 0.069 1.9 4.0 ± 0.14 3.4 ± 0.15 6.6 ± 0.16 3.52 ± 0.05 5.0 7.8 ± 0.40 8.2 ± 0.21 15.3 ± 0.63 10.7 ± 0.50 9.7 14.4 ± 0.64 15 ± 1.19 25 ± 1.94 16.6 ± 0.86 15.0 19.82 ± 0.80 22 ± 2.56 30 ± 1.95 25 ± 2.79 20.0 – – 39 ± 3.49 28.0 ± 0.97 30.0 – – 56 ± 2.23 33 ± 1.26 For higher temperatures, the NPA RPP is very similar to the force-matched RPP for low and high frequencies for all elements. For T = 0.5 eV, the dynamics predicted from the NPA model are noticeably less similar to those from KS-MD where NPA underestimates the prevalence of low- frequency modes in Au and both low and high-frequency modes in V. Interestingly, the NPA RPP captures the single-particle dynamics of low temperature C very well, but Figs. 2.5 and 2.9 indicate that this agreement comes at the expense of sacrificing the accuracy of static properties. Lastly, the TFY RPP exhibits roughly the same trends across all elements and temperatures– overestimation of the low frequency modes and underestimation of the high-frequency modes except for the case of C at 2.267 g/cm3 and T = 5 eV where excellent agreement with KS-MD is observed. 35 1.0 (a) 0.5 1 MD 0.0 DNPA/DKS 0.5 1.0 0 5 10 15 4 (b) C (2.267 g/cm3) Al (2.27 g/cm3) 3 ×1/2 V (6.11 g/cm3) DTFY/DNPA 1 Fe (7.874 g/cm3) 2 Au (19.30 g/cm3) 1 0 1 0 10 20 30 T (eV) Figure 2.12: Relative self-diffusion coefficients. The shaded region brackets the range of −0.5 and 0.5. (a) For all cases except V at T = 0.5 eV, the points fall within the bounds of the bracketed region. The NPA RPP fails to reproduce the KS-MD results at T = 0.5 eV, revealing a temperature boundary below which KS-DFT is needed. In (b) the points are within 50% of the NPA value from T = 5 eV and above for most cases. The orange point marked with an arrow has been reduced by a factor of 1/2 to improve clarity of the banded region. 2.3.5 Finite-Size Corrections Generally, thousands or even millions of atoms are needed to approximate the thermodynamic limit [88, 110]. While the KS-MD framework provides an accurate description of the electronic structure and the N-body potential is determined on-the-fly, corrections for finite-size effects must be considered. When the shear viscosity η of the system is known, finite-size corrections can be determined from [150] ξT D∞ = D N + , (2.20) 6πηL where D∞ is the self-diffusion coefficient in the thermodynamic limit, DN is the self-diffusion coefficient computed from a system of finite number of particles N, and ξ = 2.837297 for cubic 36 102 C (2.267 g/cm , T = 0.5 eV) Al (2.7 g/cm3, T = 0.5 eV) V (6.11 g/cm3, T = 0.5 eV) Au (19.30 g/cm3, T = 0.5 eV) 3 Z( ) 101 100 102 C (2.267 g/cm , T = 5 eV) Al (2.7 g/cm3, T = 5 eV) V (6.11 g/cm3, T = 5 eV) Au (19.30 g/cm3, T = 5 eV) 3 Z( ) 101 KS-MD TFY FM NPA 1010 0 1 100 101 10 1 100 101 10 1 100 101 10 1 100 101 Frequency (THz) Frequency (THz) Frequency (THz) Frequency (THz) Figure 2.13: The normalized power spectrum for C, Al, V, and Au. For C at T = 0.5 eV, the single particle dynamics are poorly described by the TFY and force-matched models but more accurately described with the NPA model. As the temperature increases from T = 0.5 to 5 eV, all models more accurately reproduce small and high frequency dynamics with the most notable improvement for C. simulation boxes with periodic boundary conditions. When η is unknown, multiple simulations of increasing particle number are carried out, and a linear fit is used to determine D∞ . Results from this procedure are shown in Fig. 2.14 where D∞ is determined via linear extrapolation to 1/L = 0. By finding the percent difference in D∞ and DN , we approximate the errors from finite-size effects in the KS-MD self-diffusion coefficient at these conditions. The approximate error in KS- MD for the case shown in Fig. 2.14, is ∼ 20%. While the error will vary with {Z, n, T }, the impact of finite-size effects is significant. From this study, the most promising approach is to fully converge the NPA MD results, using force-matched RPPs when necessary (for low temperatures T . 1 eV). Finite-size corrections allow for a direct comparison to analytic transport theories, namely the Stanton-Murillo model [151]. The Stanton-Murillo model, provides a closed form solution for ionic self diffusion by using an effective interaction potential in a Boltzmann kinetic theory framework. The major benefit of this model is that the computation of ionic transport is nearly instantaneous. However, its applicability in the cold dense matter and warm dense matter regimes is unknown. The results in Table 2.2 show that the effective interaction approach of the Stanton-Murillo 37 Number of Particles 00375 0 8 3 216 64 22 TFY FM NPA 20 D (10 3Å2/fs) 18 16 14 12 0.00 0.02 0.04 0.06 0.08 0.10 1/L (Å 1) Figure 2.14: Finite-size effect study for V at 6.11 g/cm3 and T = 2 eV. Identical MD simulations were carried out with increasing particle number. Extrapolating with a linear fit (grey dashed line) to 1/L = 0 approximates the thermodynamic limit, correcting the values in Table 2.1. model captures much of the many-body physics included in the TFY RPP results. The main weakness of the model, and also TFY, is therefore the functional form of the interaction they employ, as the differences with the force-matched and NPA columns reveal. Because self diffusion is a relatively simple transport coefficient [151], more work is needed to quantify these trends for other transport properties. With the converged self-diffusion data, we generate an effective interaction correction to the Stanton-Murillo model. The effective interaction corrected Stanton-Murillo model is DCSM = α(Z, T)DSM , (2.21) where α(Z, T) is determined by fitting the ratio of the self-diffusion coefficient from the best performing RPP model and the self-diffusion coefficient computed from the Stanton-Murillo model DSM to the functional form aerf(bT) α(Z, T) = + 1, (2.22) bT which asymptotes to DSM as T increases. Here the “best performing RPP model" refers to the RPP model that most accurately reproduced the self-diffusion coefficient computed from 64 particle 38 Element T (eV) DF M DN P A DT FY DSM DCSM C 0.47 10.55 2.14 12.66 13.08 2.14 1.0 32.44 14.21 25.57 26.11 13.87 2.0 56.70 43.12 51.14 50.53 39.23 4.9 99.51 109.55 117.88 118.34 106.76 10.0 – 169.91 210.76 217.34 206.71 15.0 – 219.99 296.21 293.54 284.10 20.0 – 256.33 342.15 356.41 348.04 28.0 – 327.32 470.10 439.44 432.07 V 0.49 4.14 1.01 5.42 6.76 4.39 1.0 8.54 8.53 11.53 12.26 7.96 2.1 15.67 18.87 22.56 23.14 15.03 4.8 28.72 31.34 42.42 46.25 30.18 9.5 49.49 54.90 73.76 77.10 51.16 14.6 66.98 74.44 99.82 99.78 67.97 20.0 – 87.90 118.24 117.89 83.15 30.0 – 105.60 143.63 141.66 106.94 50.0 – 131.84 175.35 171.07 143.02 75.0 – 178.34 202.93 194.31 172.91 100.0 – 209.50 207.20 211.86 194.36 Table 2.2: Self-diffusion coefficient in the thermodynamic limit. Both elements are at solid density (2.267 g/cm3 for C, and 6.11 g/cm3 for V). KS-MD simulations. The parameters a and b are reported in Table 2.3 for C at 2.267 g/cm3 and V at 6.11 g/cm3 , and their values vary considerably between both cases emphasizing the need for a comprehensive finite size effect study to produce correction factors for additional elements and conditions. This correction factor allows for the use of the Stanton-Murillo model in regions of previously unknown accuracy. The finite-size corrections along with the corrected Stanton-Murillo model results are shown Fig. 2.15 with the numerical values given in Table 2.2. Note that for low temperature C at 2.267 g/cm3 , the best performing RPP model was NPA (as reported in Fig. 2.11 and Table 2.1) explaining why the corrected Stanton-Murillo model tends towards the NPA RPP at low temperatures. For V at 6.11 g/cm3 , the best performing RPP model was the force-matched RPP again explaining the low temperature trend. In an attempt to summarize our work in a single figure, Fig. 2.16 shows our suggested use cases for all RPPs studied here for two relative self-diffusion accuracies computed from Table 2.1. 39 100 (a) 10 1 D (Å2/fs) SM CSM 10 2 TFY FM NPA 10 3 100 (b) 10 1 D (Å2/fs) 10 2 10 3 100 101 102 T (eV) Figure 2.15: Self-diffusion coefficient versus temperature in the thermodynamic limit. The points displayed here are taken from Table 2.2. (a) Self-diffusion coefficient for C at 2.267 g/cm3 . (b) Self-diffusion coefficient for V at 6.11 g/cm3 . The Stanton-Murillo model (denoted SM) fails for low temperature C. For V, the Stanton-Murillo model shows excellent agreement with the force- matched RPP even at low temperatures. The validity of the Stanton-Murillo model is extended to low temperatures with an effective interaction correction (denoted CSM). Element a b C (2.267 g/cm3 ) 2.198 -1.032 V (6.11 g/cm3 ) 0.03767 -0.3112 Table 2.3: Coefficients a, and b for the effective interaction correction Eq. (2.22). Note that the values of a and b vary considerably for each element. 40 When points (the average value or its uncertainty) for a given model are within the appropriate tolerance (30% for the top panel and 15% for the bottom panel), we consider the model as being accurate for that temperature and element and is denoted with a colored bar or arrow. We rank the computational expense from lowest to highest as: TFY, NPA, force matching, and KS-MD. When a computationally cheaper model is accurate, it replaces the more computationally expensive model in Fig. 2.16. Based on trends observed in Figs. 2.7, 2.12, and 2.15, we assume that the models remain accurate for higher temperatures and illustrate this by upward pointing colored arrows. For example, consider the case of Fe in the top panel of Fig. 2.16. The force-matched RPP is accurate to within 30% of the KS-MD result from T = 0.5 eV and up. The NPA model, which is computationally cheaper than the force-matched RPP, becomes accurate (within 30% of KS-MD) at T = 2 eV and up, hence the transition between the force-matched and NPA models. For Al, the NPA RPP is within 15% of KS-MD at all temperatures. However, at T = 15 eV the TFY model becomes accurate therefore replacing the NPA RPP. 2.4 Conclusions and Outlook A systematic study of various RPPs for molecular dynamics simulations of dense plasmas was performed for a wide range of elements versus temperature for solid and half-solid density cases. Of the RPPs studied here, RPPs constructed from a NPA approach come closest to accurately reproducing the transport and structural properties predicted by KS-MD. The failures of NPA for metals near T = 0.5 eV are expected: V is a polyvalent metal and s-d hybridization occurs in Au, which is not treated at all in our variant of the NPA model. Thus, it is unclear if inaccuracies in NPA reveal the need for N-body interactions or an improved NPA treatment. Moreover, finite- size corrections to KS-MD are seen to be significant; prior work on Si suggests that at least 108 particles are needed to accurately treat elements like C at low temperatures [152]. Studies on C and Si where there are transient covalent bonding at low temperatures have raised the inadequacy of the PBE XC-functional that has been used here. In [152], the SCAN functional was used showing remarkable agreement between VASP calculations and NPA results for super-cooled high-density Si. This implies that VASP calculations for systems in the low temperature warm dense matter 41 |DRPP/DKS MD 1| 0.30 T = 15 eV 101 T = 5 eV T (eV) 100 T = 0.5 eV TFY FM 10 1 NPA |DRPP/DKS MD 1| 0.15 T = 15 eV 101 T = 5 eV T (eV) 100 T = 0.5 eV 10 1 Li C Al Ar V V1/2 Fe Fe1/2 Au Figure 2.16: Suggested use cases for RPPs based on the relative self-diffusion coefficient error (between RPP-MD and KS-MD) and cheapest computation cost. The top and bottom panels correspond to a 30% and 15% relative error respectively. The elements denoted with a subscript of “1/2" corresponds to half solid density (V at 3.055 g/cm3 and Fe at 3.937 g/cm3 ). The colored bars indicate the computationally cheapest RPP that generates a self-diffusion coefficient to within the specified error tolerance available for that system based on Table 2.1. The empty space under each bar indicates regions where no KS-MD data was collected so no assessment on a RPPs accuracy can be made. 42 regime become sensitive to the choice of the XC-functional. Similarly, the XC-functionals for transition metals like V, Fe, etc., are known to need Hubbard-type corrections that are not included in our studies. Although this work does not fully resolve these issues, the trends seen for the lowest temperature for C, V and Au should be examined in detail in future work. Additionally, the NPA model is exceptionally accurate for Al. As Al is a free electron metal, its electronic structure is well described as a Fermi-Liquid, the precise physical model in which NPA performs well. In the cases where the electronic structure of the system is not well described as a Fermi-liquid, the performance of the NPA model decreases at low temperature, further emphasizing the need for a comprehensive study over a range of elements and conditions. As in previous works [89,103], the TFY model predicts the least structured g(r). Notionally, the accuracy of the TFY model appears to follow the machine learning trend of hZi/Z > 0.35 [153], although it was not possible to use all models here at high enough temperatures to be quantitative. In contrast, the NPA model with its improved Kohn-Sham treatment and use of a pseudopotential in Eq. (2.7) eliminates most of these errors except for C and V at T = 0.5 eV, elements for which we would recommend NPA for T > 2 eV. Because we examined seven diverse elements over the warm dense matter regime, the accuracy of NPA (and for moderate temperature, even TFY) suggests that no additional “short-range repulsion" [32, 103, 141–145] is needed beyond Eq. (2.7); as Eq. (2.7) does not contain core-core repulsion, the structure of the interaction is more likely to be effective core-valence repulsion captured by uei (k), as well as structure in χ(k) beyond χT F (k). However, we note, that in treating weakly ionized systems like warm-dense Ar with a mean ionization of hZi = 0.3, some 70% of the Ar atoms are neutral, while about 30% of the atoms are singly ionized. Thus, the neutrals interact via a core-core interaction screened by the free electrons. In such cases the use of Eq. (2.7) alone is inadequate. The NPA model treats such a two-component mixture using three pair potentials. In general, core-core interactions are important for weakly-ionized atoms with a large core. These core-core interactions can be readily calculated using the core- electron density obtained from the NPA Kohn-Sham calculation. As expected, the force-matched RPP reproduced the g(r) computed from KS-MD for all cases. In only one case, again C at 2.267 43 g/cm3 and T = 0.5 eV, the force-matched RPP overestimated the self-diffusion coefficient; this suggests that the spherical pair interaction isn’t applicable, and non-spherical corrections, which could include three-body contributions, are needed as suggested by the near-perfect agreement of the SNAP and KS-MD microfield of force magnitudes in Fig. 2.6. However, for all cases considered with T > 1 eV, the g(r) and self-diffusion coefficient are adequately described by a RPP. With the force-matched-validated NPA interaction, pre-computing the interaction allows for much larger pair-potential simulations. As fast analytic expressions for transport coefficients are needed for hydrodynamic modeling, we compared our self-diffusion results from all models to the Stanton-Murillo model for both C and V. In both cases, the Stanton-Murillo model was consistent with the TFY model (on which it is based) and both have agreement with force-matched-based results. The error between the Stanton- Murillo model and the force-matched results is < 65% below T = 10 eV for V and < 25% below T = 5 eV for C, adding confidence to the use of this model in hydrodynamics models above that temperature. For experiments that are rapidly heated above a few eV, little time is spent where the errors are large; because the transport coefficients are numerically very small during this transient heating, negligible transport can occur during that time. For example, note that the V diffusion coefficient varies by a factor of about 30 in the range T = 0.5 to 100 eV. Conversely, for experiments that dwell at lower temperatures, we provide a RPP-based correction factor to the Stanton-Murillo model with an error of less than 1% for C at T = 0.5 eV and 6% for V at T = 0.5 eV. Our results suggest several new avenues of investigation. From a data science perspective, larger collections of systematically-obtained simulation results would aid in better defining accuracy boundaries. In particular, more elements that produce more material types should be studied. For mixtures, N-body potentials could be explored; here, we cast all of the pair potentials as heteronuclear. Additionally, our conclusions are based on studies of the microfield distribution of forces, Einstein frequency, power spectrum, self-diffusion coefficient, and g(r), which could be extended to include other properties such as viscosities and interdiffusion in mixtures, electrical conductivity, thermal conductivity, and ion-dynamical properties like the speed of sound [109]. 44 While we focused primarily on force matching, effective interaction potentials can be obtained through “structure matching" [125,154,155]. Finally, as very large scale simulations become more common, spatially heterogeneous plasmas can be modeled; much less is known about potentials in such environments, although recent work has explored non-spherical potentials [88]. 45 CHAPTER 3 ANALYTIC MODELS FOR INTERDIFFUSION IN DENSE PLASMA MIXTURES 3.1 Introduction The contents of Chapter 2 were focused on assessing the validity of force laws for MD simu- lations of plasmas of a single ion species. In this chapter1, we extend our study to binary plasma mixtures where we focus on formulating analytic models for the interdiffusion coefficient. Interdiffusion, which is atomic-scale mixing driven by density gradients, occurs in extremely disparate physical systems. Early experimental work in alloys addressed interdiffusion at interfaces between solids [157–159]. The interpretation of these experiments led to the development of early theories in an attempt to quantify observed effects [160]. Interdiffusion remains important for industrial applications and has been studied in the context of neutral liquids [161, 162] and liq- uid metals [163–165]. In stellar environments, interdiffusion controls the distribution of elements throughout the star, impacting its evolution [166–168]. Additionally, diffusive mixing of thermonu- clear fuel in inertial confinement fusion experiments [169] can spoil the burn conditions through radiative losses [170–174]. Recent large-scale MD simulations [88] of heated plasma interfaces have exposed many complex issues: multiple ionic temperatures, jetting of light particles across the interface, uncoupled velocity fields, and intense electric fields. Experimental data for these processes is minimal, but has motivated several current experiments [175–179]. Despite progress in the theory of interdiffusion and charged particle transport [180], several gaps remain. In contrast with its one-particle counterpart, self diffusion, interdiffusion, a collec- tive property, is investigated relatively rarely. In most computational studies of interdiffusion, the thermodynamic factor is set to unity [102, 148, 181, 182]. While this may be accurate in some cases [161], a complete exploration across physical regimes for a wide range of mixtures is lacking. This has been addressed only recently in work that employed MD to create a data set of thermody- namic factors [183]. Computation of the relevant autocorrelation function, and the thermodynamic 1 The content of this chapter has been reproduced from Lucas J. Stanek and Michael S. Murillo , “Analytic models for interdiffusion in dense plasma mixtures", Physics of Plasmas 28, 072302 (2021) https://doi.org/10.1063/5.0047961" with the permission of AIP Publishing; see Ref. [156] for the full published article. 46 factor, is subject to finite-size effects [184]. These effects are particularly important when compu- tational models are very expensive, as is the case with on-the-fly potentials [87]. Because of these computational issues, there is a lack of interdiffusion data within the dense-plasma community. Seen from a data-science perspective, binary interdiffusion is at least seven dimensional: one must specify, among other choices of variables, the mean density, stoichiometry, temperature, charge, and mass of each species. If the interparticle potentials have other dependencies, then additional pa- rameters must be used. Machine learning has been applied to this setting and has shown promising results [183]. In practice, it is preferable to employ validated theoretical models that are very rapid to compute [151, 185, 186]; more work is needed to validate such models in this seven-dimensional space. We begin Sec. 3.2 by stating the definitions used here to define a binary mixture, and we show how the interdiffusion coefficient arises from hydrodynamic equations of motion. In Sec. 3.3, we derive analytic forms for the thermodynamic factor that cover a wide range of plasma conditions; these forms employ both the radial distribution function and the structure factor. Lastly, in Sec. 3.4, we derive a rapidly computable analytic expression for the interdiffusion coefficient in a binary ionic mixture (BIM). We compare this result to MD data, revealing excellent agreement in moderately and strongly coupled regimes. 3.2 Interdiffusion The description of interdiffusion, and the values of the interdiffusion coefficients, are not unique. In this section, we review various formulations of interdiffusion and establish the conventions and notations we will use. For simplicity, we examine only binary plasmas; we consider a binary plasma that contains Ni ions of each species “i, ” with charge Zi e and mass Mi . Note that in contrast to Chapter 2, here we denote the mean ionization of species i as Zi instead of hZi i. The total number of ions in the system is N = N1 + N2 . We assume that the ionic species are in thermodynamic equilibrium at inverse temperature β = 1/T and that the electrons are at inverse temperature βe = 1/Te . The average number and mass densities are ni and ρi = Mi ni , respectively, with corresponding total densities n = n1 + n2 and ρ = ρ1 + ρ2 . Additionally, we define the number 47 and mass concentrations as xi = ni /n and ci = ρi /ρ, respectively. The mean ionization and mass of the binary mixture are given as hZ α i ≡ x1 Z1α + x2 Z2α, (3.1) hMi ≡ x1 M1 + x2 M2, (3.2) where α ∈ R. From Eqs. (3.1) and (3.2), we define zi = Zi /hZi and mi = Mi /hMi. Lastly, we define the Coulomb coupling parameter of the ions as e2 Γ0 = , (3.3) aT where   −1/3 4πn a= (3.4) 3 is the total ion-sphere radius. Note that Eq. (3.3) does not depend on the mean ionization of the ions. To include the mean ionization of the ions, we define Zi Zi 0 e2 Γii 0 = = Zi Zi 0 Γ0 . (3.5) aT We now begin our discussion of interdiffusion by defining the microscopic density and velocity fields as ÕNi ni (r, t) = δ[r − ri, j (t)], (3.6) j=1 ÕNi ui (r, t) = vi, j (t)δ[r − ri, j (t)]. (3.7) j=1 In Eqs. (3.6) and (3.7), the index j refers to the jth particle of species i. The time evolution of the density Eq. (3.6), using Eq. (3.7), yields the continuity equation ∂ni (r, t) = −∇ · [ni (r, t)ui (r, t)] . (3.8) ∂t Equation (3.8) is not closed until we specify the evolution of the velocity field ui (r, t); it is such closures that yield diffusion equations. However, in many physical systems, including plasmas, the 48 flow fields can be complex and the velocity field must also be evolved as part of a hydrodynamic description. As the fluid density evolves in time, both advection and diffusion occur. To isolate diffusion from advection, we define the diffusive flux ji (r, t) relative to a reference frame as ji (r, t) = ni (r, t) [ui (r, t) − uref (r, t)] , (3.9) where uref (r, t) is a reference velocity. The choice of the reference velocity uref (r, t) is problem dependent and some common choices are listed in Table 3.1. Written in terms of the reference velocity, the continuity equation becomes ∂ni (r, t) + ∇ · [ni (r, t)uref (r, t)] = −∇ · ji (r, t). (3.10) ∂t The diffusion model and the values of the diffusion coefficients depend on the choice of reference velocity. While the reference-velocity field is assumed to be evolved by a separate equation, the closures for the diffusive flux are usually in the form of a slowly varying ansatz. Most hydrodynamic models employ mass densities rather than number densities. In this scenario, natural choices for the reference velocity and fluxes are the center-of-mass velocity (see Table 3.1) and ji (r, t) = ρi (r, t) [ui (r, t) − ucom (r, t)] , (3.11) respectively. With Eq. (3.11) as the choice for the diffusive flux, we can propose a closure of the form ji (r, t) = −Dx ∇x, (3.12) where the mass density flux is driven by forces caused by x with proportionality Dx ; again, there is considerable leeway in how these quantities are chosen. Different choices for the diffusive flux will yield different proportionality coefficients that have different physical meanings. Two of the most common choices for x are the chemical potential and the number density; these choices yield the diffusive fluxes [148, 187–189] 1 ji (r, t) = − D µ ∇µi (r, t), (3.13) T 49 and ji (r, t) = −Dn ∇ni (r, t), (3.14) respectively. For the specific choices of x in Eq. (3.13) and (3.14), the relationship between D µ and Dn is given by the thermodynamic factor [187, 190], which is addressed in Sec. 3.3. In a plasma hydrodynamics context, it is reasonable to choose to write the diffusive flux as [191] ji (r, t) = −ρ(r, t)D∇ci (r, t), (3.15) where D is the interdiffusion coefficient. Note that Eq. (3.15) assumes that particle fluxes are not driven by other gradients (e.g., temperature, pressure, electrostatic potential, etc.) or that other gradients are present but are collectively in equilibrium. Although we presume a separate time- evolution equation for uref (r, t) in Eq. (3.10), the slowly varying form of Eq. (3.15) is assumed for ji (r, t). It is worth summarizing the steps taken so far. To arrive at Eqs. (3.10) and (3.15), many non-unique choices were made. Perhaps more importantly, the form of Eq. (3.15) is an ansatz that may or may not be accurate for a given plasma scenario. The use of a chemical-potential gradient is associated with a special state referred to as “mechanical equilibrium,” [189] in which the pressure and temperature gradients have already relaxed; thus, the flux is not driven by gradients in those quantities. A generalized form for Eq. (3.15) may be needed to account for baro- and thermo- diffusion processes [70, 192, 193], for example. Nevertheless, in this work, we will proceed under the assumptions that the diffusive flux is given by Eq. (3.15). One can obtain a Green-Kubo relation consistent with these choices [188]. The interdiffusion coefficient is calculated as [181] ∫ ∞ J D= dthj(t) · j(0)i, (3.16) 3N x1 x2 0 where j(t) is the interdiffusion current defined as N1 Õ N2 Õ j(t) = x2 v1, j (t) − x1 v2, j (t). (3.17) j=1 j=1 50 Name Reference Velocity Õ ρi barycentric (center of mass) ucom = ui i=1 ρ Õ ni mean molar velocity ummv = ui i=1 n Õ mean volume velocity umvv = ρi vi ui i=1 Table 3.1: Possible choices for the reference velocity. Here, vi is the partial molar volume of species i. In reduced form, the interdiffusion coefficient is given by D∗ = D/ω p a2, (3.18) 4πnhZi 2 e2 ω2p = , (3.19) hMi where ω p is the “hydrodynamic" plasma frequency [181]. The evolution of the interdiffusion current is assumed to be stationary and to include all interparticle interactions. The brackets h· · · i represent an ensemble average over initial conditions (time and position) of the interacting plasma mixture. The prefactor J is the thermodynamic factor [191] x1 x2 J= , (3.20) Scc (k = 0) where Scc (k) is the concentration-concentration structure factor that can be decomposed into partial structure factors as √ Scc (k) = x1 x2 [x2 S11 (k) + x1 S22 (k) − 2 x1 x2 S12 (k)]. (3.21) Models for the partial structure factors are discussed in Sec. 3.3. Many researchers [186, 194–198] have explored a simplified form of Eq. (3.16) known as the Darken relation [160]. We revisit the derivation of this relation to assess its utility for modeling dense plasma mixtures. Key to obtaining the Darken relation is isolating the intraparticle contributions 51 from the interparticle contributions, as in ∫ ∞ J D= dthj(t) · j(0)i 3N x1 x2 0 ∫ ∞ *Õ N1 N1 + J x2 Õ = dt v1, j (t) · v1, j 0 (0) 3N x1 0 j=1 j =1 0 ∫ ∞ *Õ N2 N2 + J x1 Õ + dt v2, j (t) · v2, j 0 (0) 3N x2 0 j=1 j =1 0 ∫ ∞ *Õ N1 N 2 + 2J Õ − dt v1, j (t) · v2, j 0 (0) . (3.22) 3N 0 j=1 j 0 =1 Although all particles are interacting and the ensemble average h. . .i is taken over initial conditions of the interacting system, we can define a type of self-diffusion coefficient in analogy with the single-species case as ∫ ∞ *Õ Ni + 1 Di = dt vi, j (t) · vi, j (0) 3Ni 0 j=1 1 ∞ ∫ = dt vi, j (t) · vi, j (0) ( j = 1, 2, · · · , Ni ). (3.23) 3 0 Note that self diffusion refers to only intraparticle correlations (independent of species), whereas interparticle correlations (of any species) are described by the terms [188, 199] ∫ ∞ *Õ Ni Õ Ni + 1 fii = dt vi, j (t) · vi, j 0 (0) , (3.24) 3N 0 0 j=1 j , j ∫ ∞ *Õ N1 Õ N2 + 1 f12 = dt v1, j (t) · v2, j 0 (0) . (3.25) 3N 0 j=1 j =10 Note that fii and f12 are intraspecies and interspecies contributions, respectively. Finally, we arrive at the form " !# f11 f22 f12 D = J x2 D1 + x1 D2 + x1 x2 + − 2 . (3.26) x12 x22 x1 x2 The Darken relation is obtained by assuming that the third term vanishes, yielding D ≈ J (x2 D1 + x1 D2 ). (3.27) The Darken relation has direct application to systems for which experimental measurements yield D1 and/or D2 [198]; in a computational setting, no such limitation exists. 52 0.013 Full Darken 0.012 0.011 D* 0.010 0.009 0.0085000 10000 15000 20000 25000 pt Figure 3.1: The interdiffusion coefficient, Eq. (3.16), and the Darken relation, Eq. (3.27) versus time. An MD simulation was carried out for a H+ -He2+ BIM where N = 2000 particles, x1 = 0.5, ni = 1.62 × 1028 cm−3 , and T = 14.7 eV. Each point is calculated from a subset of the total simulation length (ω p t = 51946). The value of D∗ for the total simulation time is shown as a dotted (using Darken) or dashed [using Eq. (3.16)] line. The Darken relation has smaller statistical uncertainty but converges to the incorrect value. Note that use of the Darken relation does not impact MD simulations needed to create trajectory information; the Darken relation reduces the post analysis to two autocorrelation functions rather than five. Thus, the Darken relation potentially produces a less accurate result for a small improve- ment in computational cost. However, it is possible that the terms that the Darken relation retains have smaller statistical fluctuations, thereby allowing for a smaller MD trajectory calculation (as well as the faster post-MD analysis). We explore this hypothesis in Fig. 3.1 by performing MD with increasingly longer trajectories to identify when the statistical errors in the full autocorrelation, Eq. (3.26), are comparable to the Darken relation, Eq. (3.27). We find that, indeed, the full form of Eq. (3.26) has larger statistical errors and Eq. (3.27) converges more quickly. However, the mean value of the full autocorrelation is either consistent with the mean value from the Darken relation or better. Thus, suggesting that there is no penalty for using the full form even for small simulations. Moreover, converged full results are easily obtained with a modestly longer simulation. Here, we will always use the full result. 53 3.3 Thermodynamic Factor Once the values of Eqs. (3.23), (3.24), and (3.25) have been converged in particle number and simulation length, it remains to compute the prefactor J in Eq. (3.26). Typically, J is taken to be unity [102, 148, 181, 182, 200], often in analogy with neutral (Lennard-Jones) systems [161,201,202]. Other non-unity forms have been suggested [191,194] but have been explored very little. Comprehensive MD results have been obtained recently [183] in the context of the Darken relation. The thermodynamic factor can be obtained from integral equation theory, which gives the radial distribution functions [148] as gii 0 (r) = exp [−βuii 0 (r) + hii 0 (r) − cii 0 (r) + Bii 0 (r)] , (3.28) where uii 0 (r) is a pair potential [87], hii 0 (r) = gii 0 (r) − 1 are the pair correlation functions, cii 0 (r) are the direct correlation functions (DCFs), and Bii 0 (r) are the bridge functions [203]. While many choices for uii 0 (r) are possible, a BIM assumes the form of Zi Zi 0 e2 0 (r) = uiiBIM . (3.29) r The binary Yukawa model (BYM) includes effects from electron screening (e.g., Thomas-Fermi screening) and has the form Zi Zi 0 e2 −r/λT F uiiBYM 0 (r) = e , (3.30) r 4πne e2 λT−2F ≈r , (3.31)  2 Te2 + 23 EF where EF = ~2 (3π 2 ne )2/3 /2me for electron number density ne and mass me . Note that in con- trast to Chapter 2, the Thomas-Fermi screening length, Eq. (3.31), now depends on the electron temperature; here, the ions and electrons are assumed to have different temperatures. Equation (3.28) is a closure to the Ornstein-Zernicke equations (OZEs) that are given by Õ ∫ hii 0 (r) = cii 0 (r) + nk d 3r 0 cik (|r − r|0)h ki 0 (r 0). (3.32) k 54 For weakly to moderately coupled plasmas, we can employ the hypernetted chain approximation by setting Bii 0 (r) = 0 which allows us to find gii 0 (r) given uii 0 (r). After Fourier transformation, the pair correlation functions can be defined in terms of partial static structure factors [148] given by √ Sii 0 (k) = δii 0 + xi xi 0 nhii 0 (k). (3.33) For a two-component mixture, we can write hii 0 (k) in terms of the DCFs as     −1    h11  1 − n1 c11 −n2 c12 0  c11        h  =  1 − n1 c11 −n2 c12  c12  .        12   0 (3.34)        h22   0 −n1 c21 1 − n2 c22  c22              We can write the partial structure factors Sii 0 (k) (i, i0 ∈ {1, 2}) in terms of the direct correlation functions (DCFs) cii 0 (k) as [148, 204, 205] √ x1 x2 nc12 (k) S12 (k) = , (3.35) ∆(k) 1 − x2 nc22 (k) S11 (k) = , (3.36) ∆(k) 1 − x1 nc11 (k) S22 (k) = , (3.37) ∆(k) where ∆(k) = [1 − x1 nc11 (k)] [1 − x2 nc22 (k)] − x1 x2 n2 c12 2 (k). (3.38) Various approximations for the DCFs yield corresponding approximations for J . We give examples of such approximations in Secs. 3.3.1, 3.3.2, and 3.3.4 where we derive the thermodynamic factor from approximations of the DCFs for the following cases: classical weakly coupled plasmas, two-temperature electron-ion plasmas, partially degenerate dense plasmas, and strongly coupled plasmas. In Sec. 3.3.3, we derive the thermodynamic factor from estimates of the radial distribution functions (RDFs) gii 0 (r), showing equivalence to the DCF approach. When the system is at high temperature, transport coefficients will have the largest numerical value; thus, the thermodynamic factor in that limit is an important special case. Therefore, we 55 expand the exponential in Eq. (3.28) to obtain cii 0 (r) = −βuii 0 (r), (3.39) which is referred to as the mean-field limit. Substituting Eq. (3.39) into Eq. (3.34) allows us to compute hii 0 (k) directly; using hii 0 (k), J can easily be found in k-space from Eq. (3.33). Using Eq. (3.39) and setting Bii 0 (r) = 0 in the nonlinear form, Eq. (3.28), we arrive at an approximation for the RDFs in r-space gii 0 (r) ≈ exp[hii (r)]. (3.40) Now, given an interaction potential uii 0 (r), the pair correlation function hii 0 (r) in Eq. (3.40) can be obtained by solving Eq. (3.34) for hii 0 (k) and applying an inverse Fourier transform. Specifically, when uii 0 (r) = uiiBYM 0 (r), we have a non-linear (NL) model for the RDFs   Γii 0 −r k̃t gii 0 (r) ≈ exp − e , (3.41) r p where k̃t = κ 2 + 3x1 Γ11 + 3x2 Γ22 is the dimensionless total screening wavevector, and κ = a/λT F . Notice the effective potential in Eq. (3.41) depends on the total screening length from all species, which arose from solving the mixture OZE in Eq. (3.34). Effective potentials with wider applicability beyond Eqs. (3.39) and (3.34) will be discussed in Sec. 3.3.4. 3.3.1 Classical Mean-Field Approximation As mentioned in Sec. 3.3, hot plasmas that are classical and weakly coupled have the largest interdiffusion coefficients; mixing processes are rapid in such plasmas. Apart from some astro- physical plasmas, most laboratory plasmas have separate electron and ion temperatures; we allow for an independent electron temperature dependence through the effective ionic pair interaction. For two-temperature hot plasmas, we assume a standard Debye-Hückel model, which is expressed in terms of the mean-field DCF form, Eq. (3.39). For plasma mixtures, the DCF can be written in Fourier space as 4πZi Zi 0 e2 β cii 0 (k) ≈ − . (3.42) k 2 + k e2 56 Electronic screening of the ionic Coulomb interaction enters through the wavevector k e2 = 4πe2 βe ne . (3.43) Using Eq. (3.42), the partial structure factors are √ x1 x2 Z1 Z2 kr2 S12 (k) = − 2 , (3.44) k + k e2 + k 12 + k 22 k 2 + k e2 + k 22 S11 (k) = , (3.45) k 2 + k e2 + k 12 + k22 k 2 + k e2 + k 12 S22 (k) = , (3.46) k 2 + k e2 + k 12 + k22 in terms of reference and ionic species wavevectors kr2 = 4πe2 βn, (3.47) ki2 = 4πZi2 e2 βni, (3.48) respectively. Note the general trends that S12 (k) < 0 and 0 < Sii (k) < 1. Moreover, in the long-wavelength limit, there is no dependence on e2 , and when βe = β, there is no temperature dependence, except possibly through the Zi . Using the intermediate results, Eqs. (3.44) - (3.46), in Eq. (3.21), the concentration-concentration structure factor is then x1 x2 (k 2 + k e2 + kr2 hZi 2 ) Scc (k) = . (3.49) k 2 + k e2 + kr2 hZ 2 i Taking the long-wavelength limit (k → 0) of Eq. (3.49) and using Eq. (3.20), we have βe hZi + βhZ 2 i J DH = . (3.50) βe hZi + βhZi 2 Equation (3.50) is the main result of this subsection and will be derived below in Eq. (3.69) using the RDFs to show the equivalence of the two approaches. The first terms in both the numerator and denominator arise from electronic screening; while Eq. (3.43) implies that the screening is given by classical electrons, the electronic wavevector could instead be chosen to include electron degeneracy via Thomas-Fermi screening, Eq. (3.31). 57 Note that Eq. (3.50) generalizes a prior result for a one-temperature plasma [194, 206], hZi + hZ 2 i J 1T-DH = , (3.51) hZi + hZi 2 to two temperatures. Note that with this generalization we can examine the limit βe → 0 for fixed β (i.e., the limit in which the electrons have a much higher effective temperature than the ions); in that limit, Eq. (3.51) reduces to the known BIM limit [191], hZ 2 i J BIM = . (3.52) hZi 2 Unless otherwise noted, we use a Thomas-Fermi ionization model to determine the mean ionization state of each species in the mixture [151]. To assess the importance of electron screening in computing the thermodynamic factor, we compare calculations from Eqs. (3.51) and (3.52) as shown in column (a) of Fig. 3.2. At high temperatures, where ionization increases, the value of J increases. Additionally, when the number concentration differs (top and bottom rows), we find that the asymptotic value of J also differs. Column (b) shows the dependence of the value of J on ionization by using Eq. (3.51) for different combinations of fully ionized plasmas of species with charges Z1 and Z2 . We see that as the difference in charge between the ion species increases, J also increases and is maximized in the most extreme case (e.g., H-Og mixtures). Column (c) shows the value of J in the two-temperature setting via Eq. (3.50); a strong dependence of J on the electron temperature is evident. The red diagonal line indicates where T = Te , which is equivalent to using Eq. (3.51). The region above the red line (T < Te ) corresponds to plasmas produced using lasers in laboratory experiments. The region below the red line (T > Te ) shows common scenarios of plasmas produced by shocks. It is worth commenting on the ideal-gas limit of J ; the results in Fig. 3.2 reveal that J , 1. For neutral systems, such as Lennard-Jones systems, J is typically of order unity [161] and is strictly unity in the ideal-gas limit. The ideal-gas limit can be recovered by choosing cii 0 (r) = 0, which yields Scc (k) = 1. However, the plasma case is qualitatively different from neutral systems [191]. Note that Eq. (3.51), which we expect to be accurate in hot plasmas, has no temperature dependence above temperatures at which a plasma is fully ionized. At very high temperatures, J has a constant, 58 (a) (b) (c) laser produced plasmas shock produced plasmas laser produced plasmas shock produced plasmas Figure 3.2: (a) Comparison of the thermodynamic factor J for a H-Ar binary mixture at n = 1022 cm−3 . The subscript “1" denotes H, and “2" denotes Ar. In the top row, x1 = 0.5; in the bottom row, x1 = 0.6. Curves for J computed from Eqs. (3.51) and (3.52) are shown with their corresponding high temperature limiting values as dashed lines. (b) Contour lines show the thermodynamic factor for fully ionized mixtures (ranging from H to Og), calculated using Eq. (3.51). The contours show that J increases with the difference in charge between the species. The value of J is lowest for mixtures where Z1 ≈ Z2 . When x1 > x2 , J increases more quickly with Z1 than with Z2 . (c) Contours show J for the H-Ar mixture of (a) computed from the DH model Eq. (3.50), where T̃ = T/eV and T˜e = Te /eV. Note that J tends to unity as the ion temperature increases which is equivalent to setting β = 0 in Eq. (3.50). The red line along the diagonal shows where T = Te , the situation described by Eq. (3.51). The region above the red line shows a typical scenario for a laser produced plasma. The region below the red line shows typical scenarios for shock produced plasmas. 59 non-unity value unless hZ 2 i = hZi 2 . The lack of an ideal-gas limit can be traced to the fact that screening decreases with increasing temperature because the interaction strength increases with temperature; unlike Lennard-Jones systems, plasmas have explicit temperature-dependent interactions that strengthen both as ionization increases and as screening decreases, yielding a J that tends to a constant, non-unity value at high temperatures. Mathematically, this can be seen by the fact that at very high temperatures J depends on charge ratios that are inherent to the plasma composition. The exception to this rule is the two-temperature plasma case in which there are very cold electrons (βe → ∞) and very hot ions (β → 0), which yields Scc (0) → 1. (The inverse of this limit, βe → ∞, is the BIM case.) Cold electrons and hot ions could occur in a plasma shock wave; however, degeneracy plays a role in that low electron-temperature limit and predictions based on the models discussed so far are inadequate. Thus, we now turn to treating electronic degeneracy and exchange. 3.3.2 Yukawa Screening We generalize the classical result, Eq. (3.50), to dense plasmas in which the electrons can be partially degenerate. We retain the functional form of Eq. (3.42) but allow for a more general form of the electron screening length k e−1 ; we will refer to such a model generically as a “Yukawa" model. As in the previous subsection, we allow for separate electron and ion temperatures. The electronic wavevector in our Yukawa model is given by 1 2 k e−T F = , (3.53) λT2 F The electronic screening contribution now has a degeneracy correction through the Thomas-Fermi screening length λT F , Eq. (3.31), which generalizes Eq. (3.50) to βeTF hZi + βhZ 2 i J TF = , (3.54) βeTF hZi + βhZi 2 where 1 βeTF = r . (3.55)  2 Te2 + 23 EF 60 Importantly, βeTF is finite for Te → 0 (βe → ∞). We can add a finite-temperature exchange- correlation (XC) contribution [12, 104, 207] to obtain the modified Yukawa electronic wavevector 1 2 k e−XC = , (3.56) λT2 F − γ0 where γ0 is defined as ~2 β e θ γ0 ≈ [h(θ) − 2θh0(θ)], (3.57) 8me and we employ the form N (θ) h(θ) = tanh(θ −1 ), (3.58) D(θ) N (θ) = 1 + 2.8343θ 2 − 0.2151θ 3 + 5.2759θ 4, (3.59) D(θ) = 1 + 3.9431θ 2 + 7.9138θ 4, (3.60) where θ = Te /EF . These steps yield the thermodynamic factor βeXC hZi + βhZ 2 i J XC = , (3.61) βeXC hZi + βhZi 2 where our final effective inverse electron temperature is 1 βeXC = r . (3.62)  2 Te2 + 2 3 EF − 4πne e2 γ0 The XC correction has the effect of lowering the effective temperature relative to its over-estimated Thomas-Fermi value. 3.3.3 Kirkwood-Buff Approach All of the above derivations were preformed in k-space. However, approximate forms for the partial structure factors, and therefore J , can be obtained through standard approximate forms for the RDFs. The partial structure factors are related to the RDFs by ∫ √ Sii 0 (k) = δii 0 + ni ni 0 d 3r [gii 0 (r) − 1] eik·r . (3.63) In the long-wavelength limit, Eq. (3.63), reduces to ∫ ∞ √ Sii 0 (0) = δii 0 + ni ni 0 dr 4πr 2 [gii 0 (r) − 1] , (3.64) 0 61 where δii 0 is the Kronecker delta. The integral that appears here is related to the well-known Kirkwood-Buff integrals [183, 208, 209] ∫ ∞ Gii 0 = dr 4πr 2 [gii 0 (r) − 1] , (3.65) 0 which, using Eq. (3.21), yield 1 J= . (3.66) 1 + x2 n1 (G11 + G22 − 2G12 ) With approximate forms for the RDFs, we can construct predictions for J . Approximate RDFs can be constructed in different limits. Because transport coefficients tend to be largest in hot plasmas,We first consider J in the high temperature limit. For example, hot plasmas are well described by Debye-Hückel theory, which, for a BYM, yields the RDFs and Kirkwood-Buff integrals Zi Zi 0 e2 −kt r gii 0 (r) ≈ 1 − β e , (3.67) r Gii 0 = −4π βZi Zi 0 e2 kt−2, (3.68) resulting in βe hZi + βhZ 2 i J DH = . (3.69) βe hZi + βhZi 2 Note that Eq. (3.69) is identical to Eq. (3.50) but has been derived in r-space. Here, kt = q k e2 + k12 + k 22 is the total screening wavevector and k e is given by Eq. (3.43). Note that by using alternative forms of k e [e.g., Eqs. (3.53) or (3.56)] as inputs to kt , we would arrive at Eqs. (3.54) and (3.61) respectively. All results for analytic expressions of the thermodynamic factor, namely Eqs. (3.50), (3.54), and (3.61), have relied on the mean-field approximation, Eq. (3.39). While these results are applicable for hot plasmas, we wish to quantify conditions for which these analytic expression may fail. We begin by comparing estimates of the RDFs via Eq. (3.41) to results from MD simulation. Molecular dynamics simulations of a H-He BYM were carried out for a range of Γ0 with N = 10000 particles and x1 = 0.5, using standard techniques. Figure 3.3 shows the RDFs for MD simulations 62 2.5 22 = 0.14 2.0 1.5 g(r) 22 = 75 1.0 gHH gHeHe 0.5 gHHe NL NL-SC 0.00 1 2 3 4 5 r (Å) Figure 3.3: Hydrogen-helium RDFs generated from a MD simulation (solid lines) for a H-He BYM with N = 10000 particles and n = 1023 cm−3 using standard techniques. Note that Γ22 denotes the Coulomb coupling between the He ions. For the weakly coupled case (Γ22 = 0.14), the RDFs have been vertically displaced by unity for clarity. The NL model accurately reproduces the RDFs for Γ22 = 0.14, but fails at strong coupling (Γ22 = 75) whereas the NL-SC model shows reasonable agreement. corresponding to two values of Γ22 : a weakly coupled case (Γ22 = 0.14), and a strongly coupled case (Γ22 = 75). At Γ22 = 0.14, the NL approximation Eq. (3.41), shown with dashed lines, agrees with the MD results, differing most notably at small r. For the case of strong coupling (Γ22 = 75), the NL approximation fails and a strong-coupling correction is needed. Following [151], such a correction is introduced in the dimensionless total screening wavevector s Õ 3xi Γii k̃tSC = κ 2 + . (3.70) i 1 + 3 (hZi/Zi ) 1/3 Γii Together, Eqs. (3.70) and (3.41) result in a NL model with a strong-coupling correction (NL-SC) for the RDFs. We observe that the RDFs from the NL-SC model, shown with dotted lines for Γ22 = 75, show reasonable agreement with the MD results, revealing that the substantial improvement that Eq. (3.70) provides. The most notable improvement is observed in the “Coulomb hole" region. For MD simulations, finite-size corrections to Eq. (3.65) must be considered [210] yielding ∞ 3x 2 x 3 ∫   GiiFS0 = 2 dr 4πr [gii 0 (r) − 1] 1 − + , (3.71) 0 2 2 63 22 0.0 0.1 4 1.3 7.7 75 1.9 1.4 4 MD NL 1.3 NL-SC TF 1.2 1.1 1.0 0.9 10 3 10 2 10 1 100 101 102 103 0 Figure 3.4: Thermodynamic factor for a H-He BYM. Note that Γ22 denotes the Coulomb coupling between the He ions in the mixture. The details of the MD are the same as in Fig. 3.3. The thermodynamic factor was computed from MD using Eq. (3.71). The grey dashed line denotes J = 1. We note that the mean-field approximation for J , Eq. (3.54), denoted as TF, begins to fails for Γ22 > 2. where x = r/L and L is the cutoff distance. Here, we chose L to be half the side length of the cubic simulation cell. Calculations of J via Eq. (3.71) are displayed in Fig. 3.4. We find that Eqs. (3.54), and (3.41), show excellent agreement with MD at weak coupling with less than a 2% error at Γ0 = 0.01. The models remain accurate until roughly Γ0 = 11 where the error increases to 12% for Eq. (3.54) and 14% for Eq. (3.41). Thus, based on results from Figs. 3.3 and 3.4, we conclude that the mean-field approximation, Eq. (3.39) and corresponding thermodynamic factors, Eqs. (3.50) - (3.52), (3.54), and (3.61), should only be used when Γ22 < 2; for Γ22 > 2, an alternate approach to treating strong coupling, which results in an analytic form for J , is explored in the next section. 3.3.4 Strongly Coupled Plasmas In Secs. 3.3.1 and 3.3.2 we obtain results that include ion and electron screening, electron de- generacy, finite temperature exchange, separate electron and ion temperatures, and strong coupling through the electronic wavevector Eq. (3.70). However, the absence of oscillatory behavior limits 64 their applicability to moderate coupling. In this subsection, based on numerical results from the hypernetted chain approximation for a BYM, we formulate a DCF to capture oscillations in Sii 0 (k) and gii 0 (r). Our DCF model for strongly coupled plasmas is given in real space by an “empty-core” form  r < riic 0,   −βuii 0 (riic 0 ),   c (r) =  ii 0 (3.72)  −βuii 0 (r), r > riic 0,     Zi Zi 0 e2 −ke r uii 0 (r) = e , (3.73) r Γii 0 dii 0 riic 0 = . (3.74) 1 + Γii 0 where dii 0 is the distance of closest approach. To determine dii 0 , one can measure the “Coulomb hole" portion of the RDFs (see Fig. 3.3). However, in order to produce an analytic expression for J that does not rely on computing the RDFs, we approximate dii 0 ≈ aii 0 , where aii 0 = (4πnii 0 /3)−1/3 and   (ni + ni 0 )/2, i = i0,    nii 0 =  (3.75)  n, i , i . 0     Note that the DCF above could use any of the electronic wavevectors defined up to this point; therefore, we denote the electronic wavevector generically as k e . Note that Eq. (3.72) reduces to the mean-field form, Eq. (3.39), in the limit Γii 0 → 0. In Fourier space, we find that 4πZi Zi 0 e2 β cii 0 (k) = − Λii 0 (k), (3.76) k 2 + k e2 where k 2 + k e2 −ke r c 0 sin(kriic 0 ) − kriic 0 cos(kriic 0 ) k e riic 0 sin(kriic 0 ) + kriic 0 cos(kriic 0 )   Λii 0 (k) = e ii + . (3.77) kriic 0 k2 k 2 + k e2 The factor Λii 0 (k) is the strong-coupling correction factor, which has the long-wavelength limit ! k 2r c 2 e ii 0 c Λii 0 (0) = 1 + k e riic 0 + e−ke rii 0 . (3.78) 3 65 The partial structure factors are then √ x1 x2 Z1 Z2 kr2 S12 (k) = − Λ12 (k), (3.79) k 2 + k e2 + Λ11 (k)k 12 + Λ22 (k)k 22 k 2 + k e2 + Λ22 (k)k22 S11 (k) = , (3.80) k 2 + k e2 + Λ11 (k)k 12 + Λ22 (k)k22 k 2 + k e2 + Λ11 (k)k12 S22 (k) = , (3.81) k 2 + k e2 + Λ11 (k)k 12 + Λ22 (k)k22 from which the concentration-concentration structure factor can be obtained. Our final form for the thermodynamic factor that allows for two temperatures, degeneracy and strong coupling is then βeTF hZi + βT (0) J TF-SC = , (3.82) βeTF hZi + βB(0) T (0) = x1 Z12 Λ11 (0) + x2 Z22 Λ22 (0), (3.83) B(0) = x12 Z12 Λ11 (0) + x22 Z22 Λ22 (0) + 2x1 x2 Z1 Z2 Λ12 (0), (3.84) where we have used the electronic wavevector Eq. (3.31). We compare the impact of each screening model on J in Fig. 3.5 for a H-Ar mixture at density n = 1022 cm−3 for x1 = 0.5. In panel (a), screening is approximated via a Debye-Hückel formulation, Eq. (3.50). In (b), the addition of degeneracy with Eq. (3.54) changes the low-temperature behavior of J . In (c), the XC correction Eq. (3.61) once again changes the low-temperature behavior of J . The regions where βeXC < 0 or θ < 0.1 have been omitted as the XC correction either fails (in the case of negative screening) or may be inaccurate (when θ < 0.1 [207]). Additionally, the accuracy of the XC correction is unknown for θ > 12; as a result, we set γ0 = 0 and recover Eq. (3.31). Panel (d) shows the impact of strong coupling; the value of J is lower at lower ion temperatures in this case than in the cases shown in panels (a) and (b). To assess the validity of the analytic expressions for J we have derived, Table 3.2 shows a comparison of different forms of J and results from MD simulation or the hypernetted chain approximation. For the strongly coupled H+ -C4+ BYM, the strong-coupling correction, Eq. (3.82), is accurate to the hypernetted chain results to within 6%. For the moderately coupled H+ -Al6+ BYM, Eq. (3.61) is within 5% of the MD data. 66 (a) (b) (c) (d) XC correction unsuitable Figure 3.5: Comparison of J for a H-Ar plasma at n = 1022 cm−3 for x1 = 0.5. As in Fig. 3.2, T̃ = T/eV and T˜e = Te /eV. (a) The electrons are treated classically with Debye-Hückel screening, Eq. (3.50). (b) Degeneracy is included with Thomas-Fermi screening, Eq. (3.54). (c) An XC correction is added, Eq. (3.61), which fails in the region below the dashed horizontal line. The failure occurs because either θ < 0.1 (as described in Ref. [207]) or βeXC < 0. (d) Strong coupling is included, Eq. (3.82). The red line shows the case in which T = Te ; this case is equivalent to Eq. (3.51). 67 0.6 riic = 0 0 0.4 riic 0 riic, HNC 0 0.2 HNC S12(ka) 0.0 0.2 0.4 0.6 0 2 4 6 8 10 ka Figure 3.6: The mixture partial structure factor S12 (k), Eq. (3.79), of a H+ -C4+ BYM at n = 5 × 1023 cm−3 , where x1 = 0.5 and T = 1.7 eV. The hypernetted chain results, denoted as HNC, were obtained from Ref. [211]. The strong-coupling correction, Eq. (3.79), with distance of closest approach estimated from the RDFs of the HNC results is denoted as riic,HNC0 . The strong-coupling correction using Eq. (3.75), is denoted as riic 0 . When riic 0 = 0, we obtain the mean-field result, Eq. (3.44). Note that the strong-coupling correction accurately approximates k = 0 and predicts oscillations. Additionally, we compare the results of the S12 (k) with a hypernetted chain calculation [211]. In Fig. 3.6, we evaluate Eqs. (3.44) and (3.79) using Eq. (3.31). The distances of closest approach dii 0 are approximated from the corresponding RDFs in Ref. [211]. Specifically, their values are d11 = 0.5, d12 = 0.9, and d22 = 1.25. The prediction for Eq. (3.79) using these values of dii 0 is denoted as riic,HNC 0 in Fig. 3.6. For the entire domain of k, Eq. (3.79) shows reasonable agreement with the hypernetted chain calculations, and excellent agreement is achieved as k → 0. Predictions of the partial structure factor are also computed by using Eq. (3.79) with the distance of closest approach from Eq. (3.75), are denoted as riic 0 . The predictions show reasonable agreement as k → 0. The agreement in this limit increases our confidence in the resulting thermodynamic factor from Eq. (3.79). 3.4 Interdiffusion Models The previous sections have focused on approximations to DCFs and the resulting thermodynamic factors. We now turn our focus to two analytic models for the interdiffusion coefficient, including 68 an analytic expression for the BIM autocorrelation function. The models are then compared with MD data for a H+ -He2+ BIM. 3.4.1 BIM Gaussian Autocorrelation Function We begin by deriving an analytic expression for the interdiffusion coefficient of a BIM. To do this, we rewrite the Green-Kubo result of Eq. (3.16) in terms of a normalized autocorrelation function J(t) as hj(t) · j(0)i J(t) = , (3.85) hj(0) · j(0)i ∫ ∞ J D= dt J(t). (3.86) βhMim1 m2 0 Given the known behaviors of autocorrelation functions at short and long times [148, 212], we propose a Gaussian ansatz [213] for J(t) of the form   J(t) = exp −Ω2 t 2 /2 , (3.87) which satisfies d 2 J(0)/dt 2 = −Ω2 . Here, Ω is the Einstein frequency associated with interdiffusion; Hansen et al. [181] compute this quantity for a BIM and obtain s ω p x1 m12 z2 + x2 m22 z1 Ω= √ , (3.88) 3 m1 m2 which is the familiar relation between the hydrodynamic plasma frequency Eq. (3.19) and the Einstein frequency generalized to a BIM [181]. Statistical mechanics reveals that autocorrelation functions are even in time [148] and decay to zero; these requirements are approximately satisfied with a Gaussian ansatz. For a normalized autocorrelation function, this is a single-parameter model that can be computed analytically for the case of a BIM [181]. Such an ansatz is reasonably accurate for strong coupling, although in principle, a richer ansatz could be used [212] if more parameters could be determined (e.g., fit to MD data). The reduced interdiffusion coefficient in this Gaussian approximation is then √ J 6π DG AF = ∗ q . (3.89) 3hZi Γ0 x1 m1 m2 z2 + x2 m1 m2 z1 2 3 3 69 1.2 H + -He2 + , x1 = 0.5, 0 = 39.79 MD 1.0 GAF 0.8 0.6 J(t) 0.4 0.2 0.0 0.20 5 10 15 20 25 pt Figure 3.7: Autocorrelation function for a H+ -He2+ BIM versus time. The system conditions are the same as described in Table 3.2, and we compute Ω/ω p = 0.87. The GAF approximates the early-time decay of the autocorrelation function calculated from MD data. We will refer to the above model as the Gaussian autocorrelation function (GAF) model. Taking the GAF model together with the thermodynamic factors from Sec. 3.3, we have a complete, albeit approximate, analytical model of interdiffusion. We compare an autocorrelation function calculated using the Gaussian ansatz, Eq. (3.87), to one computed from MD simulation data of H+ -He2+ BIM in Fig. 3.7. We see that the early-time decay of J(t) computed using the GAF is comparable to that found with a direct computation from MD data. The GAF model, Eq. (3.89), is most accurate when the decay is roughly exponential, as is the case for strongly coupled plasmas. The GAF model relies on the cancellation of oscillations in the integral of the autocorrelation function to generate an accurate value for the interdiffusion coefficient. Such a simple result arises only for a BIM. Below, we will explore the accuracy of the GAF model compared with a transport theory that has been derived from an effective interaction potential in a Boltzmann kinetic theory framework [151]. 3.4.2 Stanton-Murillo Transport Model The analytic model, Eq. (3.89), by definition, includes no screening effects. The Stanton- Murillo transport (SMT) model, however, includes screening effects in the effective interaction potential that is used to compute the collision integrals numerically. From the SMT model [151], 70 mixture x1 Γ0 θ J BIM J DH J TF J XC J DH-SC J TF-SC J H+ -He2+ (BIM) 0.5 40 0.5 1.11 1.03a H+ -C4+ (BYM) 0.5 11 0.1 1.26 1.35 0.99 1.23 1.16b H+ -Al6+ (BYM) 0.3 1.5 0.3 1.21 1.24 1.25 1.15 1.21 1.31a Table 3.2: Comparison of the thermodynamic factor J from different models with those from MD (denoted with superscript “a”) and the hypernetted chain approximation (denoted with superscript “b”) [211] results. To compute J for cases a and b, Eq. (3.20) was employed. The MD simulations were carried out for N = 2000 particle using standard techniques. After computing the partial structure factors, an extrapolation to k = 0 was implemented. J DH-SC corresponds to the strong- coupling correction, Eq. (3.82) using (3.43). we have the following expression for the interdiffusion in a BYM: 3T 5/2 DSMT = , (3.90) 16 2π µ12 nZ12 Z22 e4 K11 (g) p where µ12 = M1 M2 /(M1 + M2 ), and Knm (g) is a fit to the collision integral for the effective coupling g. The expression for Knm (g) is given in Appendix C of [151]. For a BIM, the effective coupling g is " # 1/2 2 Õ 3xi Γii g = Γ12 . (3.91) i=1 1 + 3(xi /Zi ni )1/3 Γii We note that in Ref. [151], the numerator of the summation term was written as 3xi−1 Γii , when it should be 3xi Γii as written here. In Ref. [181], the interdiffusion coefficient was computed for H+ -He2+ mixtures for three cross-species coupling parameters Γ12 = 0.8, 8, 80. We compare the interdiffusion coefficient calculated from this data with those computed using the GAF and SMT models, and the results are shown in Fig. 3.8. We see that the simple form of the GAF model estimates the interdiffusion coefficient for moderate and strong coupling reasonably well, with errors similar to those of the SMT model. However, for weak coupling, the GAF model begins to fail, while the SMT model remains accurate. 3.5 Conclusions and Outlook In summary, we have explored interdiffusion in plasmas with a focus on the Darken relation that thermodynamic factor. The Darken relation simplifies the autocorrelation function necessary for computing interdiffusion by neglecting cross-species current correlations. By comparing the 71 103 (a) H + -He2 + , x1 = 0.5 (b) H + -He2 + , x1 = 0.632 (c) H + -D + , x1 = 0.5 (d) H + -C6 + , x1 = 0.991 102 101 D* 100 10 1 SMT 10 2 GAF MD 10 3 10 1 100 101 102 10 1 100 101 102 10 1 100 101 102 10 1 100 101 102 0 0 0 0 Figure 3.8: Normalized interdiffusion coefficient D∗ of various BIMs. The MD data in (a)–(d) was collected from Refs. [181, 214]. The GAF model accurately reproduces the MD data at moderate and strong coupling but fails at weak coupling. Darken approximation to the full result, we found that the Darken relation results in smaller statistical errors, but does not converge to the full result. To explore the possible advantages of the Darken relation, we set up a series of MD simulations of varying lengths. For short simulations, the full result had larger uncertainty but a mean value that was comparable to the Darken approximation; thus, showing that the Darken relation was not advantageous. At increasingly longer simulations, we identify where the uncertainty bands no longer overlapped, highlighting where the Darken relation fails. Further studies for a wider range of systems is warranted. Next, we turned to the development of rapidly computable analytic expressions for the thermo- dynamic factor to model plasmas in disparate regimes. We provide the derivation of Eqs. (3.52) and (3.51) revealing the physical regime in which they apply. Furthermore, we have generalized Eq. (3.51) to allow for separate electron and ion temperatures, allowing for a description of laser or shock produced plasmas. We account for electron degeneracy with Thomas-Fermi screening and finite temperature exchange. We extended these results to strong coupling in two ways: by including a modified screening length and through an empty-core DCF. Comparisons to MD and hypernetted chain calculations give confidence of their use but a more comprehensive study should be performed. We formulated a complete model for the interdiffusion of a BIM using a GAF, which is based on the short-time expansion of the interdiffusion current correlation function. A comparison of the GAF model with MD data shows reasonable accuracy in the intermediate to strong coupling 72 regime. However, at weak coupling, the GAF model begins to fail and but the SMT model remains accurate. Combining the GAF and SMT models, gives a reasonable prediction for interdiffusion across the entire coupling regime. There are many opportunities to extend the results presented here. First, the inputs to the analytic expressions derived here could be improved upon with a more robust finite-temperature exchange-correlation potential. Furthermore, numerical calculations of the hypernetted chain equations allows one to calculate quantities of interest (e.g., gii 0 (r), Sii 0 (k), J , etc.) for strongly coupled systems. Such calculations have a slightly increased computational cost relative to the results here but much less than MD.Additionally, we assumed that the pair-interaction potentials were Coulombic. That assumption may not apply to warm dense matter and a pair-interaction potential with gradient corrections [104] or one constructed from N-body MD simulations with force matching [87] may be needed. 73 CHAPTER 4 MULTI-FIDELITY REGRESSION FOR PLASMA PROPERTIES DATA 4.1 Introduction The types of plasmas that were studied in Chapters 2 and 3 were strongly coupled plasmas that spanned both the non-degenerate and degenerate regimes. In this chapter1, we expand the scope of our study and parameter regime to include weakly coupled plasmas. Our aim is to make accurate predictions of ionic transport property data across two disparate physical regimes: the strongly coupled and weakly coupled regimes. Having accurate predictions of transport coefficients and equations of state across these regimes is necessary for closures of macroscopic simulations of plasmas at large time and length scales. Due to the computational cost associated with generating microscopic data, closure data is typically precomputed and cast in the form of “look-up" tables which are used as inputs to macroscopic simulations. Because it is impossible to generate a look-up table with infinitely small resolution, interpolation between values is needed. These interpolation methods should be robust by providing uncertainty associated with the interpolation as well as numerically efficient for high-dimensional data. As mentioned in Chapter 2, the fidelity of numerical results rely directly on these look-up tables and therefore it is of interest to 1) quantify uncertainty associated with using models to generate them, and 2) create tables that are wide-ranging that agree with known results in limiting regimes (e.g., the ideal-gas law). The plasma conditions we study here are displayed in Figure 4.1 which include four elements pertinent to HED experiments across a wide temperature regime. Because out dataset is multi- modal and uses a low fidelity model that is rapidly computable anywhere in this range of conditions, we have continuous lines throughout the Γ−θ plane in Figure 4.1(a). We also note that Figure 4.1(b) highlights that our study spans well into the highly-collisional and free-flight regimes. The low-fidelity (LF) models used in this work provide a computationally efficient, wide- 1 The content described in this chapter was reproduced from Lucas J. Stanek, Shaunak D. Bopardikar, and Michael S. Murillo, “Multifidelity regression of sparse plasma transport data available in disparate physical regimes", Physical Review E 104, 065303 (2021) https://doi.org/10.1103/PhysRevE.104.065303. This article was published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license and has been modified to address the requirements of this dissertation; see Ref. [215] for the full published article. 74 Figure 4.1: (a) electron degeneracy parameter versus coulomb coupling parameter and (b) Knudsen number versus temperature for all plasma conditions of this chapter. Note that the Knudsen number ranges from the highly collisional regime to the free-flight regime. A goal of this chapter is to use high-fidelity data that is local to the highly-collisional regime and free-flight regime to make predictions across the transition and kinetic regimes. ranging prediction of transport coefficients that span disparate regimes. However the LF models lacks accuracy in certain regimes where high-fidelity (HF) data are needed. The generation HF data requires substantial resources that limit the volume of data that can be generated. Moreover, the size and scope of datasets are constrained by the experimental accessibility of physical regimes and by the applicability and efficiency of computational models. These limitations can be addressed by combining data from several sources to form a dataset that contains multiple, separated point clouds that can be “interpolated.” For example, the equation of state can be measured in one regime with a laser-heated diamond anvil cell [216] and computed in another regime with accurate electronic-structure methods [217]. Or, one may combine experimental data obtained along a shock Hugoniot with computational data available only at very low temperatures. Computational models for equations of state, atomic properties, and charged-particle transport [12, 87, 180, 218] can also be combined to create a larger dataset. Combining data sources in this way creates two challenges. First, predictions will be based on data in potentially very different physical regimes. Second, while it is natural to consider adding LF data, which can be generated cheaply, to datasets to cover a parameter space more uniformly, it 75 is not clear how to exploit such LF data in making predictions. Machine-learning (ML) methods offer promising alternative frameworks for interpolating phys- ical data [115, 219, 220]. ML treats the interpolation problem as regression in a high-dimensional space using non-traditional techniques such as neural networks. Gaussian-process regression [221] (GPR) is a nonparametric ML technique that interpolates data in multiple dimensions; importantly, GPR provides an uncertainty estimate that can be used to suggest where new data points should be acquired. Here, we will explore GPR as an approach for interpolating physical data. In particular, we will examine the situation in which there are islands of HF data in parameter space, possibly from different sources, and we will fill the space between these islands with easier-to-compute LF data. Such an approach utilizes multi-fidelity (MF) extensions [222] of GPR. Here, we use GPR to refer to the methodology described in [221], and MF-GPR to refer to its MF extensions [222–225]. The generality of MF-GPR methods enables their use in many disciplines and applications [226–233]. The original MF-GPR framework has been improved to reduce the risk of overfitting during the training procedure [225], to include nonlinear relationships between LF and HF models [223, 224], and to address concerns that arise with diverse data structures and dataset selection [234, 235]. This chapter is organized as follows. As described in the following section, we will illustrate our ML ideas using the example of ionic transport coefficients. The methods we used to generate our dataset of ionic transport coefficients are discussed in Sec. 4.2.1. In Sec. 4.2.2, we compare single- fidelity regression methods and highlight the benefit of GPR over simple cubic spline regression. We then transition to MF regression and discuss the formulation of MF-GPR, as introduced by Kennedy and O’Hagan [222], in Sec. 4.2.3. Using toy examples, we show when MF-GPR adds value over single-fidelity GPR; we also show where improvements to this formulation are needed. We conclude Sec. 4.2.3 by reporting a table of computation times and regression errors for MF-GPR and single-fidelity GPR to assess the cost-benefit trade-off for these methods. Sec. 4.3 illustrates an approach for choosing an LF model that is the most appropriate for an 76 MF-GPR setting and examines how this choice impacts the resulting MF-GPR fit. A natural choice for LF and HF models are those with the same output quantity (e.g., both predict the viscosity of a system). However, the outputs of both models need not be the same quantities. We explore the use of models in MF-GPR that have different output quantities, as well as different levels of computational complexity. In Sec. 4.4.1, we compare regression errors resulting from single-fidelity GPR and MF-GPR analyses of sparse, disparate plasma transport-coefficient datasets. We find that while MF-GPR may result in modestly smaller errors compared to single-fidelity GPR, the uncertainty of the MF-GPR prediction is consistently much smaller. Finally, in Sec. 4.4.2, we compare three approaches for sampling HF data to reduce the MF-GPR regression error. For a fixed number of HF data points, a simple approach we explored outperforms sampling from a uniform grid. We offer conclusions and discuss potential areas for future work in Sec. 4.5. 4.2 Dataset and Regression Methods In this section, we discuss our dataset and review the ML approaches we will employ in Secs. 4.3 and 4.4. We begin in Sec. 4.2.1 by describing plasma transport-coefficient data and the fidelities of several commonly used models based on the approximations they employ. In Secs. 4.2.2 and 4.2.3, we describe the GPR methodology, including standard, single-fidelity GPR and its MF generalization. Our goals in Secs. 4.2.2 and 4.2.3 are to answer the following questions: what value does GPR add compared to simpler regression methods? And, how does including data from multiple levels of fidelity impact a prediction? 4.2.1 Ionic Transport-Coefficient Dataset For our study, we chose to explore MF-GPR in the context of plasma ionic transport coefficients because plasmas span many orders of magnitude in density, temperature and nuclear charge. Plasmas can include many species, which makes it difficult to use a single (computational or experimental) method to make accurate predictions. Computational methods that are typically used can be divided into LF and HF methods by examining the underlying assumptions of the 77 models. Moreover, we can usually identify a limited parameter regime in which each model is HF. These delineations occur because the theoretical models that underpin the computational methods are known to have high accuracy only in certain limits (e.g., asymptotically at high temperature); methods that are not asymptotically accurate in a parameter regime are designated as LF there. The limiting regimes typically depend on multiple dimensionless parameters (e.g., the Coulomb coupling parameter and the degeneracy parameter) that rely on some combination of nuclear charge, density, and temperature of the system. We will use just the temperature of the system to specify the limiting regimes since models developed at extremes of temperature tend to have very different assumptions. Data in the low-temperature regime, loosely defined here as T < O(101 ) eV, and in a high- temperature regime, defined here as T > O(103 ) eV, will be generated using appropriate LF and HF models. For the self-diffusion transport coefficient D, we will use the following HF models to generate data. At low temperatures, the HF data are obtained from density functional theory molecular dynamics (DFT-MD) simulations [83, 236–238], which accurately calculate the electronic structure on-the-fly. At high temperatures, the Stanton-Murillo transport (SMT) model [151], which uses numerically computed cross-sections and an effective interaction potential, is employed. The LF model used across the entire temperature range is given by Hansen, McDonald, and Pollock (HMP) for a one-component plasma (OCP) [239]. Similarly, for viscosity η, we use one HF model at low temperatures and a different HF model at high temperatures. Once again, the HF data at low temperatures are obtained from DFT-MD simulations. We employ the Yukawa viscosity model (YVM) [140], which is based on a quasi- universal form fit to MD data, as our HF model at high temperatures. Our LF model is derived from a correspondence between an OCP system and a Yukawa system. The correspondence is obtained from the Gibbs-Bogolyubov inequality [137]; this model will be referred to as the YGBI model. The HF and LF models for the self-diffusion and viscosity coefficients in each temperature range are summarized in Table 4.1. These models are used in analyses presented in Sec. 4.4.1. 78 Coeff. T (eV) HF LF D T < O(101 ) DFT-MD [83, 236–238] HMP [239] O(101 ) < T < O(103 ) – HMP [239] T > O(10 )3 SMT [151] HMP [239] η T < O(101 ) DFT-MD [83, 236–238] YGBI [137] O(10 ) < T < O(10 ) 1 3 – YGBI [137] T > O(103 ) YVM [140] YGBI [137] Table 4.1: HF and LF models for the self-diffusion and viscosity transport coefficients in each temperature regime. Each LF model is used across the entire temperature range. 4.2.2 Single-Fidelity Regression To provide a baseline to which results of MF-GPR can be compared in later sections, we first consider approaches that require only one level of data fidelity, i.e., single-fidelity approaches. We consider cubic-spline regression and GPR. Cubic-spline regression is a parametric regression method that aims to determine the optimal parameters that define a cubic-spline fit to data. In contrast, GPR is a nonparametric regression approach that determines the optimal function that is fit to data. We begin with a brief overview of GPR that will provide a framework for understanding its MF generalization. We introduce GPR with a discussion of prior and posterior distributions. Before observing the data, we have some prior beliefs about functions that are suitable. These functions are drawn from a prior distribution: a distribution of random functions that are consistent with our prior beliefs about the data. An example of a prior distribution is one in which the distribution of functions have zero mean at each input point and vary smoothly over the entire input space. For plasma transport data, we could impose constraints on our prior distribution of functions to enforce nonnegativity and such that the functions reflect the known behaviors of different transport coefficients (e.g., increasing with temperature). After constructing a prior distribution, a posterior distribution is created by using available data to constrain the random functions by ensuring that they pass through the observed data points. As we will see, the mean and the covariance matrix of a posterior distribution are the prediction and uncertainty estimates of GPR. 79 Defining the prior and posterior distributions for GPR requires a kernel function that defines a measure of similarity among the input variables of a dataset. The kernel function determines the representation of the functions from the prior and posterior distributions (e.g., smoothness, periodicity, etc.). A common choice of kernel function, that we will use here, is the squared- exponential kernel   1 k(xi, x j ; σ , `) = σ exp − 2 k xi − x j k , 2 2 2 (4.1) 2` where for d-dimensional data, we have m points xi ∈ Rd and n points x j ∈ Rd . Evaluating the kernel k(xi, x j ; σ 2, `) gives the i jth entry of the kernel matrix (or covariance matrix) K ∈ Rm×n . The hyperparameters of Eq. (4.1) are the variance σ 2 and the length scale `; they will be compactly denoted as the set θ ∈ {σ 2, `}. These hyperparameters reveal the strength and extent of correlations in the data. As we will see, the values of the hyperparameters are particularly useful for quantifying the quality of MF-GPR methods. A single-fidelity GPR problem is posed as follows: given a set of n training points in d dimensions, represented by the columns of a matrix XSF ∈ Rd×n and the corresponding (scalar) output values y ∈ Rn of the unknown function at each training point, predict the value of the unknown function at a set of m test points X∗ ∈ Rd×m . As shown in Ref. [221], the posterior distribution of the unknown function using GPR at the new set of data points X∗ is a multivariate Gaussian with mean µ ∗ and covariance Σ∗ given by µ ∗ (X∗ ) = K(X∗, XSF ; θ)K(XSF, XSF ; θ)−1 y, (4.2) Σ∗ (X∗ ) = K(X∗, X∗ ; θ) − K(X∗, XSF ; θ)K(XSF, XSF ; θ)−1 K(XSF, X∗ ; θ), (4.3) where the hyperparameters θ of the kernel function are determined by optimizing the log-likelihood function L(y, XSF, θ), as discussed in Ref. [221]. The function L measures the likelihood that the observations are given by the values y at training locations XSF for a given value of θ. We now examine a simple example of GPR and compare with a cubic spline interpolation. Viscosity data were generated using the YVM for the element C at ni = 5.01 × 1022 cm−3 , and fits to these data using GPR [240] and cubic-spline regression [241] are shown in Fig. 4.2. The 80 Figure 4.2: Comparison of GPR and cubic-spline regression for a single-fidelity viscosity dataset using the YVM for the element C at ni = 5.01 × 1022 cm−3 . The training points (black diamonds) were fit using both GPR (blue line) and a cubic spline (grey dashed line). The shaded bands show a 95% confidence interval around the GPR fit. Locations of future HF training points are suggested by the confidence band. GPR fit, denoted as “GPR," corresponds to µ ∗ (X∗ ) from Eq. (4.2); the shaded bands around µ ∗ (X∗ ) correspond to a 95% confidence interval and are computed from Eq. (4.3). The fit generated using cubic-spline regression on the same dataset is denoted as “cubic spline." For all GPR fits, the data were first scaled to unit variance and zero mean. The hyperparameter optimization routine was carried out using the limited-memory quasi-Newton algorithm [242] with 15 random restarts, and a measurement noise with a variance of 10−6 was added to ensure that the kernel matrix K(XSF, XSF ; θ) for computing the posterior distribution would be guaranteed to be positive-definite (and therefore, invertible) during fitting. Both the cubic-spline regression and GPR methods produced accurate fits, as shown by comparison to the underlying true solution, which is denoted with a black line in Fig. 4.2 and labeled “exact." A key difference between cubic splines and GPR is that the GPR method provides a confidence interval (shaded bands) around the GPR prediction – suggesting where additional data are needed to improve the prediction. 81 4.2.3 Multi-Fidelity Gaussian-Process Regression We now turn to the case where there are two sources of data, one LF and one HF. The outputs of the LF model are denoted as y LF ∈ RNLF and are evaluated at XLF ∈ Rd×NLF . Similarly, the outputs from the HF model are denoted as yHF ∈ RNH F and are evaluated at XHF ∈ Rd×NH F . Here, the numbers of LF and HF data points are denoted as NLF and NHF , respectively. To understand how the LF data can be used in HF predictions with an MF method, consider this simple procedure with three steps. First, in step (a), we combine the LF and HF data into a single dataset with greater coverage than the HF data alone offer. In step (b), we use LF data to influence HF predictions by quantifying correlations between the LF and HF datasets with a correlation hyperparameter, ρ. Finally, step (c) of the procedure imposes a constraint that a prediction at a HF data point ignores the LF data. Each part of the above procedure is addressed by the original MF-GPR formulation proposed by Kennedy and O’Hagan [222], which begins by assuming that there is a linear mapping between fidelities that is described by the autoregressive model fHF (x) = ρ fLF (x) + δHF (x). (4.4) The function δHF (x) is to be viewed as the error or bias between the HF data and a scaled value of the LF data, where the correlation hyperparameter ρ is the scaling term. Notice that if the LF and HF data are uncorrelated, i.e., ρ = 0, then δHF = fHF . The key idea in this approach is to use the LF and HF data to learn the parameters governing the unknown functions fLF and δHF and the hyperparameter ρ to be able to predict the value of fHF at a test point x. The functions fLF and δHF are typically assumed to be realizations of independent Gaussian processes with zero mean and a kernel matrix K. This means that on a test set X∗ , fLF (X∗ ), and δHF (X∗ ) are independent Gaussian random variables that are normally distributed as per fLF (X∗ ) ∼ N [0, K(X∗, X∗ ; θ LF )], (4.5) δHF (X∗ ) ∼ N [0, K(X∗, X∗ ; θ HF )], (4.6) 82 where θ LF and θ HF denote the hyperparameters for the LF and HF models, respectively. The notation N (0, Σ) denotes a multivariate Gaussian random variable with mean 0 and covariance Σ. Because fLF and δHF are independent, it follows that 2 fHF (X∗ ) ∼ N [0, ρ2 K(X∗, X∗ ; θ LF ) + K(X∗, X∗ ; θ HF )]. (4.7) For brevity, we denote K11 (X, X 0) ≡ K(X, X 0; θ LF ), (4.8) K12 (X, X 0) ≡ ρK(X, X 0; θ LF ), (4.9) K21 (X, X 0) ≡ K12 (X, X 0), (4.10) K22 (X, X 0) ≡ ρ2 K(X, X 0; θ LF ) + K(X, X 0; θ HF ). (4.11) Equations (4.5), (4.6), and (4.7) can be jointly written as [222, 223] 0  K11 (XLF, XLF ) K12 (XLF, XHF ) K12 (XLF, X∗ )           fLF (XLF )               0 , K (X , X ) K (X , X ) K (X , X ) .          f (X ) ∼ N (4.12)  HF HF      21 HF LF 22 HF HF 22 HF ∗            fHF (X∗ )   0  K21 (X∗, XLF ) K22 (X∗, XHF ) K22 (X∗, X∗ )                  The form of Eq. (4.12) reveals how the LF and HF data are combined (i.e., through K12 and K21 ), completing step (a). Note that when the hyperparameter ρ, which couples the LF and HF models, is equal to zero, Eq. (4.12) reduces to two decoupled Gaussian processes. This means that when the LF and HF models are uncorrelated, the LF data will not influence the HF regression, resulting in one single-fidelity GPR at each fidelity level. Following the procedure for determining optimal hyperparameters θ for a kernel function, the hyperparameter ρ is also determined by optimizing a log-likelihood function, as discussed in Sec. 2.4 of Ref. [222]. As a result of this optimization procedure, if ρ turns out to have a large value, then there is substantial correlation between the LF and HF models. Otherwise, the LF and HF models are uncorrelated. Thus, the correlation hyperparameter ρ determined from the MF dataset 2 Recall that for two independent normally distributed random variables A ∼ N (µ A, σA2 ) and B ∼ N (µ B, σB2 ), then C = A + B ∼ N (µ A + µ A, σA2 + σB2 ). Also, for some constant α, a random variable D ∼ αN (µ D, σD2 ) = N (µ , α 2 σ 2 ) D D 83 directly quantifies the influence of the LF data on the HF fit, completing part (b) of the procedure mentioned above. We have shown how data from LF and HF models can be combined into a single MF dataset and how the degree of influence of LF data on fits to HF data can be quantified using the correlation hyperparameter ρ. However, we still need to show how to produce a fit to HF data using Eq. (4.12), while also completing step (c) of the procedure mentioned above. By conditioning the joint Gaussian prior distribution, Eq. (4.12), the predictive mean and the covariance matrix are obtained from the Gaussian posterior distribution f∗,HF |X∗, XLF, XHF, y ∼ N [K∗ K−1 y, K22 (X∗, X∗ ) − K∗ K−1 K∗T ], (4.13) where f∗,HF denotes the posterior distribution of the HF data, and    y LF  y≡  , (4.14)   y   HF    h i K∗ ≡ K21 (X∗, XLF ) K22 (X∗, XHF ) , (4.15)    K11 (XLF, XLF ) K12 (XLF, XHF )  K≡  . (4.16)   K (X , X ) K (X , X )  21 HF LF 22 HF HF    We note that the hyperparameters θ LF and θ HF of the kernels and ρ are all determined simultane- ouslyby optimizing the log-likelihood function, as discussed in Refs. [222–224]. From Eq. (4.13), the MF predictive mean and covariance for the HF data are µ ∗,HF (X∗ ) = K∗ K−1 y, (4.17) Σ∗,HF (X∗ ) = K22 (X∗, X∗ ) − K∗ K−1 K∗T . (4.18) Note that when X∗ = XHF , we have µ ∗,HF (XHF ) = yHF 3, which guarantees that the regression will pass through the HF data. This satisfies the constraint imposed in step (c) of the procedure and is due to the independence assumption of fLF and δHF , as discussed in Ref. [243]. 3 This can be most easily seen by using a single test point x∗ = x H F and single training points x LF and x H F 84 To highlight how the MF-GPR approach given by Eq. (4.4), which we denote as “linear MF- GPR," may add value over single-fidelity GPR, we consider the pedagogical case where the LF and HF models have the form y LF (x) = sin(2πx), (4.19) 1 yHF (x) = sin(2πx), (4.20) 3 for x ∈ [0, 4]. Note that the LF and HF models are linearly related by the factor of 1/3 in Eq. (4.20). Predictions from single-fidelity GPR and linear MF-GPR are shown in Fig. 4.3, with NHF = 6 and NLF = 22. For all MF-GPR and GPR fits, the data were first scaled to unit variance and mean zero. The hyperparameter optimization routine was carried out using the limited-memory quasi-Newton algorithm for 15 random restarts, and a measurement noise with a variance of 10−6 was added to each kernel matrix to ensure a positive-definite matrix during fitting (see Ref. [244] for more information on the numerical implementation used here). In Fig. 4.3, the linear MF-GPR fit, denoted by a purple solid line, corresponds to µ ∗,HF (X∗ ) from Eq. (4.17); the confidence bands around µ ∗,HF (X∗ ) were computed from Eq. (4.18) and are approximately the width of the thickness of the purple line. The GPR fit, denoted by a blue solid line, corresponds to µ ∗ (X∗ ) from Eq. (4.2), and the shaded confidence bands around µ ∗ (X∗ ) are computed from Eq. (4.3). We see that inclusion of the LF data leads to a more accurate prediction, as linear MF-GPR recovers the exact HF solution. The GPR result, which is fit to only the HF data, is unable to recover the HF true solution. In addition, the 95% uncertainty band reported in Fig. 4.3 around the fit is much narrower with linear MF-GPR than with GPR, and the agreement of the linear MF-GPR fit with the HF true solution persists even beyond the last HF data point. It is important to note that all regression methods based on GPR will generate a fit that will regress to the mean of the data when the distance between a new test point and an HF training point is greater than the length-scale of the kernel(s). This particular example can also be viewed through an information-theoretic lens. Observe that the LF and HF models have the same period of 1 s and therefore, according to the Nyquist- Shannon sampling theorem [245, 246], the sampling period must be less than 0.5 s to reconstruct 85 Figure 4.3: Comparison of linear MF-GPR and single-fidelity GPR for a linear mapping between fidelities. The shaded bands represent a 95% confidence interval around a fit. The single-fidelity GPR result is shown as a blue line; single-fidelity GPR is used to fit only the HF data and does not recover the exact HF solution. The linear MF-GPR result is shown in purple; linear MF-GPR accurately predicts the exact HF solution by using the LF data in addition to the HF data, and this result overlaps the exact HF solution. The confidence interval for the linear MF-GPR fit is approximately the width of the thickness of the purple line. the HF model with sufficient accuracy. Note that the HF data by themselves do not satisfy the Nyquist-Shannon sampling rate. Thus, a GPR fit to the given HF data will be unable to recover the exact HF solution. If the LF model is sampled sufficiently to satisfy the Nyquist-Shannon sampling theorem, then it allows the linear MF-GPR model to recover the exact HF solution. If the LF model is not sampled sufficiently or if the LF model has a different frequency than the HF model, then the LF model is uncorrelated with the HF and, therefore, does not add any new information to the MF-GPR. Lastly, we note that the LF model introduces bias in the resulting MF regression, and the MF-GPR fit is dependent on the choice of LF model; we compare various choices of LF models and their impact on MF-GPR in Sec. 4.3. Fig. 4.3 illustrates how the autoregressive model, Eq. (4.4), results in more accurate fits to HF data when the LF and HF models are related linearly. However, in many cases, the LF and HF 86 models may be related nonlinearly, and schemes beyond the original MF-GPR approach [222] are needed. In recent years, there have been many improvements to the original MF-GPR approach that explore more efficient numerical schemes [247], transform input data to more accurately predict discontinuities in HF data [223], have the ability to learn a nonlinear mapping between LF and HF models [224], and more accurately propagate uncertainty between fidelity levels [225]. The approach proposed in [224] goes beyond the linear autoregressive scheme, Eq. (4.4), by allowing for a spatially dependent nonlinear mapping between fidelities; we denote this mapping as z(·). Following [224], the modified autoregressive equation that includes this mapping is fHF (x) = z[x, fLF (x)] + δHF (x), (4.21) where z(·) is sampled from of a Gaussian process. Note that z[x, fLF (x)] is now a Gaussian process of a Gaussian process and is referred to as a “deep GP" [248, 249]. While the form of Eq. (4.21) has been shown to provide improvements over simpler models [248], computing the mean and covariance of the posterior distribution corresponding to Eq. (4.21) is often computationally intractable [249]. To address this intractability, the Gaussian-process prior fLF (x) is often replaced with the corresponding posterior distribution f∗,LF (x) [247], resulting in a recursive multi-fidelity model (i.e., performing GPR at each fidelity level separately and then propagating the results to each successive level of fidelity). Replacing fLF (x) with f∗,LF (x) in Eq. (4.21) and using the independence assumption of z[ fLF (x), x] and δHF (x) results in a compact recursive multi-fidelity formulation [224] fHF (x) = g[x, f∗,LF (x)], (4.22) where the prior distribution g includes dependencies of both x and f∗,LF (x). It is shown in Ref. [224] that this recursive multi-fidelity model Eq. (4.22) can be modeled by using a kernel of the form k g (xi, x j ) = k ρ (xi, x j ; θ ρ ) · k f [ f∗,LF (xi ), f∗,LF (x j ); θ f ] + k δ (xi, x j ; θ δ ). (4.23) In contrast with the linear autoregressive model Eq. (4.4), the kernel k ρ is now a spatially-dependent scaling factor responsible for measuring the correlations between the LF and HF models, k f 87 measures the correlations of the outputs of the GPR performed on the LF data, and k δ accounts for the bias between the LF and HF data; in this work, each term in Eq. (4.23) is represented by a kernel of the form in Eq. (4.1). The set of hyperparameters (variance and length scale) for each kernel is denoted by θ ρ, θ f , and θ δ , respectively. Importantly, unlike the linear autoregressive formulation Eq. (4.4) where all hyperparameters at all fidelity levels are trained simultaneously, the hyperparameters at each fidelity level using the recursive formulation Eq. (4.22) are trained separately. This aspect greatly reduces computation costs associated with hyperparameter estimation. When the correlations between the LF and HF data are small, the product k ρ k f will be close to zero, and the MF-GPR fit approximately recovers the GPR fit to the HF data. Recall that this was also the case for the correlation hyperparameter ρ in Eq. (4.4). The product k ρ k f in Eq. (4.23) is plotted in Sec. 4.3 to reveal the effectiveness of different choices of LF models. Next, we turn to the three steps for making an MF prediction using Eq. (4.22) with kernel Eq. (4.23). These are discussed in detail in Ref. [224]; for completeness, we summarize them here. Step 1 involves performing GPR on the lowest-fidelity data. This includes optimizing the kernel hyperparameters using the LF data. Step 2 takes as input the trained GPR model from Step 1, together with the HF data, to construct the posterior distribution according to the kernel in Eq. (4.23) (see Eq. (2.14) of Ref. [224]). The last step, Step 3, calculates the predictive mean and covariance by sampling the posterior distribution using numerical integration techniques (e.g., Monte Carlo [224, 225]). Numerical integration is necessary because unlike the prior distributions of single-fidelity GPR and linear MF-GPR, the prior distribution in Eq. (4.22) ,ay not be Gaussian. As a result, we will be unable to express its posterior distribution as a Gaussian. More details of the MF-GPR approach used in this work and its numerical implementation can be found in Refs. [224, 225, 244]. Recall that the LF and HF models given by Eqs. (4.19) and (4.20) are linearly related, with the quantity in Eq. (4.20) equal to the quantity in Eq. (4.19) multiplied by a coefficient of 1/3. To highlight the limitations of the linear MF-GPR approach given by Eq. (4.4), we now consider LF 88 and HF models of the form y LF (x) = sin(8πx), (4.24) yHF (x) = x sin(8πx), (4.25) for x ∈ [0, 1]. Note that the coefficient by which Eq. (4.24) is multiplied to get Eq. (4.25) has been changed from 1/3 to x. As a result of this mapping, we expect that predictions made using Eq. (4.17) will be of poor quality. This expectation is verified in Fig. 4.4, which shows a comparison between predictions from Eqs. (4.4) and (4.22), with NHF = 8 and NLF = 30. We find that MF-GPR not only exhibits excellent agreement with the exact HF solution but also has far smaller confidence bands than those obtained with the linear MF-GPR model. Fig. 4.4 illustrates the ability of MF-GPR to produce accurate results with limited HF data by incorporating additional data from an LF model that is not linearly related to the HF model. It would be undesirable to restrict MF-GPR approaches to plasma transport-coefficient data to linear relationships alone, as such data are known or derived to be accurate in certain physical regimes that need not be related linearly. The plasma transport coefficients we are considering illustrate this point; they are obtained using a variety of methods (recall Sec. 4.2.1) that have no simple, prescribed relationship to each other. Thus, we will use the nonlinear formulation of MF-GPR, Eq. (4.22), throughout the reminder of this work, referring to it simply as “MF-GPR." 4.2.4 Error Calculations and Computation Cost We have shown the benefit of MF regression over single-fidelity techniques by considering toy examples. However, the computational cost of MF-GPR over single-fidelity GPR can not be disregarded. Thus, we would like to determine the cost-benefit trade-off for using MF-GPR over single-fidelity GPR. To begin, we define an error metric to measure the regression error between the HF test set and MF-GPR/GPR predictions. The metric we use is the root-mean-square error (RMSE) v u t Ntest 1 Õ RMSE = k yi,true − yi,pred k 2, (4.26) Ntest i=1 89 Figure 4.4: Comparison of two MF-GPR approaches. One MF-GPR approach assumes a linear relationship between the fidelity levels [see Ref. [222] and Eq. (4.4)] and is denoted as “linear MF- GPR." The other approach assumes a nonlinear mapping between fidelity levels [see Ref. [224] and Eq. (4.21)] and is denoted as “MF-GPR"; the shaded bands represent a 95% confidence interval around the fit. The MF-GPR approach that assumes a nonlinear mapping between fidelities (orange solid line) is able to recover the underlying exact HF solution, in contrast to the MF-GPR approach that assumes a linear mapping between fidelity levels (purple solid line). where i denotes the location of a test point, Ntest is the total number of test points, yi,true is the true solution at location i, and yi,pred is the value of the fit (MF-GPR or GPR) at location i. Table 4.2 compares the computational costs, which includes the costs of both hyperparameter training and predictions, and regression errors for the GPR and MF-GPR methods using the LF and HF models Eqs. (4.24) and (4.25). We find that while MF-GPR is roughly six to ten times more expensive than single-fidelity GPR, the MF-GPR method results in regression errors that are often a couple orders of magnitude lower than those obtained with single-fidelity GPR. 4.3 Multi-fidelity Regression of Plasma Transport-Coefficient Data In Secs. 4.2.2 and 4.2.3, we have demonstrated the effectiveness and limitations of single- fidelity GPR and different MF-GPR approaches using toy examples. Additionally, in Sec. 4.2.4, we assessed the cost-benefit trade-off between GPR and MF-GPR approaches. We illustrated the fact that relative to single-fidelity GPR, MF-GPR increases computation cost but decreases prediction 90 NLF NHF ∗ TGPR TMF-GPR /TGPR RMSEGPR RMSEMF-GPR 30 8 1 6±1 3.2[−1] 8.7[−3] 34 9 1.0 ± 0.1 6±1 3.5[−1] 1.5[−2] 38 10 1.0 ± 0.1 5±1 3.8[−1] 9.4[−4] 43 11 1.3 ± 0.7 6±3 3.5[−1] 1.1[−3] 50 13 1.1 ± 0.1 6±1 3.6[−1] 2.0[−3] 60 15 1.1 ± 0.2 8±2 1.1[−1] 1.9[−4] 75 19 1.4 ± 0.2 7±2 9.0[−3] 1.5[−4] 100 25 1.2 ± 0.2 8±2 5.4[−3] 9.9[−5] 150 38 1.7 ± 0.3 7±2 3.1[−4] 7.7[−5] 300 75 2.1 ± 0.3 10 ± 2 2.1[−4] 6.5[−5] Table 4.2: Average computation time and regression errors for single-fidelity GPR and MF-GPR, Eq. (4.22) fits using the LF and HF models Eqs. (4.24) and (4.25). Each entry is an average over ten fits, and the hyperparameters for each fit were trained using the limited-memory quasi-Newton algorithm with 15 random restarts. For the RMSE values, the numbers in brackets denote the power of ten that the value in front of the brackets is multiplied by (e.g., 3.2[−1] = 0.32). The column ∗ labeled TGPR shows the computation time for single-fidelity GPR normalized by the computation time when NHF = 8. The computational cost of single-fidelity GPR increases by a factor of two when the number of HF training points increases by roughly ten. We note that when NLF = 50 and NHF = 13, MF-GPR is six times more expensive than single-fidelity GPR but reduces the regression error by more than two orders of magnitude. error. While these toy examples were useful for building intuition and providing a baseline for computation-cost and error estimates, we will now consider real data generated for ionic plasma transport coefficients; we will begin by analyzing what role the choice of LF model plays in MF-GPR. The LF and HF models will be chosen from those listed in Table 4.1. We first consider two choices for the LF model for predicting the viscosity for the element C at ni = 5.01 × 1022 cm−3 , as shown in Fig. 4.5. MF-GPR fits produced using the LF SMT model are shown in panel (a) of the figure, and fits produced using the LF YGBI model are shown in panel (b). The HF training data were computed from the YVM. The inserts in Fig. 4.5 show the kernel matrix corresponding to k ρ k f in Eq. (4.23). In panel (a) of Fig. 4.5, with the SMT model used as the LF model, we see that the only nonzero values of the kernel matrix occupy the diagonal and quickly decay to zero a short distance from the diagonal, corresponding to a small length scale for the kernel. Because the entries of the kernel matrix have nearly zero magnitudes, the MF-GPR fit is nearly equivalent to the fit obtained by 91 performing GPR on the HF data alone; this equivalence explains the overlap of the fits produced by GPR and MF-GPR. In panel (b) of Fig. 4.5, with the YGBI model used as the LF model, two findings are of note. The first is that there are regions where the MF-GPR and GPR fits do not overlap; this is most clearly seen around T = 0.2 eV. Second, the entries of the kernel matrix are nonzero away from the diagonal, implying substantial correlations between the LF and HF data. However, the values are nearly constant throughout the matrix, differing from each other by at most by 1%. Thus, in contrast with the MF-GPR fit shown in panel (a), the MF-GPR fit shown in panel (b) includes information from the LF data and suggests correctly that the LF and HF data differ by an approximately constant shift. A comparison of the sizes of the confidence bands for the MF-GPR results in panels (a) and (b) in Fig. 4.5 shows that the MF-GPR fit in panel (b) is superior to that in panel (a). The choice of the YGBI model as the LF model for MF-GPR in panel (b) results in a superior fit because the YGBI model provides additional information that is used to improve the fit. This additional information can be seen in the kernel matrix computed from k ρ k f ; an LF model for which kernel entries off the diagonal are non-zero improved the MF-GPR fit over the GPR fit more than an LF model for which the kernel entries are close to zero. Thus, we have found that the kernel matrix computed from k ρ k f is a natural indicator of when an LF model is insufficient for MF-GPR and that a different, or more precise, LF model is needed to impact the MF-GPR fit. When kernel matrix entries decay rapidly to zero off the diagonal, it would be best to consider alternative LF models. In Fig. 4.5, we considered LF and HF models that both predict the same quantity. ML models have been developed in which the LF and HF models do not predict the same quantity; for example, the prediction of rainfall using an elevation model has been examined [250, 251]. As discussed in [250, 251], a large amount of elevation data are available, but only a minimal amount of rainfall data are available; together, these data have been used to construct MF rainfall models. Similarly, a large amount of self-diffusion coefficient data and a minimal amount of viscosity data are available, and MF models of plasma transport coefficients could be constructed using both data sources. 92 Figure 4.5: MF-GPR prediction of the viscosity of the element C at ni = 5.01 × 1022 cm−3 versus temperature. In both (a) and (b), GPR was performed using the HF training data computed from the YVM, and the GPR results are compared with those of an MF-GPR model constructed using data from both an HF model and an LF model. (a) The LF model is given by the SMT model. (b) The LF model is given by the YGBI model. The inserts in (a) and (b) display the kernel matrix from k ρ k f with optimized hyperparameters. In (a), little correlation is found between the HF and LF models, as the only non-zero entries of the kernel matrix are on, or close to, the diagonal; in (b), however, the correlation is substantial, as demonstrated by the extent of the non-zero values off the diagonal of the kernel matrix, as shown in the insert. 93 Figure 4.6: Using self diffusion as the LF model to predict viscosity. The reduced transport coefficients φ ∈ {D∗, η∗ } are shown for the element C at ni = 5.01 × 1022 cm−3 versus temperature. In both (a) and (b), GPR was performed using the HF training data computed from the YVM, and the GPR results are compared with those of an MF-GPR model constructed using data from both an HF model and an LF model. (a) The LF model is the reduced self-diffusion coefficient D∗ from the HMP model. (b) The LF model is D∗ computed from the SMT model. The inserts display the kernel matrix from k ρ k f with optimized hyperparameters. Thus, we also consider LF and HF models that do not predict the same quantity. In particular, we assess the validity of using the self-diffusion coefficient (LF model) as a predictor for the viscosity (HF model). MF-GPR fits for viscosity of the element C at ni = 5.01 × 1022 cm−3 using self-diffusion data for the LF model and viscosity data for the HF model are shown in Fig. 4.6; two different LF models for predicting the self-diffusion coefficient are considered. In both panels (a) and (b), HF data were calculated from the YVM model. In panel (a), the LF data were computed using the HMP model, and in panel (b), the LF data were computed from the SMT model. The transport coefficients have 94 been reduced such that D∗ = D/ω p ai2 and η∗ = η/mi ni ω p ai2 . Here, ω p = (4πni Z 2 e2 /mi )1/2 is the ion plasma frequency, and ai = (4πni /3)−1/3 is the ion-sphere radius, where ni is the ion number density, Z is the mean ionization state, e is the elementary charge, and mi is the ion mass. The inserts once again show the kernel matrix k ρ k f . Fig. 4.6 demonstrates that using self-diffusion coefficient data as our LF model and viscosity as our HF model substantially improves the MF-GPR model of the viscosity compared to using viscosity data for both models. What we mean by this is that the LF data used in both panels of Fig. 4.6 are more strongly correlated with the HF data than the LF data used in panel (a) of Fig. 4.5 are. A comparison between the kernel matrices shown in the insert of panel (a) of Fig. 4.5 and in the insert of both panels of Fig. 4.6 demonstrates this point; in contrast to panel (a) of Fig. 4.5, the kernel matrices shown in both panels of Fig. 4.6 have a non-zero value away from the diagonal. This means that the LF data used in both panels of Fig. 4.6 have a larger contribution to the MF-GPR model than the LF data used in panel (a) of Fig. 4.5 does. Also note that the entries of the kernel matrix in both panels of Fig. 4.6 are not a constant value, in contrast with the entries in the kernel matrix shown in the insert in panel (b) of Fig. 4.5. Therefore, the LF and HF data used in both panels of Fig. 4.6 are not related by a shift but rather by a nonlinear relationship. Comparisons of the kernel matrix k ρ k f provide valuable insight into the effectiveness of an LF model in an MF-GPR framework by quantifying the spatial extent of correlations and type of relationship between the low- and high-fidelity models e.g., linear or nonlinear. In particular, these comparisons revealed the effectiveness of using self-diffusion LF data to predict viscosity HF data. As self-diffusion data are more readily available and are cheaper to compute than viscosity data, Fig. 4.6 illustrates how MF-GPR provides improved estimates of viscosity at low computational cost where it has not been measured. In addition to selecting a sufficient LF model for MF-GPR, it is imperative to include data in the LF and HF datasets that capture essential special features of a physical system. For example, it is possible that neither the LF data nor the HF data include information about features such as sudden changes (i.e., a jump discontinuity). For plasma transport coefficient data, sudden changes 95 Figure 4.7: MF-GPR and GPR fits, with 95% confidence intervals (shaded bands), of the self- diffusion coefficient versus temperature for multiple elements. The models used to generate this data are given in Table 4.1. Panels (a) through (d) show MF-GPR and GPR fits obtained using a portion (filled diamonds) of the HF data (all diamonds). Panels (e) through (h) compare MF-GPR and GPR fits obtained using all of the available data; GPR is fit to only HF data, whereas MF-GPR uses both the LF and HF data. In general, the MF-GPR fit is less prone to spurious oscillations than the GPR fit, and the size of the uncertainty band is much smaller with MF-GPR than with GPR. in quantities such as the electrical conductivity may result from a phase transition. In the absence of such data, MF-GPR is incapable of predicting a discontinuity. If this behavior is known in advance, then the LF and HF models should be sampled accordingly to ensure that the MF-GPR framework has sufficient training data near the discontinuity; then, an MF-GPR approach capable of handling a discontinuity, such as that described in Ref. [223], can be used. 4.4 Regression of Sparse Disparate Data In this section, we will use MF-GPR to predict transport coefficients when HF data are available in disparate physical regimes. We will consider a transport-coefficient dataset, which has “gap" regions, i.e., temperature ranges in which no HF data are available, as shown in Table 4.1. This section is organized as follows. First, we use MF-GPR to fit gapped transport-coefficient data as a function of temperature. Then, we consider a higher-dimensional feature space of ion number density and temperature. We conclude by varying the approach used to sample the HF 96 Figure 4.8: MF-GPR and GPR fits, with 95% confidence intervals (shaded bands), of the viscosity coefficient versus temperature for multiple elements. The models used to generate this data are given in Table 4.1. Panels (a) through (d) show MF-GPR/GPR fits obtained using a portion (filled diamonds) of the HF data (all diamonds). Panels (e) through (h) show MF-GPR fits obtained using all of the data. In general, the MF-GPR fit is less prone to oscillations than is the GPR fit which uses only HF data. dataset. We find that using a low-discrepancy sequence 4 to select data-sampling locations yields smaller regression errors than does sampling data on a uniform grid. 4.4.1 Self-diffusion and Viscosity Predictions versus Temperature We apply MF-GPR to gapped transport-coefficient data for the elements H, He, Be, and Fe. We first consider an HF training set that consists of only four data points – two points at both high and low temperatures – and thus features a large gap between the patches of HF data. Then, this gap is reduced in size by including all HF points in the training set. This approach is illustrated in Fig. 4.7, which shows MF-GPR and GPR fits for the self-diffusion coefficient. In the top row, we note that multiple inflection points in the GPR predictions for He, Be, and Fe can be seen, while the MF-GPR fits are monotonically increasing. For Fe, a large oscillatory pattern is seen in the GPR fit. These oscillations are not physical and are likely due 4A sequence of points is said to be low-discrepancy if the proportion of points in the sequence falling into an arbitrary set is (on average) near-proportional to the measure of that set. 97 to the hyperparameters responsible for specifying the length scale of the kernel Eq. (4.1). With MF-GPR, oscillations do not appear, as the three terms in Eq. (4.23) do not restrict the form of the fit to a single length scale. Similar patterns are observed for the viscosity in Fig. 4.8. 4.4.2 Viscosity Predictions versus Temperature and Number Density Because only a small amount of HF data were used in the work described in Sec. 4.4.1, a well-defined error metric could not be reported. Therefore, we constructed an HF dataset in the ni − T plane containing 900 points sampled on a grid. The data were generated using the YVM for H and Fe, and these data will act as a test set for the results described in this section. The dataset spans a temperature range of T = 101 − 104 eV and an ion number density of ni = 1018 − 1026 cm−3 . Next, we constructed an MF training dataset. When this MF training dataset is used together with the HF test set described above, we will be able to compute regression errors for GPR and MF-GPR using Eq. (4.26), now in ni −T space. With the view of mimicking the scenario of datasets containing “gaps," as discussed in Sec. 4.4.1, an MF training dataset was constructed to contain a region lacking HF data. We chose the YGBI model as our LF model and assumed that this LF model can be evaluated everywhere in the domain. Fig. 4.9 illustrates the concept of physical regimes that include an area or “gap" in which no HF data exist. The figure shows three regions, labeled as “1," “2," and “3." In regions 1 and 3, both the HF (YVM) and LF (YGBI) models can be evaluated. The area between the red dashed lines, denoted with a 2 and labeled as “no HF data," shows the region where no HF data are available. We refer to this region as the “gap region." A summary of the choices of LF and HF models for all of the regions shown in the figure are given in Table 4.3. Having defined the models used to generate the test and training datasets, we will describe, below in Sec. 4.4.2.1, the three HF sampling approaches we used to create the MF training dataset. 4.4.2.1 Sampling Methods for HF Data We used three approaches to sample the HF gapped dataset initially: an evenly spaced grid, a low-discrepancy sequence, namely a Halton-23 sequence [252], and a hybrid method that used 98 Region HF LF 1 YVM [140] YGBI [137] 2 – YGBI [137] 3 YVM [140] YGBI [137] Table 4.3: HF and LF models for the viscosity used in the temperature/number-density regions shown in Fig. 4.9. The same LF model is employed across all regions. Figure 4.9: The regions of the temperature/number-density space where HF and LF models were used to generate viscosity data for the MF training dataset. The red dashed lines indicate the divisions between the regions. In the regions labeled as “1" and “3," both HF and LF data are available. In the region labeled as “2," only LF data are available. The models used for each region are listed in Table 4.3. both approaches. For the LF data, we restricted the sampling approach to an evenly spaced grid. The details of each sampling approach are discussed below and summarized in Table 4.4. To place data on an evenly spaced grid, we first specify the total number of HF data points (e.g., NHF = 100). Then, the grid spacing is computed by xu − x` ∆x = √ , (4.27) NHF where x ∈ {ni, T }, and the subscripts “u” and “`” denote the upper and lower bounds of x, respectively. Using Eq. (4.27) to determine the spacing between HF points is straightforward; however, we note that in order to refine the grid spacing by a factor of two, four times as many 99 HF data points are needed. As a result, the evenly spaced grid approach becomes increasingly computationally expensive as the dimension of the input space increases. Instead of restricting the locations of HF data to points on an evenly spaced grid, their locations may be determined randomly. However, two HF points chosen in this way could be extremely close together, and in such a circumstance, calculations would be repeated at roughly the same location in parameter space. By enforcing a constraint on the minimum distance between two points, calculating HF data at close locations can be avoided. An alternative to enforcing a constraint on the distance between data sampling locations is to use a low-discrepancy sequence to determine sampling locations; this is the second of our sampling methods. Low-discrepancy sequences consist of “quasi-random" numbers that are generated deterministically, and points constructed using these numbers as coordinates cover a domain more quickly and evenly than do points constructed with random numbers as coordinates. Here, we use a Halton-23 low-discrepancy sequence [252]. In the name “Halton-23," “23" denotes the bases 2 for dimension ni ) and 3 (for dimension T); the bases 2 and 3 were chosen as they are mutually prime, which results in a uniform, limiting density of the points in the sequence [252]. While use of only a low-discrepancy sequence to determine HF sampling locations reduces the chance of performing repeated calculations, the edges of the domain may not be included in an HF dataset constructed in this way. If a specific domain is desired, it is necessary to augment the low-discrepancy sequence locations with data along the domain boundary. To ensure coverage in a fixed domain, we used a hybrid sampling method. In this hybrid method, the four extreme corners of the domain of the HF dataset are sampled first. Then, the remainder of the allocated HF data points are sampled using a low-discrepancy sequence. The three sampling approaches we used are summarized in Table 4.4. In Fig. 4.10, we compare the MF-GPR prediction of the viscosity using each of these sampling methods, for NLF = 100 and NHF = 12. In the top row, we show the locations in the ni − T plane where the HF data, indicated by filled black diamonds, and the LF data, indicated by open blue circles, were sampled using each method. The bottom row shows heat maps of the absolute error between the prediction and the 100 Figure 4.10: MF-GPR prediction of the viscosity of the element Fe using NLF = 100 and NHF = 12. The HF data were sampled (a) on a uniformly spaced grid, (b) using a Halton-23 sequence, and (c) using a hybrid method. Top row: The locations of the HF training data (filled black diamonds) and LF training data (open blue circles) used to construct the MF training dataset are shown. The red dashed lines denote the boundaries between the regions shown in Fig. 4.9. Bottom row: The absolute differences between the predicted viscosities η pred and the true viscosities ηtrue are shown. The hybrid sampling approach improves the prediction in the gap region between the dashed white lines. Note that the failure of the Halton-23 sampling approach to include the boundaries of the HF data in the training set results in large errors at the boundaries. true solution, for each sampling method; differences between the sampling methods are apparent. In particular, the regression error in the gap region is substantially smaller with the hybrid method than with the grid method. 4.4.2.2 Regression Error Fits produced using GPR and MF-GPR are shown in Fig. 4.11, with the HF sampling approach varied as described in Table 4.4. Each point in the figure shows an average of 10 fits, with error bars indicating one standard deviation from the average. For MF-GPR, the LF data were sampled from a grid, and the cases NLF = 25 and 400 are shown. The GPR fits were carried out using only the HF data from the MF dataset. We see that the RMSE decreases as NHF increases for all 101 HF Sampling LF Sampling Description HF and LF data were sampled on an evenly spaced grid Grid Grid in ni and T. HF data were sampled using a Halton-23 sequence. Halton-23 [252] Grid LF data were sampled on an evenly spaced grid in ni and T. The HF dataset includes the four extreme corners of the domain and data sampled using a Halton-23 sequence. Hybrid Grid The LF data were sampled on an evenly spaced grid in ni and T. Table 4.4: Sampling approaches used to sample the MF training dataset. The LF data were always evaluated on a grid; sampling methods for the HF data varied. methods, and that the MF-GPR fit yields smaller RMSE values than does GPR. In almost all cases, MF-GPR performs at least as well as GPR. We next computed the RMSE of fits to HF viscosity data for different combinations of NHF and NLF for H and Fe. The results for H are displayed in Fig. 4.12, and for Fe, in Fig. 4.13; each column in the figure corresponds to a different HF sampling method. We note that the RMSE values for NHF = NLF = 4 should be the same in columns (a) and (c), because the hybrid method first samples the four corners from the grid and then adds points sampled using the Halton-23 sequence. The average value of the RMSE for NHF = NLF = 4 in column (a) is within one standard deviation of the average value of the RMSE for the same case in column (c), and vice versa. Therefore, we do not consider these differences to be statistically significant. As shown in Figs. 4.12 and 4.13, fits generated using the hybrid sampling approach result in smaller RMSE values overall than do those generated using a simple grid approach. It is also worth noting that the pure Halton-23 method often produced higher RMSE values than did the grid method; this is because the boundaries of the domain were not sufficiently sampled in the HF training set. As a result, the MF-GPR fit tends to the mean of the HF data, and the largest errors are incurred near the boundaries, as shown in Fig. 4.10. 102 Figure 4.11: The RMSE of log10 (η) for GPR and MF-GPR fits for the element H using different HF sampling methods. We sampled NHF and NLF points from the gapped dataset shown in Fig. 4.9. The models used to generate the data are specified in Table 4.3. Each RMSE value was determined from an average of ten fits, and standard deviations for the values are shown as error bars. The HF data were sampled (a) on an evenly spaced grid, (b) using a Halton-23 sequence, and (c) using a hybrid method. We note that in most cases, MF-GPR outperformed GPR. Figure 4.12: The RMSE of log10 (η) for the element H using MF-GPR with different MF training sets constructed using various HF sampling approaches. (a) The HF data were sampled on a grid. (b) The HF data were sampled using a Halton-23 sequence. (c) The HF data were sampled using our hybrid approach. 103 Figure 4.13: The RMSE of log10 (η) for the element Fe using MF-GPR with different MF training sets constructed using various HF sampling approaches. (a) The HF data were sampled on a grid. (b) The HF data were sampled using a Halton-23 sequence. (c) The HF data were sampled using our hybrid approach. 4.5 Conclusions and Outlook We have investigated the use of MF-GPR to interpolate plasma transport data over a wide parameter space in which HF data are available in localized patches. We have examined the improvements in both the predicted mean and the predicted uncertainty that MF-GPR provides over GPR. We have seen that in most cases, MF-GPR results in a lower uncertainty than does single- fidelity GPR, sometimes by an order of magnitude. Examining the hyperparameters governing the structure of the k ρ k f kernel reveals the improvement in the mean and uncertainty, or lack thereof, given by the LF data. As a “black-box" regression method, MF-GPR provides increased reliability over single-fidelity methods, as trends from LF models are used during regression where HF data are sparse; the use of such LF trends enables MF-GPR to reduce the occurrence of nonphysical oscillations or inflection points that occur with single-fidelity GPR. In addition, confidence bands generated by MF-GPR and GPR suggest where additional HF data are needed once a fit has been produced; simpler regression methods do not offer this benefit. From an experimental-design perspective, HF data are often sampled on a grid that is refined 104 uniformly when finer resolution is needed [253]. We found that when performing MF-GPR, sampling HF data on a uniformly spaced grid can bias length-scale hyperparameters and results in larger regression errors. Therefore, we developed a simple hybrid approach for initially sampling HF data that combines sampling both on a grid and using a low-discrepancy sequence, resulting in smaller regression errors. The results here can be expanded upon in multiple ways. For example, the MF-GPR framework could be extended to include physically motivated constraints, such as enforcing non-negativity [254]. Additionally, we restricted the work here to the self-diffusion and viscosity transport coefficients, but other transport coefficients, such as the thermal conductivity, the resistivity, and the interdiffusion coefficient in plasma mixtures, can also be investigated. The sampling methods described here can also be improved upon greatly and optimized for higher-dimensional feature spaces to avoid the curse of dimensionality. However, our approaches offer a starting point that highlights the importance of avoiding regressing beyond the bounds of available data in a GPR/MF- GPR setting. Through the confidence intervals it provides, the GPR approach suggests where it would be most useful to generate additional data; the confidence of a fit would be improved most by obtaining additional HF data points in regions with the greatest uncertainties. Comparing GPR and MF- GPR results show the utility of generating LF data in parallel with HF datasets. In addition, it could be possible to improve the ML approach itself by developing customized kernels for this application [255–257]. 105 CHAPTER 5 KINETIC MODELING OF STRONGLY COUPLED PLASMA MIXTURES 5.1 Introduction In Chapter 4, we highlighted how machine learning can be used for interpolating multi-modal plasma transport data in disparate physical regimes. Much of the data in Chapter 4 were obtained for dense plasmas using methods like MD or analytic models that have known accuracy in certain limits. We change focus in this chapter to applications of ultracold neutral plasmas (UNPs) which have many orders of magnitude lower number density and ion temperature than dense plasmas; this property of UNPs is illustrated in Table 1.1 and Table 5.1. From these tables, we see that UNPs have strongly coupled ions and are non-degenerate. Moreover, UNPs are highly-collisional and close to equilibrium. The method that we develop and employ in this chapter stems from kinetic theory and aims to characterize all of the aforementioned details. The organization of this chapter is as follows. First, In Sec. 5.2, we provide a brief review kinetic theory where we present the governing equations and specific choices surrounding the collision operator. Then, we derive expressions for the entropy generation in a system from the viewpoint of kinetic theory which will be used to assess the time-reversibility of multi-species element ni (cm−3 ) Ti (K) Te (K) Γ θ Kn Ref. Ca 1.8×1010 1 96 7.1 3.4×103 6.3×10−5 [15] Yb 2×109 1 96 3.4 1.5×104 1.5×10−3 [15] Sr 5×109 1.4 70 3.3 5.7×105 4.5×10−4 [258] Sr 1×109 1 100 2.7 2.4×106 1.1×10−3 [259] Ca 3.4×109 1.8 96 2.3 1.9×104 1.1×10−3 [260] Yb 1.9×10 9 1 96 3.3 1.5×104 6.0×10−4 [260] Xe 2×1010 1 500 7.3 3.2×103 5.7×10−4 [261] Table 5.1: Comparison of non-dimensional parameters for UNPs. To compute the Knudsen numbers, a reference length-scale of L = 1 mm was used. While the ion species in UNPs are typically laser-cooled to temperatures on the order of micro-Kelvin (µK), disorder-induced heating increases the ion temperature to the order of Kelvin (K). Based on the magnitude of the non- dimensional parameters, we classify UNPs as strongly coupled, non-degenerate, highly-collisional systems. 106 UNPs. Next, we derive the Vlasov equation in the Cartesian, cylindrical [262], and spherical coordinate system showing that additional acceleration terms appear as a result of the coordinate transformation. We present the conservative form of the 1D-1V Vlasov equation that can be used to model plasma with certain symmetries, like the radial symmetry found in Gaussian UNPs. In Sec. 5.6, we detail the numerical methods used to solve the equations derived in Sec. 5.2. The basis of our numerical scheme consists of a fourth-order finite-volume method [262]. We show numerical results for single- and multi-species UNPs as well as multi-species high-energy-density (HED) plasmas. We find that the amount of entropy generated in multi-species UNPs is largely controlled by the initial conditions of the plasma. Additionally, we use our numerical method to simulate the interfacial mixing in multi-species HED plasmas. We interrogate our numerical results to find that the dominant drivers of diffusion across the interface is dominated by the self-consistent electric field. We conclude by summarizing the results and discuss future work in Sec. 5.7. 5.2 Kinetic Theory Non-equilibrium statistical mechanics or kinetic theory [263, 264] is a vast field of physics that has many applications. The theoretical basis for kinetic theory stems from the Liouville equation which describes the phase-space evolution of the N-particle distribution function. Through statistical averaging, we obtain the so-called “BBGKY" hierarchy which is the starting point for “kinetic equations." Kinetic equations are defined as a closed equation of motion for f1 , the one- particle distribution function. In this work, we will omit the subscript “1" and simply use fi to denote the one-particle distribution function of species i. A generic set of kinetic equation for N species has the form N ∂ fi Õ + v · ∇r fi + ai · ∇v fi = Q( fi, f j ), i = 1, 2, · · · , N, (5.1) ∂t j=1 where, in a Cartesian coordinate system, the phase-space distribution for species i is fi = fi (r, v, t), for r = (x, y, x), v = (v x, v y, vz ), and the acceleration (or force) acting on species i is ai = (ai,x, ai,y, ai,z ). The term Q( fi, f j ) is the collision operator between species i and j which introduces microscopic information by way of particle collisions. There are many treatments for Q( fi, f j ) 107 which all stem from the Boltzmann equation. Some examples include the so-called Fokker-Planck, Lenard-Balescu, Lenard-Bernstein, and Bhatnagar-Gross-Krook collision operators; we explore three versions of the collision operator in this work and they are described in Sec. 5.2.1. A benefit of kinetic equations for modeling plasmas is that they have fewer degrees of freedom than MD, which has 6N degrees of freedom, but more degrees of freedom than hydrodynamic models, which have 3 degrees of freedom. The saying that “a distribution function is worth a thousand macroscopic variables" [263] encapsulates this relation of kinetics and hydrodynamics. In essence, you can derive hydrodynamic equations and macroscopic variables by taking specific moments of the distribution function f . Specifically, the first three moments yield the number density, bulk velocity, and temperature of species i ∫ ni (r, t) = fi (r, v, t) dv, (5.2) ∫ ni (r, t)ui (r, t) = v fi (r, v, t) dv, (5.3) ∫ 3 mi ni (r, t)Ti (r, t) = [v − ui (r, t)]2 fi (r, v, t) dv. (5.4) 2 2 In Sec. 5.2.1 we present three distinct collision operators that we will employ in this chapter. We note that all collision operators stem from approximations to the Boltzmann equation. In particular an assumption of two of the collision operators we discuss is that the system is near equilibrium. 5.2.1 Collision Operators The simplest choice for the collision operator consists of settting Q( fi, f j ) = 0 and replacing the acceleration term ai in Eq. (5.1) with a force computed from the self-consistent electric field (the mean-field approximation) obtained from Poisson’s equation. These choices result in the “Vlasov-Poisson" equation, or simply the “Vlasov" equation ∂ fi Zi eE + v · ∇r fi + · ∇v fi = 0, (5.5) ∂t mi E = −∇ϕ, (5.6) N ∫ 1 Õ − ∇2 ϕ = Zi e fi (r, v, t) dv. (5.7) 4π i=1 108 It is important to note that here, we have assumed that the system is electrostatic and thus no magnetic fields are generated from electric currents. More generally we would replace ai = Zi e(E+v×B)/mi which is the Lorentz force with the magnetic field B; in this case, obtaining E and B would require solving Maxwell’s equations. We note that while the Vlasov equation is often referred to as “collisionless" the particles still interact via the self-consistent electric field. For our study of UNPs using kinetic equations, we will require the use of a collision operator as UNPs are highly collisional (see Table 5.1). One choice of non-zero collision operator that applies to systems near equilibrium is the Bhatnagar, Gross, and Krook (BGK) [265] collision operator. We begin our discussion of collision operators for a single species kinetic equation (dropping all subscripts so that fi = f ). For a single species, the BGK collision operator is QBGK ( f ) = ν [M(v; n, u, T) − f ] , (5.8) where ν is the ion collision frequency and M(v; n, u, T) is a Maxwellian distribution that is param- eterized by moments of f [see Eqs. (5.2) - (5.4)].  3/2 m [v − u(r, t)]2    m M(v; n, u, T) = n(r, t) exp − . (5.9) 2πT(r, t) 2T(r, t) The choice of M(v; n, u, T) depending on the moments of f ensures the conservation of mass, momentum, and energy. We stress that the BGK collision operator is not to be confused with a strikingly similar collision operator proposed by Krook which replaces M(v; n, u, T), with some arbitrary equilibrium distribution f0 independent of the moments of f so that QKrook ( f ) = ν( f0 − f ). (5.10) Since f0 in Eq. (5.10) does not rely on moments of f conservation of mass, momentum, and energy is not guaranteed. The BGK model has the effect of relaxing the distribution function to equilibrium in phase-space at a time scale determined by ν. We note that the BGK collision operator is computationally simple: it consists of a subtraction and a multiplication. However, because of the underlying assumptions 109 resulting its simplistic nature, it may fail to accurately model certain plasmas. Another option for the collision operator is one proposed by Lenard and Bernstein (LB) [266]. The LB collision operator is derived from a linearization of the Fokker-Planck equation, which is obtained in the “small-angle" or “weak-scattering" approximation of the Boltzmann equation. The LB equation includes information about gradients in velocity space viz. Q LB ( f ) = ν∇v · (v − u0 ) f + v0,th 2 ∇v f ,   (5.11) where u0 is the initial bulk/drift velocity and v0,th = p 3T0 /m is the initial root-mean-square speed of the ions. Much like the Krook version of the BGK operator, Eq. (5.11) is non-conservative. To remedy this, conservation is imposed by the form Q LBD ( f ) = ν∇v · [v − u(r, t)] f + vth 2 ∇v f ,  (5.12) where u0 had been replaced with the first moment of f and vth = p 3T(r, t)/m is the root-mean- square speed that now depends on the second central moment of f . To extend the collision operator to N-species, we need to make modifications to the expressions given in Sec. 5.2.1. For example, the multi-species BGK operator between species i and j is QBGK ( fi, f j ) = νi j Mi j (v; ni, ui j , Ti j ) − fi .   (5.13) Equation (5.13) now depends on a Maxwellian distribution that is parametrized by the mixture quantities ui j and Ti j . There are non-unique choices for these quantities and the versions we implement in this chapter are derived in Ref. [267] and ensure that the multi-species BGK model is conservative and entropic. The mixture bulk velocity is ρi νi j ui + ρ j ν ji u j ui j = , (5.14) ρi νi j + ρ j ν ji and the mixture temperature is ni νi j Ti + n j ν ji T j ρi νi j (vi − νi j ) + ρ j ν ji (v j − νi j ) 2 2 2 2 Ti j = + . (5.15) ni νi j + n j ν ji 3(ni νi j + n j ν ji ) Note that both the BGK and LBD operators depend on the same input: the collision frequency νi j . There are many choices for obtaining the collision frequency, and the specific choices used in 110 this chapter are described in Secs. 5.2.2 and 5.2.3. Since we are interested in simulating UNPs, we chose a collision rate model that is accurate for strongly coupled plasmas. We note that most collision rate models are typically accurate for weakly coupled plasmas. 5.2.2 Ion-Ion Collision Rate As shown in Table 5.1, UNPs are moderately-coupled systems; therefore, the collision rate νi j must be accurate in these regimes. There are many collision rate models that exist [268, 269] that often make approximations to circumvent issues associated with computing the necessary collision integrals. These approximations introduce the so-called “Coulomb logarithm" which is known to fail at strong coupling. Here, we employ a collision rate model derived in Ref. [267] which is valid for moderately-coupled systems and does not rely on a Coulomb logarithm. The specific collision rate we use in this chapter is the the rate derived from temperature relaxation rate denoted as “ET" in Ref. [267]. The collision rate between species i and j is given by 1 256π 2 ni n j (mi m j )1/2 (Zi Z j e2 )2 νi j = K11 (gi j ), (5.16) ni 3(2π)3/2 (mi T j + m j Ti )3/2 Zi Z j e2 (mi + m j ) gi j = . (5.17) λeff (mi T j + m j Ti ) Where K11 is a fit to the collision integrals with an effective interaction potential  Í    − 1 ln 5 a g k , g<1   4 k=1 k K11 (g) =  (5.18)  b0 + b1 ln(g) + b2 ln2 (g)  g ≥ 1, 1 + b3 g + b4 g 2    where the a coefficients are a1 = 1.4660, a2 = −1.7836, a3 = 1.4313, a4 = −0.55833, and a5 = 0.061162 and the b coefficients are b0 = 0.081033, b1 = −0.091336, b2 = 0.051760, b3 = −0.50026, and b4 = 0.17044; these coefficients are provided in Appendix C of Ref. [151]. Here, the plasma parameter gi j appears in Eq. (67) of Ref. [151] but with a modified temperature to account for multiple ion temperatures. The effective screening length is given by both the ions and electron species as " N !# −1/2 1 Õ 1 1 λeff = + , (5.19) λe i=1 λi 1 + 3ΓiIS 111 where the electron and ion screening lengths are r  2 Te2 + 2 3 EF λe2 = , (5.20) 4πe2 ne Ti λi2 = , (5.21) 4πZi2 e2 ni and the modified Coulomb coupling parameter is given by (Zi e)2 ΓiIS = , (5.22) aiISTi   1/3 3Zi e ai = IS , (5.23) 4πρtot ÕN ρtot = Zi eni . (5.24) i=1 We will use Eq. (5.16) in all kinetic simulations where ion-ion interactions are present. Ion-ion kinetic simulations are carried out for HED plasmas in Secs. 5.6.1.1 and 5.6.2.3 and for UNPs in Secs. 5.6.1.2 and 5.6.2. If we are interested in simulating electron-electron collisions, modifications to the collision rate need to be made. We describe eight collision rate models for electron-electron interactions in Sec. 5.2.3. 5.2.3 Electron-Electron Collision Rate To compute the electron-electron collision rate, we implement eight collision rate models, seven of which require the use of a Coulomb logarithm. The seven models that require a Coulomb logarithm are the Landau-Spitzer (LS) model [270], which assumes a straight-line trajectory and binary interactions, and six models proposed by Gericke, Murillo, and Schlanges (GMS) [269], three of which make modifications to the LS straight-line trajectory assumption and three additional models that introduce the concept of hyperbolic trajectories and binary collisions. The electron- electron collision rate νee can be obtained from [151, 271] √ 8 πne e4 ln Λ νee = √ , (5.25) 3 meTe3/2 Where ln Λ is the Coulomb logarithm. Expression for the Coulomb logarithm can be found in Refs. [270] and [269]. The final model used for obtaining collision rates does not depend on 112 a Coulomb logarithm as proposed by Haack, Hauck, and Murillo (HHM) [44, 267]. The HHM collision rate includes effects of particle correlations beyond a binary scattering viewpoint by numerically evaluating the collision integral with an effective interaction potential [151]. The HHM collision rate is given by √ 32 πne e4 νee HHM = √ K11 (γee ), (5.26) 3 meTe3/2 e2 γee = , (5.27) Te λTF 4πe2 ne λTF 2 = 2 , (5.28) Te + ( 23 EF )2 where EF = ~2 (3π 2 ne )2/3 /2me is the Fermi energy. Table 5.2 shows a comparison of the eight collision rate models described here applied to an electron UNP. In Sec. 5.6.1.3 we will use these electron-electron collision rates to simulate the tail-filling in an electron UNP. Tail filling is the process by which "depleted" high-energy tails of a velocity distribution for a system of particles equilibrates. That is, the plasma equilibrates to equilibrium on some time-scale determined by the collision rate – “replenishing" the tails. In general, a system that is in equilibrium does not generate entropy. In Sec. 5.2.4, we mathematically define the process of entropy generation starting from a kinetic theory viewpoint. 5.2.4 Entropy Generation and Time-Reversibility Ultimately, the role of a collision operator is to drive a system to equilibrium. To quantify when a system has equilibrated, we consider the amount of entropy being generated in a system. When entropy is no longer being generated, the system has reached thermodynamic equilibrium. Later in this chapter, we will show that the initial conditions of a plasma mixture dictate the amount of entropy that is generated. We begin with an equation for the entropy density of an N-species system [189] ÕN ∫ ρs = −k B fi (ln fi − 1)dv. (5.29) i=1 113 Taking the time derivative of Eq. (5.29) we get N ∂ ρs ∂ fi Õ ∫ = −k B ln fi dv ∂t i=1 ∂t N ∫  N  Õ Z i eE Õ = kB ln fi v · ∇r fi + Q( fi, f j ) dv   · ∇v fi − i=1  mi j=1    N ∫ N ∫ Õ Õ Zi eE = k B ∇r · v[ fi (ln fi − 1)]dv + k B · ln fi ∇v fi dv i=1 i=1 mi Õ N ∫ − kB Q( fi, f j ) ln fi dv, (5.30) i, j=1 where we have used the relation that ln fi ∇r fi = ∇r [ fi (ln fi − 1)]. Integration by parts on the second term in Eq. (5.30) yields zero and holds for any arbitrary non-velocity dependent force. Therefore we have N N ∫ ∂ ρs Õ ∫ Õ = k B ∇r · v[ fi (ln fi − 1)]dv − k B Q( fi, f j ) ln fi dv. (5.31) ∂t i=1 i, j=1 Adding and subtracting the barycentric velocity defined as N 1Õ u= ρi ui, (5.32) ρ i=1 ÕN ρ= mi ni, (5.33) i=1 we obtain N N ∫ ∂ ρs Õ ∫ Õ = k B ∇r · (v − u + u)[ fi (ln fi − 1)]dv − k B Q( fi, f j ) ln fi dv. (5.34) ∂t i=1 i, j=1 Rearranging the first term we have N ∫ N ∫ ∂ ρs Õ Õ = k B ∇r · u fi (ln fi − 1)dv + k B ∇r · (v − u)[ fi (ln fi − 1)]dv ∂t i=1 i=1 Õ N ∫ − kB Q( fi, f j ) ln fi dv. (5.35) i, j=1 114 Using Eq. (5.29) and defining the entropy flux and entropy source term as ÕN ∫ Js = −k B ∇r · (v − u)[ fi (ln fi − 1)]dv, (5.36) i=1 Õ N ∫ σ = −k B Q( fi, f j ) ln fi dv, (5.37) i, j=1 respectively, Eq (5.35) becomes ∂ ρs = −∇r · (ρsu + Js ) + σ. (5.38) ∂t Equation (5.38) is simply an advection equation plus a source term for the entropy density or in other words, a time- and space-dependent equation for the second law of thermodynamics. We stress that the entropy source term σ is characterized purely by the collision operator. In the case of Vlasov, no entropy will be generated in the system implying that the system is not evolving to equilibrium. In fact, solutions of the Vlasov equation, as well as MD, are completely deterministic and time- reversible. The fact that MD is also time-reversible should be alarming since fundamentally, our universe is made up of atoms with interactions that could be described by Newton’s second law. The time-reversibility of Newton’s second law implies that our universe is time-reversible and that after some amount of time, we would expect to return to our “initial condition." While mathematically true, the Poincaré recurrence theorem suggests that the time that would need to pass in order for that to happen would be nearly infinite as there are essentially infinite number of particles in our universe. 5.2.5 Entropy Density in Thermodynamic Equilibrium Consider the case where of thermodynamic equilibrium where the distribution function has the form of a Maxwellian mi (v − u)2  m  3/2   eq i fi = ni exp − . (5.39) 2πT 2T We can re-write the above expression by substituting in the thermodynamic potential for an ideal gas mixture   T 3 2πT µi = ln ni − ln , (5.40) mi 2 mi 115 we get that m µ   m  3/2 i i i exp = ni , (5.41) T 2πT which we can substitute into Eq. (5.39) to obtain m µ   mi (v − u)2  eq i i fi = exp exp − T 2T " # µi − 2 (v − u) 1 2 = exp mi . (5.42) 2T Using Eq. (5.42) into Eq. (5.29) gives " # Õ N ∫ eq µi − 21 (v − u)2 ρs = −k B fi mi − 1 dv, (5.43) i=1 2T which reduces to ! N kB Õ ρs = − ρi µi − ρu − p . (5.44) T i=1 Equation(5.44) shows that the kinetic theory opinion of the entropy density in equilibrium is con- sistent with the thermodynamic expression. Using the Gibbs relation, we arrive at an expression for the entropy source term in terms of the diffusion flux, heat flux, and pressure tensor. Decomposing the entropy source term this way allows for an analysis of the dominant terms that generate entropy. That is, we can connect quantities like interdiffusion, thermal conductivity, and viscosity to the amount of entropy production in a plasma. 5.3 The Vlasov Equation in Conservative Form Before beginning our discussion on the numerical methods we employ to solve our kinetic equation, we wish to represent our kinetic equation in conservative form. Doing so will allow us to utilize numerical methods for conservation laws, in particular, finite-volume methods. Returning to the single-species Vlasov equation, which will be the left-hand-side for all of our spatially dependent kinetic equations in this chapter, we expand Eq. (5.5) as ∂f ∂ ∂ ∂ ∂ ∂ ∂ + vx f + vy f + vz f + a x f + ay f + az f = 0. (5.45) ∂t ∂x ∂y ∂z ∂v x ∂v y ∂vz Note that if the a x, a y, and az are determined from the self-consistent electric field [see Eq. (5.6)], then both the velocity and acceleration components commute with the partial derivatives and 116 Eq. (5.45) becomes ∂f ∂ ∂ ∂ ∂ ∂ ∂ + (v x f ) + (v y f ) + (vz f ) + (a x f ) + (a y f ) + (az f ) = 0. (5.46) ∂t ∂x ∂y ∂z ∂v x ∂v y ∂vz Equation (5.46) can be written in a compact notation as ∂f + ∇X · (X Û f ) = 0, (5.47) ∂t where X = [x, y, z, v x, v y, vz ]T , XÛ = [v x, v y, vz, a x, a y, az ]T . (5.48) We note that Eq. (5.47) is written in conservative form, i.e., the time derivative of the distribution function changes due to the divergence of fluxes X Û f. Representing Eq. (5.5) in conservative form was a trivial task in Cartesian coordinates. The triviality was a direct consequence of the fact that the velocity and acceleration components do not depend on the partial derivatives, allowing them to commute. In Secs. 5.3.1 and 5.3.2, we derive the conservative form of the Vlasov equation in both cylindrical and spherical coordinate systems. The cylindrical form was discussed in detail in Ref. [262] and we follow the same derivation procedure for spherical coordinates. For completeness, we repeat the derivation for cylindrical coordinates carried out in Ref. [262] here. Our ultimate goal is to numerically implement our kinetic model in spherical coordinates. The reason the spherical coordinate system is desirable is due to the fact that UNPs are often initialized as spherically symmetric clouds which then expand radially outward into the surrounding vacuum. Additionally, many diagnostics used in UNP experiments report the radial expansion of the plasma and we wish to compare our work against available experimental data. 5.3.1 Cylindrical Coordinates In cylindrical coordinates, a vector r is defined by (ρ, φ, z) where r = ρê ρ + φêφ + zêz, (5.49) 117 where êξ denote unit vectors point along the coordinate ξ. The relation of unit vectors in cylindrical coordinates to the unit vectors in Cartesian coordinates by ê ρ = cos φê x + sin φê y, (5.50) êφ = − sin φê x + y cos φê y, (5.51) êz = zêz . (5.52) The relation of a vector in cylindrical coordinates to Cartesian coordinates is given by r ρ   ρ (x + y 2 )1/2       2        ≡ rφ  = φ =  arctan x  .        rcyl      y (5.53)        rz   z   z              The velocity in cylindrical coordinates is found via d vcyl = ρê ρ + zêz = ρeÛ̂ ρ + ρê Û ρ + zeÛ̂ z + zÛêz  dt = ρê Û ρ + ρφê Û φ + zÛêz . The velocity vector in cylindrical coordinates is now related to Cartesian coordinates by v ρ   ρÛ        xv x + yv y   xv x + yv y               (x 2 + y 2 )1/2   (x 2 + y 2 )1/2     xv y − yv x   xv y − yv x          ≡ vφ  =  ρφÛ = (x + y ) = .      2 2 1/2 vcyl (5.54)           x 2 + y 2   (x 2 + y 2 )1/2           vz   zÛ   vz vz                    Now define F̄ ≡ X Û f . The goal is to convert the divergence of the fluxes F̄ into a new coordinate system which can be done by using the following relation:   1 ∇X · F̄ = J∇ξ · JF̄ . (5.55) J Where J denotes the Jacobian matrix with determinant J. The Jacobian matrix is  ∂rcyl ∂rcyl    ∂r ∂vcar   car J=  , (5.56)  ∂vcyl ∂vcyl    ∂r ∂vcar   car 118 where  ∂r ρ ∂r ρ ∂r ρ   x y  0  ∂x ∂y ∂z   ρ ρ      ∂rcyl  ∂r ∂rφ ∂rφ   −y    φ x = = 0 ,  (5.57) ∂rcar  ∂x ∂y ∂z   x 2 + y 2 x2 + y2      ∂r ∂rz ∂rz     z   ∂x ∂y ∂z   0 0 1       ∂r ρ ∂r ρ ∂r ρ   ∂v x ∂v y ∂vz     ∂rcyl  ∂r  φ ∂rφ ∂rφ  =  = 0, (5.58) ∂vcar  ∂v x ∂v y ∂vz     ∂rz ∂rz ∂rz    ∂v ∂v y ∂vz   x  ∂v ρ ∂v ρ ∂v ρ   y(yv x − xv y ) x(xv y − yv x )   ∂x 0 ∂y ∂z        ρ3 ρ3   ∂vcyl  ∂v  φ ∂vφ ∂vφ   y(xv x + yv y ) x(xv x + yv y )   = = − 0 , (5.59) ∂rcar  ∂x ∂y ∂z   ρ 3 ρ3      ∂v  z ∂vz ∂vz    0 0 0   ∂x ∂y ∂z        ∂v ρ ∂v ρ ∂v ρ   x y   ∂v x  ∂v y ∂vz   ρ   0  ρ  ∂vcyl  ∂v  φ ∂vφ ∂vφ   −y x   = = 0 . (5.60) ∂vcar  ∂v x ∂v y ∂vz   ρ ρ      ∂vz ∂vz ∂vz    0 0 1    ∂v ∂v y ∂vz    x  119 The Jacobian transformation matrix from Cartesian to cylindrical coordinates is  x y   0 0 0 0   ρ ρ   −y x  0 0 0 0   ρ2 ρ2         0 0 1 0 0 0 J =  .  (5.61)  y(yv x − xv y ) x(xv y − yv x ) x y  0 0 ρ3 ρ3 ρ ρ     y(xv x + yv y ) x(xv x + yv y )   −y x   − 0 0   ρ3 ρ3 ρ ρ    0 0 0 0 0 1      We note that the determinant of J is J = 1/ρ. Computing the matrix-vector product defined in Eq. (5.55), we have  ∂      x y vx f    ∂ρ        0 0 0 0      ρ ρ         ∂         −y x  vy f            ∂φ  0 0 0 0         ρ 2 ρ2         ∂                   v f     ∂z     0 0 1 0 0 0  z   1   ∇X · F̄ =  ∂  · ρ   ,          (5.62) ρ   y(yv x − xv y ) x(xv y − yv x ) x y    a x f   0 0   ∂v ρ      ρ3 ρ3 ρ ρ              ∂     y(xv x + yv y ) x(xv x + yv y )          −y x    a y f   − 0 0   ∂vφ       ρ 3 ρ3 ρ ρ               ∂         a f   0 0 0 0 0 1        z   ∂vz                   120 which simplifies to  ∂  ρv    ∂ρ  f   ρ        ∂           ∂φ     vφ f     ∂        ρvz f    ∂z    1   ∇X · F̄ =  ∂    .  ·  (5.63)   ρ   ∂v ρ    ρar + v 2 f     φ  ∂          ρaφ − v ρ vφ f      ∂vφ         ∂      ρaz f    ∂vz            Where we have made the definitions that xa x + ya y aρ ≡ , (5.64) ρ xa y − ya x aφ ≡ . (5.65) ρ Now that we have an expression for the divergence of the fluxes in the cylindrical coordinate system, we use Eq. (5.47) to obtain the Vlasov equations in cylindrical coordinates ∂f 1 ∂ 1 ∂ + (ρv ρ f ) + (vφ f ) ∂t ρ ∂ρ ρ ∂φ 1 ∂ + (ρvz f ) ρ ∂z 1 ∂ h i 1 ∂   1 ∂ + (ρa ρ + vφ2 ) f + (ρaφ − v ρ vφ ) f + (ρaz f ) = 0, (5.66) ρ ∂v ρ ρ ∂vφ ρ ∂vz which is the same result obtained in Refs. [262, 272]. We can further simplify Eq. (5.66) by canceling the pre-factor of ρ to yield vφ2 " ! # ∂f 1 ∂ 1 ∂ ∂ ∂ + (ρv ρ f ) + (vφ f ) + (vz f ) + aρ + f ∂t ρ ∂ρ ρ ∂φ ∂z ∂v ρ ρ ∂ ∂    v ρ vφ + aφ − f + (az f ) = 0. (5.67) ∂vφ ρ ∂vz We note that in contrast to the conservative form in the Cartesian coordinate system, additional acceleration terms in Eq. (5.67) appear. Specifically, these additional acceleration terms correspond 121 to the acceleration from the Coriolis force −vr vθ /r and the centripetal force vφ2 /r. We will repeat this procedure for the spherical coordinate system in Sec. 5.3.2 and show that these additional acceleration terms also appear. 5.3.2 Spherical Coordinates A vector in spherical coordinates is described at a point (r, θ, φ) so that r = r êr + θ êθ + φêφ, (5.68) where the unit vectors in spherical coordinates are related to the unit vectors in Cartesian coordinates by êr = sin θ cos φê x + sin θ sin φê y + cos θ êz, (5.69) êθ = cos θ cos φê x + cos θ sin φê y − sin θ êz, (5.70) êφ = − sin φê x + cos φê y . (5.71) The relation of a vector in spherical coordinates to Cartesian coordinates is given by (x 2 + y 2 + z2 )1/2        rr   r            z ≡ rθ  =  θ  = arccos .      rsph (5.72)        (x + y + z ) 2 2 2 1/2   y   rφ  φ       arctan  x        The velocity of a point in spherical coordinates is computed by taking a time derivative d vsph ≡ rêr = r eÛ̂ r + rÛêr dt = rÛêr + r θÛêθ + r φÛ sin θ êφ . 122 The velocity vector is related to Cartesian coordinates by      xv x + yv y + zvz   vr   rÛ         (x 2 + y 2 + z2 )1/2        + 2 + 2       xzv x yzv y − (x y )v z ≡  vθ  =  r θÛ  =   vsph (x 2 + y 2 + z2 )1/2 2  (5.73)               (x + y 2 )1/2 (x 2 + y 2 + z 2 )        h  i −yv + xv    x y vφ  r sin θ φÛ (x 2 + y 2 + z2 )1/2 sin arccos (x 2 +y2 +z2 )1/2     z       x +y 2 2     xv x + yv y + zvz     r    xzv x + yzv y − (x + y )vz  2 2   =  . (5.74)  (x 2 + y 2 )1/2r   −yv x + xv y         (x 2 + y 2 )1/2   The Jacobian for representing how vectors in a spherical coordinate system vary with respect to vectors in a Cartesian coordinate system is  ∂rsph ∂rsph    ∂r ∂vcar   car J=  , (5.75)  ∂vsph ∂vsph    ∂r ∂vcar   car 123 where  ∂rr ∂rr ∂rr   x y z   ∂x ∂y ∂z        r r r   ∂rsph  ∂r  θ ∂rθ ∂rθ     xz yz −x − y2 2  = = 2 ,  ∂rcar  ∂x ∂y ∂z   (x + y 2 )1/2r 2 (x 2 + y 2 )1/2r 2 (x + y ) r  2 2 1/2 2  ∂rφ ∂rφ ∂rφ      −y x    ∂x 0   ∂y ∂z   x 2 + y 2   x2 + y2   ∂rsph = 0, ∂vcar  ∂vr ∂vr ∂vr   ∂x ∂y ∂z     ∂vsph  ∂v  θ ∂vθ ∂vθ  = ∂rcar  ∂x ∂y ∂z    ∂vφ ∂vφ ∂vφ      ∂x ∂y ∂z    (y2 +z2 )vx −x yvy −xzvz −xyvx +(x 2 +z2 )vy −yzvz −xzvx −yzvy +(x 2 +y 2 )vz     r3 r3 r3      =  αvx2 +βv2 y3/2 +γvz βvx +ρvy +ηvz (x 2 +y 2 )1/2 (xvx +yvy +zvz )  ,   (x +y ) r3 (x 2 +y 2 )3/2 r 3 r3    x yvx +y 2 vy −x 2 vx −x yvy   0     (x 2 +y 2 )3/2 (x 2 +y 2 )3/2    ∂vr ∂vr ∂vr   x y z   ∂v x ∂v y ∂vz        r r r   ∂vsph  ∂v  θ ∂vθ ∂vθ   xz yz −(x 2 + y 2 )   = = 2 , ∂vcar  ∂v x ∂v y ∂vz   (x + y 2 )1/2r (x 2 + y 2 )1/2r (x 2 + y 2 )1/2r   ∂vφ ∂vφ ∂vφ      −y x    ∂v 0   x ∂v y ∂vz   (x 2 + y 2 )1/2 (x + y 2 )1/2 2   where α = −z(x 4 − y 4 − y 2 z2 ), β = −x yz(2x 2 + 2y 2 + z2 ), γ = −xz 2 (x 2 + y 2 ), ρ = z(x 4 + x 2 z2 − y 4 ), and η = −yz 2 (x 2 + y 2 ). We note that the determinant of J is J = 1/r 2 sin θ and thus Eq. (5.55) 124 becomes  ∂      ∂r    vx f                       ∂    v f     ∂θ      y                 ∂         v f     ∂φ     z  1   ∇X · F̄ = 2 · r sin θ J   .     2         (5.76) r sin θ  ∂   a f  ∂vr   x                    ∂       ∂vθ     a y f                           ∂     a f     ∂vφ      z             Computing the matrix-vector product defined in Eq. (5.76) and using Eq. (5.55), we have that the Vlasov equation in conservative form in spherical coordinates is v φ 1 ∂(r 2 sin θ r sin θ f ) v ∂f 1 ∂(r 2 sin θvr f ) 1 ∂(r 2 sin θ rθ f ) + + 2 + 2 ∂t r 2 sin θ ∂r r sin θ ∂θ r sin θ ∂φ n h  i o n h  i o 1 ∂ r sin θ ar + r vθ + vφ f 1 ∂ r sin θ aθ − r vr vθ − cot θvφ f 2 1 2 2 2 1 2 + 2 + 2 r sin θ ∂vr r sin θ ∂vθ 1 ∂ r 2 sin θ aφ − r vr vφ + cot θvθ vφ f 1    + 2 = 0. (5.77) r sin θ ∂vφ Simplifying the pre-factors yields ∂f 1 ∂(r 2 vr f ) 1 ∂(sin θvθ f ) 1 ∂(vφ f ) + 2 + + ∂t r ∂r r sin θ ∂θ r sin θ ∂φ nh  i o nh  i o ∂ ar + r vθ + vφ f 1 2 2 ∂ aθ − r vr vθ − cot θvφ f 1 2 + + ∂vr ∂vθ ∂ aφ − 1r vr vφ + cot θvθ vφ f   + = 0. (5.78) ∂vφ The Vlasov equation in conservative form in spherical coordinates also include acceleration terms that are a result of the coordinate system transformation – just like in cylindrical coordinates. These acceleration terms correspond to acceleration due to the Coriolis effect that manifests itself 125 as a product of radial and angular velocities – the terms vr vθ and vr vφ . Additional acceleration terms also appear due to centripetal acceleration which appear as squared velocities. We reiterate that only the ar , aθ, and aφ terms in Eq. (5.78) that are due to the Lorentz force; the additional acceleration terms are purely a result of the coordinate transformation. In Sec. 5.3.3, we present the 1D-1V representation of the Vlasov equation in the Cartesian, cylindrical, and spherical coordinate system. These 1D-1V representations are the starting point for the numerical methods we use for solving kinetic equations as discussed in Sec. 5.5.1. 5.3.3 The 1D-1V Vlasov Equation in Cartesian, Cylindrical, and Spherical Coordinates In this section we show the comparison of 1D-1V representations for Eqs. (5.47), (5.67), and (5.78). Equation (5.47) is the easiest to reduce to 1D-1V as the velocity and acceleration components do not depend on the differentiation variable. Thus, the 1D-1V conservative form of Eq. (5.47) is ∂f ∂ ∂ + (v x f ) + (a x f ) = 0. (5.79) ∂t ∂x ∂v x Mathematically, by 1D-1V we are enforcing that f (x, y, z, v x, v y, vz, t) = f (x, v x, t) and as a result, many of the partial derivatives equate to zero. In cylindrical coordinates, we must explicitly carry out the product rule as some of the terms in the fluxes depend on the differentiation variable. Equation (5.67) becomes vφ2 ∂ ! ∂ f vρ ∂ vφ ∂ ∂   + ρ f+f + f + vz f + a ρ + f (5.80) ∂t ρ ∂ρ ρ ∂φ ∂z ρ ∂v ρ ∂ ∂ ∂   vρ + aφ f− vφ f + f + az f = 0, (5.81) ∂vφ ρ ∂vφ ∂vz which in 1D-1V reduces to vφ2 ∂ ! ∂f ∂ + vρ f + aρ + f = 0, (5.82) ∂t ∂ρ ρ ∂v ρ which can also be readily expressed in conservative form: vφ2 " ! # ∂f ∂ ∂ + (v ρ f ) + aρ + f = 0. (5.83) ∂t ∂ρ ∂v ρ ρ 126 We follow a similar procedure for spherical coordinates where Eq. (5.78) becomes vθ2 + vφ2 ∂ ! ∂f ∂ 2vr f vθ ∂ cot θvθ f vφ ∂ ∂ + vr f+ + f+ + f + ar f+ f ∂t ∂r r r ∂θ r r sin θ ∂φ ∂vr r ∂vr vr f cot θvφ ∂ f 2 ∂ vr vθ ∂ ∂ vr vφ ∂ + aθ f+ f− + + aφ f− f ∂vθ r ∂vθ r r ∂vθ ∂vφ r ∂vφ vr f cot θvθ vφ ∂ cot θvθ f − − f− = 0, r r ∂vφ r where the underlined terms cancel. In 1D-1V the above equation reduces to vθ2 + vφ2 ∂ ! ∂f ∂ + vr f + ar + f = 0, (5.84) ∂t ∂r r ∂vr which can also be readily expressed in conservative form: vθ2 + vφ2 " ! # ∂f ∂ ∂ + (vr f ) + ar + f = 0. (5.85) ∂t ∂r ∂vr r The 1D-1V representations of the Vlasov equation in conservative form lend themselves nicely to numerical methods derived for hyperbolic conservation laws. Moreover, the assumed symmetries in 1D-1V result reduce the dimensionality of the problem from 6D to 2D decreasing the computational demands associated with numerically solving the Vlasov equation. In the case of the spherical coordinate system, radial symmetries are often observed in the expansion of UNPs into a vacuum when the UNP is initialized with a Gaussian density profile. In Sec. 5.4, we derive analytic models that approximate the radial expansion of a Gaussian plasma into a vacuum. our ultimate goal is to compare numerical simulations of the 1D-1V Vlasov equations to these analytic models to test their validity for single-species UNPs. Moreover, as no analytic models exist for describing the expansion of a multi-species UNP, these 1D-1V forms provide means of computing the evolution of UNP mixtures. 5.4 Plasma Expansion Into a Vacuum The density profiles of UNPs in the absence of magnetic fields are often initialized as a spherically symmetric Gaussian profile. Additionally, UNPs are created in a vacuum in which they expand radially outward. One goal of this chapter is to compare analytic models to numerical 127 simulations of Gaussian UNPs. If the analytic models derived below are valid, we avoid the computation cost associated with the kinetic simulations for these plasmas. In what follows, we will derive expressions for the time evolution of the ion density and electron temperature in two cases: an isothermal case, and an adiabatic case. The derivations presented here have been originally presented in Ref. [273]; for completeness, we provide them here and provide our own insights. To begin, we start with an expression for a Gaussian profile of the ion density ! r2 ni (r, t = 0) = ni0 exp − 2 , (5.86) σ0 where ni0 is the “peak" ion number density and σ0 characterizes the initial width of the plasma. We now consider the conservation laws for the ion species where we assume that the plasma is isotropic such that we can describe the expansion in one-spatial dimension. The continuity and momentum equation for the ion species are ∂ni ∂ + (ni ui ) = 0, (5.87) ∂t ∂x ∂ui ∂ 1 ∂ F + ui ui + p= , (5.88) ∂t ∂x mi ni ∂ x mi where ni = ni (x, t) and ui = ui (x, t). Note that, p = ni Ti is the ideal gas pressure where Ti = Ti (x, t) is the ion temperature in energy units and F is an internal or external force. For an initially cold plasma, we approximate Ti = 0 which also means that p = 0. We also assume that there are no external forces acting on the ions and that F describes the self-consistent electric field. That is ∂ F = Zi eE = −Zi e ϕ, (5.89) ∂x where ϕ = ϕ(x, t) is the electrostatic potential. The momentum equation with the above approxi- mations becomes ∂ui ∂ Zi e ∂ + ui ui = − ϕ. (5.90) ∂t ∂x mi ∂ x The momentum equation for the electrons is ∂ue ∂ 1 ∂ e ∂ + ue ue + (neTe ) = ϕ. (5.91) ∂t ∂x me ne ∂ x me ∂ x 128 Now we assume that the electron temperature is independent of space so that Te = Te (t). By quasineutrality we have ne = Zi ni . From the continuity equation from the electrons and ions we find that ui = ue . With these assumptions, we have that Eq. (5.91) is ∂ui ∂ e ∂ Te ∂ + ui ui = ϕ− ne . (5.92) ∂t ∂x me ∂ x me ne ∂ x Subtracting Eqs. (5.92) and (5.90) re-arranging terms we get ∂ Zi me e ∂ϕ   ne = ne 1 + . (5.93) ∂x mi Te ∂ x The solution of the above equation is    Zi me eϕ ne (x, t) = ne0 exp 1+ , (5.94) mi Te where ne0 = ne (x, t = 0). Since mi  me , we arrive at the familiar Boltzmann equilibrium distribution for the electron number density   eϕ ne (x, t) = ne0 exp . (5.95) Te We can once again make use of the quasineutral assumption that ne = Zi ni and Eq. (5.93) to write ∂ϕ Te ∂ = ni . (5.96) ∂ x ni e ∂ x Substituting the above equation into Eq. (5.90), we get ∂ui ∂ ∂ + ui ui = −cs2 ni, (5.97) ∂t ∂x ∂x where cs = (Zi Te /mi )1/2 and is known as the ion acoustic velocity. Equation (5.97) admits a self-similar [274] solution of σ0 x2   ni (x, t) = ni0 exp − 2 . (5.98) σ(t) σ (t) From Eq. (5.98), the quasineutrality assumption, and Eq. (5.95) we get that σ0 x2     eϕ ne0 exp = ne0 exp − 2 . (5.99) Te σ(t) σ (t) 129 By solving for ϕ we obtain Te x 2 σ0   Te ϕ = − 2 + ln 2 . (5.100) eσ (t) e σ (t) By substituting Eq. (5.98) into Eq. (5.87) we get ∂ σ0 x2 ∂ σ0 −x 2        ni0 exp − 2 + ni0 exp 2 ui = 0. (5.101) ∂t σ(t) σ (t) ∂x σ(t) σ (t) Applying the product rule we get x2 2x 2 dσ(t) x 2 dσ(t)       1 1 ni0 σ0 exp − 2 − 2 exp − 2 σ(t) σ (t) σ 3 (t) dt σ (t) σ (t) dt ni0 σ0 x 2 ∂ui x2      2xui + exp − 2 − exp − 2 = 0. σ(t) σ (t) ∂ x σ 2 (t) σ (t) Dividing the exponential terms and using the fact that σ −1 (t)dσ(t)/dt = d ln σ(t)/dt, we have ni0 σ0 2x 2 d ln σ(t) d ln σ(t) ni0 σ0 ∂ui     2xui − + − = 0. (5.102) σ(t) σ 2 (t) dt dt σ(t) ∂ x σ 2 (t) ∂ui d ln σ(t) Note that from the above equation, ∂x = dt implies that d ln σ(t) ui = x . (5.103) dt Substituting Eq. (5.103) into Eq. (5.102) yields ni0 σ0 2x 2 d ln σ(t) d ln σ(t) ni0 σ0 d ln σ(t) 2x 2 d ln σ(t)     − + − 2 = 0, (5.104) σ(t) σ 2 (t) dt dt σ(t) dt σ (t) dt which is a solution to the above equation. Substituting Eq. (5.103) into Eq. (5.90) gives d 2 ln σ(t) d ln σ(t) ∂ d ln σ(t) Zi e ∂ x +x x =− ϕ, (5.105) dt 2 dt ∂x dt mi ∂ x which by using Eq. (5.100) reduces to 2 d 2 ln σ(t) d ln σ(t)  Zi e Te 2x x 2 + = , (5.106) dt dt mi eσ 2 (t) which is 2 d σ(t) σ(t)    Û Û Zi e Te 2x x + = . (5.107) dt σ(t) σ(t) mi eσ 2 (t) 130 Û ≡ dσ(t)/dt. Taking another time derivative gives where σ(t) σ(t)σ(t) Ü −σ Û 2 (t) σ Û 2 (t) Zi e Te 2x x + = , (5.108) σ 2 (t) σ 2 (t) mi eσ 2 (t) which simplifies to d 2 σ(t) cs2 = 2 , (5.109) dt 2 σ(t) or d 2 σ(t) Zi Te (t) =2 , (5.110) dt 2 mi σ(t) where the time dependence of the electron temperature has been made explicit. In the sections that follow, we solve Eq. (5.110) for two approximations: and isothermal approximation and an adiabatic approximation. 5.4.1 Isothermal Expansion Isothermal expansion is one in which the temperature of a system remains constant during the expansion. This means that for a gas to expand and keep a constant temperature, an external energy source would need to provide energy to the expanding gas. Therefore, the statistical ensemble that this gas would occupy would be the canonical ensemble (NVT). In the context of the self-similar expansion model shown above, this would imply that the electron temperature Te is independent of space and time. With this assumption, we can integrate Eq. (5.110) from 0 to t gives  2  2 1 dσ(t) 1 dσ(0) − = 2cs2 ln σ(t) − 2cs2 ln σ(0). (5.111) 2 dt 2 dt Since the ions are initially at rest, dσ(0)/dt = 0 and Eq. (5.111) becomes s  σ(t)  dσ(t) = 2cs ln . (5.112) dt σ0 Solving Eq. (5.112) for dt and integrating both sides gives ∫ σ(t) 1 1 t= r . (5.113) 2cs σ0 h σ(t) i ln σ0 dσ(t) 131 Carrying out the integral in Eq. (5.113) results in √ s  πσ0 σ(t)    t=    erfi ln 2cs  σ0    √   1/2   3/2 ! πσ0 1 σ(t) σ(t)     2 = √ 2 ln + ln +··· . (5.114) 2cs π σ0 3 σ0 Retaining only the first term in the Taylor expansion of Eq. (5.114) gives   1/2 σ0 σ(t)   t= ln . (5.115) cs σ0 Solving for σ(t) in Eq. (5.115) yields ! cs2 t 2 σ(t) = σ0 exp . (5.116) σ02 For small t (i.e., t  σ0 /cs ), Taylor expansion of the exponential gives ! cs2 t 2 σ(t) = σ0 1 + 2 . (5.117) σ0 Using Eq. (5.117) in Eq. (5.98) gives an analytic expression for the self-similar evolution of ions in an isothermal system. 5.4.2 Adiabatic Expansion In adiabatic expansion, the total energy in the system in conserved. During plasma expansion in a vacuum there is a transfer of energy between the electrons to the ions. To account for this in our determination of σ(t), we begin with the temperature equation for the electrons in 1D 1 ∂(neTe ) ∂ ∂ ∂   + (neTe ue ) + qe + pe ue = 0. (5.118) 2 ∂t ∂x ∂x ∂x where qe and pe denote the heat flux and pressure of the electrons respectively. If we assume that qe = 0 and pe = neTe , Eq. (5.118) becomes 1 ∂(neTe ) ∂ ∂   + (neTe ue ) + neTe ue = 0. (5.119) 2 ∂t ∂x ∂x After selectively expanding out the derivative terms in the brackets and multiplying by 1/2, we get ∂Te ∂ne ∂ ∂Te ∂ ne + Te + Te (ne ue ) + ne ue + 2neTe ue = 0. (5.120) ∂t ∂t ∂x ∂x ∂x 132 The second and third term are zero from the continuity equation which reduces Eq. (5.120) to ∂Te ∂Te ∂ ne + ne u e + 2neTe ue = 0. (5.121) ∂t ∂x ∂x Also note that by the continuity equation Eq. (5.121) becomes ∂Te ∂Te ∂ne ne ne + ne u e − 2Te − 2Te ue = 0, (5.122) ∂t ∂x ∂t ∂x which can be written as 1 ∂Te ∂Te 2 ∂ne ∂ne     + ue = + ue . (5.123) Te ∂t ∂x ne ∂t ∂t Equation (5.123) can be solved to obtain a spatio-temporal description of the electron temperature. If we assume that the electron temperature is constant in space, Eq. (5.123) reduces to 1 dTe ∂ui = −2 , (5.124) Te dt ∂t where we have yet again used the continuity equation along with the relation ue = ui . From Eq. (5.103), Eq. (5.124) becomes 1 dTe 2 dσ =− . (5.125) Te dt σ dt Rearranging Eq. (5.125) and integrating from τ = 0 to t yields Te (0)σ 2 (0) Te (t) = . (5.126) σ 2 (t) Substituting Eq. (5.126) into Eq. (5.110) yields 2 σ2 cs0 dσ(t) = 2 3 0, (5.127) dt 2 σ (t) where we have defined cs0 ≡ (Zi Te0 /mi )2 . Integrating Eq. (5.127) yields σ 2 " #  2 dσ(t) = 2cs02 1 − 20 , (5.128) dt σ (t) which has a solution of σ(t) = (σ02 + 2cs0 t ) . 2 2 1/2 (5.129) 133 Using Eq. (5.126) we get that σ02 Te (t) = Te0 . (5.130) σ02 + 2cs0 2 t2 By substituting Eq. (5.129) into Eq. (5.98) we have an analytic model for the adiabatic expansion of a plasma along with an expression for the time evolution of the electron temperature. In Sec. 5.6.2, we compare numerical solutions of 1D-1V kinetic equation and compare our numerical results with the analytic formulae given by Eq. (5.98) using Eq. (5.117) and (5.129). We show that the expansion of Gaussian single-species UNPs are well-described by the adiabatic model. The focus of Sec. 5.5 is on the presentation of the numerical methods we use to solve the 1D-1V equations. 5.5 Numerical Methods for Kinetic Equations We numerically solve Eq. (5.1) in steps; the first step considers the advection terms on the left-hand side and the second step considers the collision operator on the right-hand side. We refer to the two steps as the “advection step" and the “collision step." We treat the advection step by using a second-order operator split method (Strang split) [275] to first advect the phase-space distribution function in physical space. Next, we advect the phase-space distribution function in velocity space and the advection step is complete1. After the advection step, we carry out the collision step which, for the BGK operator, consists of solving a time dependent ordinary differential equation. In the sections that follow, we detail the specifics of how we numerically treat each step. In Sec. 5.5.1, we discuss how we treat the advection step using a high-order finite volume method [262, 272, 276] (FVM). The component of the advection step that advects the phase-space distribution function in velocity space requires the determination of a self-consistent electric field. In Sec. 5.5.2, we describe an approach for obtaining the electrostatic potential – and thus the self- consistent electric field – via Poisson’s equation. In most cases studied here, we do not explicitly model the electron species with a kinetic equation. Instead, we approximate the electron number density with a linear Poisson-Boltzmann or non-linear Poisson-Boltzmann approach. In the linear 1 To be more precise, we advect in physical space with time step ∆t/2, use the results to advect in velocity space for ∆t, and then finally use that result and advect for ∆t/2. 134 case, the system is solved exactly via matrix inversion; in the non-linear case, we employ a Newton iteration. In both cases, the Poisson equation is discretized to second-order using a simple second order central finite-difference. In practice, one could explicitly simulate the electrons as their own species but due to their small mass, a much smaller simulation time step is needed to retain numerical stability. Therefore, unless otherwise mentioned, the electron species is assumed to be treated with a non-linear Poisson solve. In this way, the interactions between the electrons and ions are implicitly accounted for in calculation of the self-consistent electric field through the total charge density. The numerical treatment of the collision step varies in different stages of this work. For the 0D- 1V simulations, the method of lines [277] is employed to allow the use of high-order implicit time integrators that are accessible in extant computational libraries. For the 0D-3V BGK simulations, we employ an explicit fourth-order Runge-Kutta time integrator. For the 1D-1V simulations, we employ a simple first order explicit time step for the BGK operator. 5.5.1 Finite Volume Methods In contrast to finite difference methods (FDMs), FVMs are derived for numerically solving conservation laws – the governing equation of this chapter. We begin our discussion of FVMs by considering a conservation law in one space dimension that is in differential form ∂q(x, t) ∂ + f [q(x, t)] = 0, (5.131) ∂t ∂x where q(x, t) is some conserved quantity (e.g., total number density of a system), and f (q) is a flux. Finite difference methods numerically discretely approximate the derivatives in Eq. (5.131) using “stencils" – the choice of stencil distinguishes FDMs. In contrast, FVMs instead work with conservations laws in integral form where we integrate Eq. (5.131) over some volume Vi = (xi−1/2, xi+1/2 ) that yields ∂ ∫ ∫ d q(x, t) dV + f [q(x, t)] dV = 0, (5.132) dt Vi Vi ∂x 135 which reduces to ∫ d q(x, t) dV + f [q(xi+1/2, t)] − f [q(xi−1/2, t)] = 0. (5.133) dt Vi With the definition that ∫ 1 Qi = q(x, t) dV, (5.134) Vi Vi we have d 1  Qi = f [q(xi−1/2, t)] − f [q(xi+1/2, t)] = 0. (5.135) dt Vi Equation (5.135) represents the time evolution of average flux of q in some volume Vi . Integrating Eq. (5.135) over a time interval of tn to tn+1 gives ∆t Qin+1 = Qin − Fi+1/2 − Fi−1/2 ,  (5.136) ∆x where we have assumed a simple Forward Euler time integration and have made the definition that ∫ tn+1 Fi+1/2 = f [q(xi+1/2, t)]dt. (5.137) tn It is the choice of how we approximate the flux Fi+1/2 , with an appropriate numerical flux that distinguishes FVMs from one another. Different choices of numerical fluxes usually result in methods of varying orders of accuracy or methods which attempt to control oscillations that occur in discontinuous solutions. A similar choice appears in FDMs where instead of choosing a numerical flux, we choose a numerical derivative (i.e., first-order forward, second-order central, etc.). Additionally, one could choose a different time integration for example, Backward Euler or a fourth-order Runge-Kutta method. In this work, we implement three different choices for the numerical flux function. The first is the first-order upwinding flux [276], a second-order superbee flux limiter [276], and a fourth-order upwinding flux [262,272]. A convergence study for each of these choices is displayed in Figure 5.1 for the linear advection equation  ∂u(x, t) ∂u(x, t) +v = 0 x ∈ [0, 1],    ∂t ∂x   (5.138)  u(x, 0) = exp −32(x − 0.5)2 cos[16π(x − 0.5)],       136 Figure 5.1: Comparison of different numerical flux functions for the linear advection equation. (a) numerical solution of the advection equation for a wave packet initial condition [Eq. (5.138)] with periodic boundary conditions. The wave passes two times through the domain and comparisons are shown with first, second, and fourth order methods; The first order method severely damps the numerical solution. (b) grid resolution convergence test. The first, second, and fourth order methods scale appropriately with grid resolution. where we set v = 1 and we assume periodic boundary conditions. For the first and second-order method, a forward Euler time step was implemented, for the fourth-order upwinding flux a fourth- order Runge-Kutta time integrator was used as described in Ref. [262,272]. Figure 5.1(a) shows the numerical solution for each choice of numerical flux function with 2 passes through the domain. In contrast to the second- and fourth-order methods, we see that the first-order upwind method severely dampens the solution. Figure 5.1(b) shows a grid resolution convergence study using the L∞ -norm for each method; we have added three lines with slopes 1, 2, and 4, to guide the eye. We see that for smooth problems, each choice of numerical flux gives a reasonable approximation to the true solutions albeit in the first-order case, the numerical solution may be severely damped. For non- smooth problems, we cannot use the fourth order upwinding flux as it will introduce oscillations near the discontinuity. The appearance of oscillations from using the fourth-order upwinding flux 137 Figure 5.2: Comparison of numerical fluxes for a square pulse initial condition. Note that the fourth order upwinding scheme introduces oscillations in the numerical solution near the discontinuities; the first-order method does not introduce unphysical oscillations but smears the numerical solution. is shown in Fig. 5.2 where we assume  ∂u(x, t) ∂u(x, t) +v =0  x ∈ [0, 1],   ∂t ∂x        u(x, 0) = 1 |x| ≤ 0.5, (5.139)     u(x, 0) = 0 otherwise,     where again we let v = 1 and assume periodic boundary conditions. We observe that the first- order upwinding and second-order superbee numerical fluxes do not introduce oscillations near the discontinuity in contrast to the fourth-order upwinding flux. The comparison of the different numerical fluxes informs us on their appropriateness for solving kinetic equations. We conclude that if we expect the phase-space distribution function to be smooth for all time, a high-order upwinding method is appropriate for treating the advection terms on the left-hand side of Eq. (5.1). The smoothness of the phase-space distribution function is largely set by the initial condition but discontinuities may appear as a result of self-consistent forces or collisions in the plasma. For UNP simulations, the phase-space distribution function remains smooth for the problems studied in this chapter and we employ the fourth-order upwinding flux. However, if the problem of interest 138 includes a phase-space distribution with discontinuities, as is the case when there is an interface between two plasma species, a slope limiter methods, essentially non-oscillatory methods, or weighted essentially non-oscillatory, methods [278] are necessary to achieve higher than first-order accuracy. 5.5.2 Poisson’s Equation in 1D Cartesian Coordinates In 1D Cartesian coordinates Poisson’s equation has the form d2 ϕ(x) = −4πρ, (5.140) dx 2 where ρ = ÍN i Zi eni is the total charge density of the system. We use a second order central difference approximation to Eq. (5.140) so that at some grid point i = 0, 1, · · · , N, we have ϕi−1 − 2ϕi + ϕi+1 = −4πρi . (5.141) ∆x 2 If we impose periodic boundary conditions at i = 0 and i = N, then we have that ϕ0 = ϕN and construct a linear system that is solved to obtain ϕ. Specifically, for a 6 cell system the linear system takes the following form:  ϕ0   ρ0      −2 1 0 0 0 1              ϕ  ρ     1  1  1 −2 1 0 0 0       ϕ2  ρ          = −4π∆x 2  2  .    0 1 −2 1 0 0  (5.142)  ϕ3   ρ3           0 0 1 −2 1 0              ϕ  ρ     4  4  1 0 0 0 1 −2      ϕ5   ρ5            139 Since the above system is singular, an additional constraint specifying that the average value of the potential is zero is introduced which results in the modified linear system  ρ0    −2 1 0 0 0 1  ϕ0                ρ1   1 −2 1 0 0 0  ϕ      1    ρ2         0 1 −2 1 0 0  ϕ2           = −4π∆x 2  ρ  . (5.143)  0 0 1 −2 1 0  ϕ3      3          ρ4   1 0 0 0 1 −2 ϕ       4   ρ      5  1 1 1 1 1 1   ϕ5            0   Equation (5.143) can be solved using various linear algebra techniques that exist in extant computational libraries. If using the electron density as defined in Eq. (5.94), a non-linear approach must be used since the total charge density will non-linearly depend on ϕ; in this work, a Newton iteration was implemented to determine ϕ. To avoid a non-linear solve, one can linearize Eq. (5.94) which results in a linear system that can be solve exactly; Figure 5.3 shows a comparison of a numerical solution from the linear case of Eq. (5.143) with a manufactured analytic solution. We note that the linearization holds in the high electron-temperature limit. Additional approaches for treating the electrons as quantum mechanical with Fermi-Dirac statistics can be found in Ref. [44]. The self-consistent electric field can be obtained via Eq. (5.6), where second-order central difference has been employed. 140 Figure 5.3: Electrostatic potential ϕ(x) computed from Eq. (5.143) and compared against a manu- factured analytic solution ϕ(x) = sin(2πx) with ∆x = 0.001. 5.5.3 Poisson’s Equation in 1D Spherical Coordinates To convert Eq. (5.7) to spherical coordinates, we transform the gradient into spherical coordi- nates to obtain   1 d 2d r ϕ = −4πρ. (5.144) r 2 dr dr To discretize Eq. (5.144), we use a FVM framework. Importantly, the choice of FVM over FDM circumvents singularities at r = 0. Additional numerical implementations as well as extensions to higher dimensions can be found in Refs. [279,280]. Following the procedure outlined in Ref. [281], we obtain the discretization of Eq. (5.144) (r 2 F̃r )i+ 1 − (r 2 F̃r )i− 1 2 2 = −4πρi . (5.145) ∆Vr,i where we define the flux F̃r ≡ dϕ/dr and ∆Vr,i = (ri+1/2 3 3 − ri−1/2 )/3. Evaluating the fluxes at the cell interfaces we have r2 r2 i+ 21 i− 12 4πρi F̃r,i+ 1 − F̃r,i− 1 = − . (5.146) r3 1 − r3 1 2 r3 1 − r3 1 2 3 i+ 2 i− 2 i+ 2 i− 2 We approximate the fluxes with an upwinding numerical flux so that Eq. (5.146) becomes r2 ϕ r2 1 i+ 12 i+1 − ϕ i  i− 2 ϕ − ϕ  i i−1 4πρi − 3 =− . (5.147) r3 1 − r3 ∆r r 1 − r3 ∆r 3 i+ 2 i− 12 i+ 2 i− 21 141 Multiplying by ∆r and grouping terms of ϕ with the same index we get r2 r2 − r2 1 r2 1 i− 12 © i+ 12 i− 2 ª i+ 2 4πρi ∆r ϕi−1 + ϕ i + ϕi+1 = − . (5.148) r3 3 3 3 3 3 ­ ® −r 1 r 1 −r 1 r 1 −r 1 3 i+ 12 i− 2 « i+ 2 i− 2 ¬ i+ 2 i− 2 The boundary conditions we use at the left, and right endpoints of the grid are chosen to be dr ϕ r=0 = 0 at the left boundary and an outflow boundary at the right boundary. Together, d Eq. (5.148) and these boundary conditions generate the matrix (for a 6 cell system) r2 r 21 − 21   2  r 3 −r 3 r 31 −r 3 1 0 0 0 0   1 −1 − ϕ0   ρ0        2 2 2 2   r 21 r 23 +r 21 r 23             323 − 2 2 2 0 0 0  ϕ  ρ   r 3 −r 1 r 31 −r 3 1 r 33 −r 31   1  1  2 2 − 2 2 2 2       r 23 r 25 +r 23 r 25  ϕ2     ρ2      = − 4π∆r   2 2 2 2  0 r 35 −r 33 − r 33 −r 31 r 35 −r 33 0 0    (5.149)  ϕ3    3  ρ3    2 2 2 2 2 2   r 25 r 27 +r 25 2        r7       0 0 0 2 − r 32 −r 32 2  ϕ  ρ   r 37 −r 35 r 37 −r 35   4  4  7 5     2 2 2 2 2 2  ϕ5   ρ5        r 29 −r 211 r 211 +r 29  2 2 2 2   0 0 0 0      r 311 −r 39 r 311 −r 39     2 2 2 2  We validate our numerical method against the test case shown in Appenxdix A of constant charge density inside a sphere. In Fig. 5.4, we see a comparison of the analytic solution to our numerical solution for the electric field. 142 Figure 5.4: Electric field E(r) computed the numerical method Eq. (5.5.3) and compared against the analytic solution Eq. (A.17). This case considers a sphere of size R = 2, with constant density of ρ0 = 1, and ∆r = 0.01, all with arbitrary units. We see that for r ≤ 2, the numerical method correctly predicts the linear electric field and then transitions to the quadratic decay for r > 2. 5.6 Numerical Results Using the numerical methods described in Sec. 5.5, we simulate a range of problems related to UNPs and HED plasmas. We begin in Sec. 5.6.1 by simulating plasmas with zero spatial dimension where we focus on three distinct applications. The first application focuses on the verification of our numerical implementation in the context of temperature and momentum relaxation for HED plasmas. The second application focuses on the validation of our numerical implementation by simulating tem- perature relaxation in multi-species UNPs where we find that our kinetic model agrees well with experimental and MD simulation data. The third application is concerned with simulating the velocity distribution tail filling rate in an electron UNP where we compare two different collision operators. In Sec. 5.6.2, we extend the scope of our numerical study to include one spatial dimension with two particular applications. Namely, the expansion of single- and multi-species UNPs into a vacuum, and the diffusion of materials across an interface in HED mixtures. First, we find that the single-species UNP expansion is adiabatic and that the plasma is so quickly driven to 143 equilibrium by collisions, that the expansion is well described by the Vlasov equation. Because of this, our simulations suggest that the expansion of a single-species UNP is time-reversible. Next, we compare results from our numerical simulations to results from the analytic models derived in Sec. 5.4 and find that although the assumption that the electron temperature is spatially independent is invalid for the cases studied here, the overall dynamics of the UNP do not appear to be significantly impacted by this choice. Then, for the case of multi-species UNPs, we find that our multi-species BGK simulations disagree with multi-species Vlasov simulations which implies that thermodynamic forces such as heat flux, viscous flux, and diffusive fluxes generate entropy in the plasma. Lastly, we carry out numerical simulations of diffusive mixing in HED experiments. We find that the dominant drivers of diffusion in our simulations are caused by electrodiffusion: diffusion driven by electric fields. The kinetic calculations in all sections of this chapter except for those reported in Sec. 5.6.2.3 were numerically implemented in the “Python" programming language; the specific version used here is Python 3.7. To increase the performance of our Python code, which is an interpreted lan- guage, we (i) utilized numerically efficient pre-compiled libraries like NumPy [282] and SciPy [283] and (ii) compiled non-compiled portions of the code separately using the Numba just-in-time com- plier [284]. For the latter, we saw drastic performance gains; in many cases the computation cost was reduced by orders of magnitude. All simulations, except for those reported in Sec. 5.6.2.3, were carried out on a single core of a 2.7 GHz Core i7-8559U CPU. In Sec. 5.6.2.3, a variant of the multi-component BGK code2 – which is written in the “C" programming language – was used. 5.6.1 0D Kinetic Simulations Kinetic equations are composed of advection terms – the left-hand side of Eq. (5.1) – and a collision operator – the right hand-side of (5.1). We begin our presentation of numerical results by first verifying the multi-species BGK collision operator is conservative and also validate it against experimental data. By reducing Eq. (5.1) to 0D-3V, i.e., retaining only the velocity degrees of freedom, we carry out a direct numerical study of the collision operator. For the multi-species 2 See: https://github.com/lanl/Multi-BGK. 144 BGK collision operator, Eq. (5.1) becomes N dfi Õ = νi j [Mi j (v; ni, ui j , Ti j ) − fi ], j = 1, 2, · · · , N. (5.150) dt i=1 5.6.1.1 Verification of the Multi-Species BGK Operator To verify the multi-species BGK collision operator, Eq. (5.150), we compare our implementation to one reported in Ref. [267]. We compare our numerical results to a test case of a C6+ -H+ mixture where nC = nH = 1 × 1023 cm−3 , the initial temperatures of each species are TC = 10 eV, and TH = 12 eV, and their initial velocity vectors are uC = (1.26 × 105, 0, 0) cm s−1 , and uH = (0, 0, 0) cm s−1 . For the electron species, we set ne = 2 × 1023 cm−3 and Te = 11 eV. The equilibrium temperature and momentum can be calculated by via the conservation of energy and momentum. The equilibrium velocity of the mixture is m1 n1 u12 + m2 n2 u22 2 ueq = , (5.151) ρ which for the above conditions is ueq 2 ≈ 1.162 × 105 cm s−1 . Similarly, for the temperature of the system, conservation of energy yields n1T1 + n2T2 + m1 n1 u12 + m2 n2 u22 − ρueq 2 Teq = , (5.152) n which for the conditions above is Teq ≈ 11.0 eV. In our implementation, Eq. (5.150) is numerically solved using a fourth-order explicit Runge- Kutta time integrator with a time step of ∆t = 1 × 10−16 s. The 3D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 4 × T/m cm/s with a grid resolution of ∆v = 0.2 p cm/s in each velocity direction. The simulations took approximately 40 minutes to complete. We note that these particular simulations are highly resolved in time in order to compare our results to data in the literature; a time step that is a factor of 10 larger appears to be sufficient for obtaining data and a simulation using this time step takes roughly 4 minutes to complete. We compare our results with those reported in Ref. [267]. Our comparison is displayed in Figure 5.5 where we compare our calculations with the results provided in Figure 2 of Ref. [267]; we denote their data 145 Figure 5.5: 0D-3V multi-species BGK simulations of a C6+ -H+ mixture with plasma conditions described in Sec. 5.6.1.1. The collision rate model used here is given by Eq. (5.16). (a) temperature relaxation and (b) shows momentum relaxation between the species. Note that in the absence of electron screening (1/λe = 0), the system equilibrates more quickly because the collision rate is larger. Our numerical results without electron screening match data provided in Ref. [267]. The equilibrium velocity and temperature values are computed from Eq. (5.151) and (5.152) respectively; our results show that the multi-species BGK operator approaches the correct result; this is not the case for some multi-species BGK operators [267]. by “HHM." We find that by setting the inverse electron screening length to zero in Eq. (5.19), we recover their results. By including a non-zero inverse electron screening length (solid lines), we see that in Figure 5.5(a) and Figure 5.5(b), the relaxation time is longer because the collision rate decreases due to the presence of electron screening. Electron screening plays an important role in many HED and UNP experiments and we include it by using Eq. (5.20) in Eq. (5.19) for all results shown in this chapter. 5.6.1.2 Validation of the Multi-Species BGK Operator Now that we have verified our numerical scheme, we validate it with data of temperature relaxation in UNP mixtures. In Figure 5.6, we compare results from our multi-species BGK model to experimental and simulation data for a UNP Ca+ -Yb+ mixture at two different conditions (see Refs. [260] and [24]). In both simulations, the 3D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 4 × T/m cm/s with a grid resolution of ∆v = 0.2 cm/s in p each velocity direction; an explicit fourth-order Runge-Kutta time integrator was employed with 146 Figure 5.6: 0D-3V multi-species BGK simulations of a Ca+ -Yb+ UNP mixture with plasma conditions described in Sec. 5.6.1.2. The collision rate model used here is the temperature relaxation rate given in Eq. (5.16). We show the temperature relaxation predicted by the BGK model compared to (a) experimental data from Ref. [260] and (b) MD data obtained from Ref. [24]. We find that the BGK model accurately predicts the temperature relaxation rate compared to both experimental and MD data. These results provide confidence in the use of the BGK model for modeling UNP mixtures beyond the disorder-induced heating region which is labeled as “DIH" and denoted with a grey shaded rectangle in (a) and (b). time step of ∆t = 1 × 10−7 s. In Fig. 5.6(a) we set the ion densities to be nCa+ = 2.9 × 109 cm−3 and nYb+ = 1.9 × 109 cm−3 . The initial ion temperatures are computed from the average of the experimental data from t = 0.4 to 0.5 µs which results in TCa+ = 1.9 K, and TYb+ = 0.8 K. The electrons have a temperature of Te = 96 K and we assume the electron temperature decreases according to a modified form of the adiabatic model Eq. (5.130) which is Te0 Te (t) = , (5.153) 1 + t 2 /τ̄ 2 where Te0 = Te (0) and τ̄ ≡ σ̄0 /c̄s0 . We define the mixture width and sound speed parameters σi,0 ni + σj,0 n j σ̄0 = , (5.154) n hZ 2 /mi i 2 c̄s0 = Te0 i , (5.155) hZi i where n = ni + n j , hAiα i ≡ xi Aiα + x j Aαj , and xi = ni /n denotes the number concentration of species i. Lastly, we assume that the drift velocities are uCa+ = uYb+ = 0. We see that our numerical results well approximate experimental data for temperature relaxation and can be used to extrapolate past 147 the disorder induced heating region denoted as "DIH." In Fig. 5.6(b) we compare our multi-species BGK results against temperature relaxation data from an MD simulation. For this case, the ion densities are nCa+ = 4.3 × 109 cm−3 and nYb+ = 1.3 × 1010 cm−3 . The initial ion temperatures are determined from the average of the MD data from t = 0.4 to 0.5 µs which results in TCa+ = 3.27 K, and TYb+ = 1.97 K. The initial electron temperature is Te0 = 100 K and its time dependence is given by Eq. (5.153). Additionally, we set uCa+ = uYb+ = 0. For this case, we see that our multi-species BGK model accurately predicts the equilibrium temperature to within 4%; this discrepancy is due to uncertainties from the approach used to compute the initial ion temperatures. To account for these uncertainties, an ensemble of simulations could be carried out with different initial ion temperatures. These temperatures could be generated by sampling from a normal distribution of temperatures with mean and variance determined by experimental or MD data. Additionally, we can measure the rate at which equilibration of these simulation occurs by computing the entropy source term in Eq. (5.37). Figure 5.7 shows the entropy source term for both cases in Fig. 5.6. It is important to note that by using a kinetic simulation in place of MD, we drastically reduce the dimensionality of the problem – from 6N dimensional to 3 dimensional – which decreases the computation time by a factor of roughly 3000. Specifically, the MD results reported above state an MD run – using the Sarkas MD code [285] – took approximately 20 hours to complete on a single core of an Intel Core i7-8700K CPU for a total simulation length of approximately 4 µs. In contrast, our 0D-3V kinetic simulations took approximately 3 minutes and 51 seconds for a total simulation time length of 100 µs. 5.6.1.3 Tail Filling in UNPs The previous sections were concerned with multi-species ion plasmas. In this section, we change our focus to simulating the equilibration of a velocity distribution of a single-species electron plasma with conditions relevant to UNPs. Specifically, we consider an electron plasma with a truncated distribution function that is nearly zero over some range of velocities: a depleted tail. An example of physical systems in which this type of distribution function occurs is in situations where evaporation occurs. In this scenario, the highest energy particles escape and leave 148 Figure 5.7: Entropy source term calculated from Eq. (5.37) from the simulations displayed in Fig. 5.6. (a) corresponds to the plasma conditions in Fig. 5.6(a) and (b) corresponds to the plasma conditions in Fig. 5.6(b). We see that certain terms correspond to a larger amount of entropy than others, specifically, the Yb+ -Yb+ and cross-species collisions. Since entropy is being generated, the system is irreversible. The source of irreversibility is associated with transport coefficients like the interdiffusion, thermal conductivity, and viscosity of the plasma. behind particles with a missing high-energy tail. Thermonuclear fusion also results in a distribution function with a depleted tail by converting the fast tail particles into different species through a nuclear reaction. The question we aim to answer with kinetic theory is: how long does it take for these distribution functions to become Maxwellian? Ultracold neutral plasmas provide a unique laboratory for studying the rate at which the tails of the velocity distribution repopulate because their low densities effectively slow physical processes to a measurable time scale. To examine just the physics of the forms of the collision operators, we consider a kinetic model with zero spatial dimensions and one velocity dimension (0D-1V). Numerical results are obtained for spatially homogeneous tail filling. In 0D-1V, Eq. (5.1) has the form ∂ fe = Q( fe, fe ), (5.156) ∂t where fe ≡ fe (v, t) is the distribution function of an electron species. We will consider two forms for the collision operator in this section: the BGK and LBD operators which in 0D-1V have the 149 form Q BGK = νee [M e (v; ne, ue, Te ) − fe ] , (5.157) ∂ 2 ∂   Q LBD = νee (v − ue ) fe + vth fe . (5.158) ∂v ∂v We discretize each of the collision operators using the method of lines approach allowing us to use high-order time integrators3 in extant software libraries [283]. The initial Maxwellian distribution function is shown in Fig. 5.8 and is given by   1/2   me −me M e (v) = ne exp (v − ue ) . 2 (5.159) 2πTe 2Te Here, we set ne = 3 × 107 cm−3, ue = 0 cms−1 , and Te = 3.5 K which are conditions relevant to UNPs. We “truncate" the tails of the equilibrium distribution to remove the fastest 1% of electrons on either side as shown in Fig. 5.8; this reduces the electron temperature to roughly Te = 3.1 K. Using the truncated distribution function as our initial condition, we carry out simulations using the BGK and LBD operator to measure to rate at which the tails of the truncated distribution replenish/fill. For these simulations, the distribution function was first truncated, and the collision rate νee is calculated using eight distinct models – see Table 5.2. We simulate the expansion of the electron distribution function for a total simulation length of 0.2 µs with a time step of ∆t = 0.002 µs. The 1D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 4 × T/m cm/s with a grid resolution of ∆v = 0.026 cm/s. p Figure 5.8 shows the initial condition at t = 0 and at the final time t = 0.2µs. We see that the BGK and LBD operators equilibrate to the same equilibrium distribution to within 1% at the peak. The distribution does not return to its initial condition prior to truncation because of evaporation: the high-energy tails have been removed; this results in an equilibrium distribution function that has a larger magnitude at v/vth = 0. In Fig. 5.9(a), we compare the time evolution of the distribution function using the BGK and LBD collision operators with the HHM collision rate given in Table 5.2. By plotting the difference of the distribution functions, we highlight the impact each collision operator has on the tail-filling rate. Because the LBD operator incorporates 3 We used the “Radau" time integrator for this work. 150 Model ln Λ νee (Hz×10−7 ) Description Ref. LS 1.41 3.96 Straight-line trajectories [270] GMS-1 1.41 3.96 Straight-line trajectories [269] GMS-2 1.71 4.80 Straight-line trajectories [269] GMS-3 2.00 5.62 Straight-line trajectories [269] GMS-4 1.44 4.04 Hyperbolic trajectories [269] GMS-5 1.72 4.84 Hyperbolic trajectories [269] GMS-6 1.72 4.84 Hyperbolic trajectories [269] Correlations beyond HHM – 3.41 [267] binary interactions Table 5.2: A comparison of the numerical values of the electron-electron collision rate νee . All values of νee were obtained via Eqs. (5.25) and (5.26) with values of ne = 2.9 × 107 gcm−3 and Te = 3.1 K; these values of ne and Te correspond to the electron density and temperature of the truncated distribution shown in Fig. 5.8. velocity gradients in its functional form, the tails fill more quickly than the BGK operator. We see that after t ≈ 0.12 µs, the difference between the LBD and BGK operator is small and the truncated distribution function has equilibrated. To measure the rate of tail filling, we compute the ratio of the number of electrons in the truncated regions to the number of electrons in the tail of the full distribution N0 given by Eq. (5.159); results using both the BGK and LBD operator using all eight of the collision frequencies given in Table 5.2 are shown in Fig. 5.9(b). Because the initial condition of the distribution function contained essentially infinite gradients at the truncated regions, we expect the LBD operator to drive the distribution function to equilibrium more quickly than the BGK operator which relaxes the the distribution function to equilibrium by the difference of Me and fe . At a high level, we have found that two models result in two different tail-filling rates which is not surprising. However, one model – the LBD operator – predicts a four times quicker tail filling rate than the BGK operator. The difference in the tail-filling rate predicted by the BGK and LBD models is non-negligible and should be compared to experimental data in order to appropriately simulate these processes using kinetic models. Additionally, by increasing the electron temperature, we find that the collision frequencies begin to converge to the same results in contrast to the results shown in Fig. 5.9(b). To 151 Figure 5.8: Initial velocity distribution of electrons with ne = 3 × 107 cm−3 , and Te = 3.5 K. The “full" and “truncated" distributions overlap until |v/vth | ≈ 2.3, where the tails of the full distribution have been truncated beyond this amount; this removes the fastest 1% of the electrons on either side of the distribution. We show the equilibrated velocity distribution function using the BGK and LBD collision operators with the HHM collision rate; both collision operators results in the same equilibrated distribution function. Note that the distribution function does not return to its original shape because of evaporation. maximize the benefit of this analysis, an ensemble of experiments should be carried out at different temperatures to assess the validity of different collision operators and collision rate models. 5.6.2 1D Kinetic Simulations We now extend our kinetic simulations to include one spatial dimension where we focus our simulations on quantifying (i) the expansion of single-species UNPs, (ii) the expansion of multi- species UNP, and (iii) the diffusive mixing in HED plasma mixtures. 5.6.2.1 Expansion of Single-species UNPs In this section, we answer the following three distinct questions: (1) can we approximate the electron species with analytic formulae avoiding the need for explicit calculations [See Eqs. (5.130) and (5.94)] (2) what is the role of a collision operator for single-species UNPs and (3) are single- species UNPs time-reversible? For the simulations carried out here, the initial ion density is 152 Figure 5.9: (a) Difference of distribution functions using the BGK and LBD operator with HHM collision rate versus velocity and time. The dashed vertical lines mark that locations where the full distribution was truncated (see Fig. 5.8). The negative values show where the distribution function evolved using the LBD operator is smaller than the corresponding BGK operator opinion. Large gradients at the truncation locations cause the LBD operator to quickly fill the tail regions in contrast to the BGK operator which does not use information of velocity gradients. (b) tail filling rate using the BGK and LBD collision operators in 0D-1V dimensions (spatially homogeneous) with the collision rates given in Table 5.2. N0 denotes the number of electrons originally in the tails of the full distribution before truncation (see Fig. 5.8). Approximately 60% of the original number of electrons replenish the tail after equilibration which occurs at t ≈ 0.12 µs. assumed to be Gaussian and given by ! x2 ni (x, t = 0) = ni,0 exp − 2 , (5.160) 2σi,0 where ni,0 is the initial “peak" ion density and σi,0 is the initial width of the plasma. We begin by answering the question: can we approximate the electron species with analytic formulae and avoid the need for explicit calculations? The main assumptions we will check in deriving the analytic formulae given by Eqs. (5.130) and (5.94) is the spatial independence of the electron temperature and that the electron density can be well-approximated by Eq. (5.94). To do this, we carry out a multi-species Vlasov simulation of a UNP mixture of Ca+ ions and an electron species where we approximate the electron’s mass to be me = mCa+ /183. While this is not the true proton-electron mass ratio, it should be a good approximation for the electron species in that the electrons respond relatively quickly to the dynamics of the ion species. Additionally, since the electrons in an UNP are classical – see Table 5.1 – we can directly use Eq. (5.1) without introducing 153 Figure 5.10: Density profiles for a Ca+ -e UNP computed from a multi-species Vlasov simulation. Two simulation snapshots are shown: one at t = 0 µs and another at t = 2 µs. The density profile obtained from the adiabatic self-similar hydrodynamics model [Eq. (5.129)] is also shown. The density evolution of the Ca+ species agrees with the self-similar hydrodynamics model to sub-percent accuracy. quantum statistics in the distribution function. We initialize our Ca+ -e mixture with conditions relevant to ongoing UNP experiments. For the case shown here, we initialize the peak ion density to be nCa+,0 = 1 × 109 cm−3 with a width of σCa+,0 = 3 mm and an ion temperature of TCa+ = 1 K. The initial density of the electron species is calculated self-consistently from the ion density with an initial temperature of Te = 100 K. We then solve the multi-species Vlasov equation [Eq. (5.5)] where the self-consistent electric field is obtained with a fast-Fourier transform [262]. The total simulation length time was 2µs with a Courant–Friedrichs–Lewy (CFL) value of 0.4. The 1D spatial grid is initialized with an upper (and lower) bound of magnitude |xmax | = 0.5 cm with a grid resolution of ∆x = 0.001 cm. The 1D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 40 × T/m cm/s with a grid resolution of ∆v = 0.02 cm/s. For the advection p terms, we use a fourth-order upwinding FVM reconstruction. The simulation took approximately 60 hours to complete. 154 Figure 5.11: Velocity profiles of a Ca+ -e UNP computed from a multi-species Vlasov simulation at various time steps. The dotted lines denote T = 0.03 µs, the dashed lines denote t = 0.08 µs, and the solid lines denote t = 0.95 µs. After roughly t = 0.1 µs, the velocity profiles of Ca+ and the electrons “lock" together i.e., the process of momentum relaxation. Since the momentum relaxation occurs on a timescale much quicker than the ion dynamics, the assumption of instantaneous momentum relaxation – the basis for the derivation of the self-similar hydrodynamics models – is valid. The simulation results of the Ca+ -e expansion are shown in Fig. 5.10. Initially, the electrons and ions differ the most in the tails as shown by the insert of Fig. 5.10. At t = 2µs, the ions agree with the adiabatic expansion model to within 0.5% at the peak density (x = 0 cm). Moreover, the electron density retains the form of Eq. (5.94) from t = 0 to 2 µs suggesting that the Poisson-Boltzmann form [Eq. (5.94)] is sufficient in describing the electron density in an expanding UNP. Figure 5.11 shows the velocity profiles for different times throughout the simulation. We see that momentum relaxation between the ions and electrons occurs roughly on the order of 0.1 µs which is much faster than the dynamics of the ion species. We note that the Poisson-Boltzmann approximation essentially enforces instantaneous momentum relaxation which appears to be a valid assumption for the case of the Ca+ -e UNP mixture. Lastly, we assess the temperature evolution of the electron species – the input to the Poisson- Boltzmann expression Eq. (5.94) – and compare our simulation results to the adiabatic model Eq. (5.130). The comparison of the simulation and the adiabatic expansion model Eq. (5.130) is 155 Figure 5.12: Electron temperature of a Ca+ -e UNP computed from a multi-species Vlasov sim- ulation. (a) the electron temperature at the center of the plasma compared to the self-similar hydrodynamics model Eq. (5.130). The electron temperature is within ∼2% for all time as shown by the insert. (b) the electron temperature versus space and time. The adiabatic √ self-similar hydro- dynamics model Eq. (5.130) agrees well with an isothermal model for 2σ0 /cs0  1. Although there is a spatial dependence of the electron temperature, 90% of the plasma is within a region of space for which the electron temperature is within 10% of the self-similar hydrodynamics model Eq. (5.130). This substantiates the approximation of a spatially independent electron temperature in the derivation of the self-similar hydrodynamics models. displayed in Fig. 5.12. Figure 5.12(a) shows that that the electron temperature from the Ca+ -e simulation results agree to within ∼2% of the adiabatic model at the peak region (x = 0 cm). In Fig. 5.12, we see that there is indeed a spatial dependence of the electron temperature however, the adiabatic model for the electron temperature holds to within ∼10% for all time between the range of x ∈ [−0.07, 0.07] cm. We note that only ∼10% of the ion and electron density occupy the region outside of this range of x. Therefore, although the electron temperature differs from the adiabatic model, the majority of the plasma occupies the range where Eq. (5.130) is accurate. We also note that the isothermal and adiabatic models are approximately equivalent when the parameter √ 2σ0 /cs0  1. We plot a horizontal line in Fig. 5.12(b) that validates this approximation. That is, below the horizontal line, the electrons are essentially at a constant temperature. The Ca+ -e simulations verify that for a single species UNP, the adiabatic expansion model is accurate for approximating the expansion of an UNP modeled by the Vlasov equation. However, the analytic formulae were derived under the assumption of a collisionless plasma. Since UNPs are highly 156 Figure 5.13: Comparison of the (a) isothermal electrons (b) adiabatic electrons. The relative error at the peak between the BGK simulations and the self-similar hydrodynamics model is ∼ 1%. Note that the density profiles from the BGK and Vlasov simulations agree nearly exactly for all time. collisional – see Table 5.1 – we extend the numerical study of single-species UNPs to include collisions by introducing a BGK collision operator with collision rates given by Eq. (5.16). We carry out single-species BGK simulations of Ca+ and treat the electrons in two ways: isothermally with density given by Eq. (5.94) and adiabatically with density given by Eq. (5.94) and temperature given by Eq. (5.130). For these simulations, we pick plasma conditions relevant to ongoing UNP experiments. Specifically, we set nCa+,0 = 3.4 × 109 cm−3 with width σCa+,0 = 0.57 mm and TCa+ = 1.9 K. For the isothermal case we set Te = 96 K for all time and for the adiabatic case, we set Te0 = 96 K and calculate the temperature profile from Eq. (5.130). The total simulation time is 8µs with a CFL of 0.7. The 1D spatial grid is initialized with an upper (and lower) bound of magnitude |xmax | = 0.5 cm with a grid resolution of ∆x = 0.00125 cm. The 1D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 50 × T/m cm/s with a grid p resolution of ∆v = 0.2 cm/s. We carry out two simulation for both the isothermal and adiabatic cases. One simulation is collisionless (i.e., the Vlasov equation) and another is with the BGK collision operator. Each of the four simulations took approximately 1 hour to complete. In contrast to the explicit Ca+ -e simulations, the Ca+ simulations with an implicit electron species decreased computation time by roughly a factor of 240. 157 The density profiles of the simulation results are displayed in Fig. 5.13. In Fig. 5.13(a) we show results from assuming the electrons are isothermal, and in Fig. 5.13(b) we show results from assuming the electrons are adiabatic. In the isothermal case, we see that both the Vlasov and BGK data disagree with the isothermal self-similar expansion model Eq. (5.117). In the adiabatic case, we see that the Vlasov, BGK, and adiabatic self-similar model [Eq. (5.129)] all agree implying that the plasma expansion is adiabatic. Moreover, in both cases, the Vlasov and BGK models agree almost exactly – to within a few percent for the first three moments of the distribution function – implying that the dynamics of the single-species UNP are time-reversible since the Vlasov equation does not generate entropy. The above simulations showcase interesting features of single-species UNPs. Namely, we have verified that the electron and ion species are well-approximated by a self-similar hydrodynamic model. While other work has been done to validate this point Ref. cite, we have quantified the validity of various approximations in the derivation of the self-similar hydrodynamics models with kinetic simulations. Specifically, we have shown that although the electron temperature is not spatially independent, the errors associated with that approximation are on the order of a few percent. Additionally, we have assessed the role that collisions have in the expansion of UNPs. We have found that the Vlasov and BGK simulations agree to sub-percent accuracy in the first three moments and within 1% of the self-similar expansion model Eq. (5.129). Therefore, the UNP expansion is adiabatic and little to no entropy is being generated by collisions. Because of this, single-species UNPs should be time-reversible which perhaps can be verified experimentally. We note that the self-similar model is based around a Gaussian UNP which may not be the case in the presence of magnetic fields. Thus, if the initial condition is not carefully controlled, the UNPs may no longer be time-reversible. We will show in the next section, that UNP mixtures do not follow the same self-similar expansion properties of single-species UNPs and that the amount of entropy generated in them can be mitigated by the initial plasma conditions. 158 5.6.2.2 Expansion of Multi-species UNPs In this section, we extend our numerical study to plasmas UNP mixtures where we answer the question: can the UNP mixture conditions be chosen to minimize the amount of entropy being generated? Recently the capability to create UNP mixtures has provided the ability to validate models for plasma mixtures and allowed for measurements of process otherwise unattainable such as temperature relaxation. We begin this section by studying the impact that changing the plasma conditions has on entropy generation. We compare the numerical results of four different plasma conditions. The first case we will analyze is a Ca+ -Yb+ mixture with plasma conditions relevant to ongoing UNP mixture experiments [24]. Specifically, the plasma conditions here are nCa+,0 = 1.4 × 1010 cm−3 , and nYb+,0 = 2.7 × 1010 cm−3 with widths σCa+,0 = 0.76 mm and σYb+,0 = 0.44 mm. The initial ion temperatures are taken to be TCa+ = 2 K and TYb+ = 1 K. The second case we study is the same as the first case but instead with σCa+,0 = σYb+,0 = 0.5 mm. For the third case we now assume that both species share the same peak density and width. Specifically, nCa+,0 = nYb+,0 = 2 × 1010 cm−3 with widths σCa+,0 = σYb+,0 = 0.5 mm. The initial ion temperatures are taken to be TCa+ = 2 K and TYb+ = 2 K with the initial electron temperature is Te0 = 96 K. The fourth and final case is instead a UNP mixture of Ca+ -K+ with nCa+,0 = nK+,0 = 2 × 1010 cm−3 with widths σCa+,0 = σK+,0 = 0.5 mm. The initial ion temperatures are taken to be TCa+ = 1 K and TK+ = 1 K. For all cases the initial electron temperature is Te0 = 96 K and we assume that the electron temperature profile follows Eq. (5.130). The plasma conditions for all four cases are summarized in Table 5.3; each simulation took approximately 2.5 hours to complete. For all simulations, the following conditions are the same: (i) the total simulation time is 8 µs with a CFL of 0.6, (ii) the 1D spatial grid is initialized with an upper (and lower) bound of magnitude |xmax | = 0.5 cm with a grid resolution of ∆x = 0.001 cm, (iii) the 1D velocity grid is initialized with an upper (and lower) bound of magnitude |vmax | = 40 × T/m cm/s with a grid p resolution of ∆v = 0.1 cm/s, and (iv) a fourth-order upwinding FVM reconstruction is used with a Newton iteration to obtain the electron density. To assess the timer-reversibility of each case given 159 Case Species 1 Species 2 n1 (cm−3 ) n2 (cm−3 ) σ1,0 (mm) σ2,0 (mm) T1 (k) T2 (K) (a) Ca+ Yb+ 1.4 × 1010 2.7 × 1010 0.76 0.44 2 1 (b) Ca+ Yb+ 1.4 × 1010 2.7 × 1010 0.5 0.5 2 1 (c) Ca+ Yb+ 2 × 1010 2 × 1010 0.5 0.5 2 2 (d) Ca+ K+ 2 × 1010 2 × 1010 0.5 0.5 1 1 Table 5.3: Plasma conditions of four UNP mixtures. By varying the initial conditions of the plasma mixture, we aim to minimize the amount of entropy generated in the system. Numerical results for cases (a) - (d) are displayed in the corresponding panels of Fig. 5.14. We see that by selecting certain plasma parameters, the Vlasov and BGK numerical results agree suggesting that specific UNP mixtures are time-reversible. in Table 5.3, both a Vlasov and BGK simulation were carried out. The initial condition and final density profiles are for each case in Table 5.3 are displayed in Fig. 5.14. In Fig. 5.14(a), we see that the Vlasov and BGK results differ greatly. The role of collisions in this case prevent the Ca+ species from being forced away from the more massive Yb+ species. In an attempt to control the spread of the Ca+ species in the Vlasov simulations, we make the initial width of Ca+ and Yb+ the same which is shown in Fig. 5.14(b). The Vlasov results show that the Ca+ ions still spread away from the Yb+ species but the Ca+ species retains an overall Gaussian profile. Next, we make the densities and temperatures of the Ca+ and Yb+ the same and the numerical results are shown in Fig. 5.14(c). We observe that for the Ca+ species, the Vlasov results are in better agreement to the BGK results. Lastly, we change the second species to K+ so that both species have similar mass. These results are shown in Fig. 5.14(d). We see that the Vlasov and BGK results agree to within sub-percent accuracy and retain an overall Gaussian shape. Our numerical results suggests that of all the cases given in Table 5.3 case (d) is the most likely multi-species UNP to be time-reversible. It is worth summarizing the results of comparing the Vlasov and BGK results for the different cases of UNP mixtures. From our numerical results, we see that enforcing the same width parameter results in Gaussian profiles for both species; this suggests that the expansion of the UNP mixture may be self-similar although there is currently no self-similar expansion models for plasma mixtures. Next, by enforcing that the number densities of both species are the same, we find that the expansion of the Ca+ species is less than the case where both species only have the same width parameters. 160 Figure 5.14: Number density profiles for UNP mixtures for the plasma conditions given by cases (a)-(d) in Table 5.3. Lastly, we see that by picking both species to have roughly the same mass, number density, and temperature, the Vlasov and BGK agree nearly exactly. Unfortunately, no experimental data exist to validate these numerical simulations. As was the case of the time-reversible single-species UNP, it may be possible to “reverse" the dynamics of UNP mixture experiments; our numerical results suggest that the highest probability of success for time-reversible UNP mixtures is case (d) of Table 5.3. The claim of time-reversibility of the UNP mixture with conditions given by case (d) of Table 5.3 is substantiated by calculating the total entropy that is generated during these simulations. Figure 5.15 displays the total entropy source term given by Eq. (5.37) for each of the four cases in Table 5.3. Noting the scale of the colorbar for each panel, we see that the cases (a) and (b) generate roughly the same amount of entropy. In contrast, cases (c) and (d) generate considerably 161 Figure 5.15: Total entropy source [Eq. (5.37)] for cases (a)-(d) in Table 5.3. Note the scale on the colorbar for each case. Cases (a) and (b) generate roughly the same amount of entropy, whereas (c) and (d) generate considerably less entropy. The amount of entropy being generated in case (d) suggests that the dynamics of the UNP mixture are time-reversible. This is further confirmed by the agreement of the Vlasov and BGK results in Fig. 5.14. less entropy with the latter case generating roughly 4 orders of magnitude less that cases (a) and (b). 5.6.2.3 Diffusive Mixing in HED Plasmas As illustrated by Fig. 1.1, HED plasmas are much higher in density and temperature than UNPs. However as previously mentioned there is a crossover regime at which UNPs and HED plasmas roughly share the same dimensionless parameters. Using the multi-species BGK model4, we simulate a system that is pertinent to ongoing experiments on the Z machine at Sandia National 4 For this section, the 1D-3V code used to simulate this work can be found at: https://github.com/lanl/Multi-BGK. 162 Laboratory5. The main goal of the experiment and the simulations of the experiments is to quantify atomic-scale mixing across an interface of between a material with high nuclear charge and a material with low nuclear charge. The initial condition is displayed in Fig. 5.16. We assume the system is periodic and is comprised of a region made up of the elements V and Al in the center that is 50 µm wide. The VAl region is surrounded on either side by a 50 µm plastic region made up of the elements C,H, and O. Figure 5.16: Initial condition for the 1D-3V multi-species BGK simulation. The dotted vertical lines denote the VAl/CHO interface and the number density of O has been multiplied by 10 for visual clarity. To account for partially-degenerate electrons, we employ a linear Thomas-Fermi model [44] instead of the classical Possion-Boltzmann formula Eq. (5.94). As previously mentioned a main benefit of kinetic models is that they return an infinite number of macroscopic variables via moments of the distribution function. Additionally, in contrast to many hydrodynamic models, 5 Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This section describes objective technical results and analysis. Any subjective views or opinions that might be expressed in this section do not necessarily represent the views of the U.S. Department of Energy or the United States Government: SAND2022-16353 T. 163 kinetic models allow for each species to have distinct moments which results in density, velocity and temperature fields for each species in the system. In interface mixing in particular, the need for multiple continuity, momentum, and temperature fields is crucial as the velocity fields of each species become uncoupled [44, 88]. The work done in this section is part of a much larger simulation effort at Sandia National Lab- oratories that also consists of radiation-hydrodynamic simulations6. The radiation-hydrodynamics simulations include models of radiation that ultimately turns the initial condition shown in Fig. 5.16 into a plasma. Since the current kinetic models do not directly treat radiation, we use data from the radiation-hydrodynamics simulations to estimate an electron temperature ramp that acts as an external (radiation) energy source in the kinetic simulations. The electron temperature ramps are determined from the ion temperature of the radiation-hydrodynamics simulations and are shown in Fig. 5.17. To obtain a functional form for the electron temperature that can be evaluated at any time, we fit ion temperature data of the VAl portion of two radiation-hydrodynamics simulations. The data follow linear trends (see Fig. 5.17) and a linear best-fit function is used as the electron temperature ramp for the electron temperature model. 6 The radiation-hydrodynamics simulations were carried out using the ALEGRA code [286]. 164 Figure 5.17: Temperature of ion species calculated from two radiation-hydrodynamics simulations. The data are shown as solid lines with linear fits denoted by dashed lines. We assume that the temperature of the electrons follows the linear trend and the linear fits are used as the electron temperature ramp in the multi-species BGK simulations. For brevity, we denote the temperature ramp with the larger magnitude as the “high-temperature ramp" (HTR), and the temperature ramp with the lower magnitude as the “low-temperature ramp" (LTR). The kinetic simulations have simulation length of 12ns and use a first-order upwinding flux; a CFL number is chosen to ensure stability. A Newton iteration was employed for determining the self-consistent electric field. For the BGK operator, we employ a collision rate given by Eq. (5.16). The density profiles at t = 12 ns using both the LTR and HTR are displayed in Figs. 5.18(a) and 5.18(b). We see qualitative differences between the density profiles computed with the low- and high-temperature ramps but the overall number density of each species appears to be the same in each region of both cases. For example, the amount of V in the region from x ∈ [25, 75]µm appears to be roughly equivalent in Figs. 5.18(a) and 5.18(b). 165 Figure 5.18: Simulation snapshot at t = 12 ns of the density profile with the (a) the LTR shown in Fig. 5.17 and (b) the HTR shown in Fig. 5.17. The initial condition is denoted by horizontal dotted lines and the location of the interface is shown as vertical black dotted lines. To quantify the amount of diffusion that occurs throughout the simulation time, we introduce a metric for computing the amount of diffusion across the interfaces at x = −25 µm and x = 25 µm. The total number density of N ion species in the region of x1 and x2 is computed via N ∫ Õ x2 A(t) = ni (x, t) dx. (5.161) i x1 Using Eq. (5.161) we compute the total amount of V and Al in the region of x ∈ [25, 75]µm; the initial density of V and Al in this region is zero. We see that for the case of the HTR, the amount of V and Al diffuses in to the CHO region more quickly than does the case where we use the LTR. This is consistent with the fact that for higher temperatures, the diffusion will be greater. Between the time range of t =∼ 9 to 12 ns, we see that the amount of V and Al in the CHO region decreases and is approximately the same as the amount of diffusion using the LTR. This phenomenon is due to the periodic nature of the system; the V and Al expand outwards into the CHO region and then return to the VAl region more quickly in the HTR case. 166 Figure 5.19: Total density of V and Al in the CHO region using the HTR and LTR. We see that the HTR results in more diffusion of the VAl species than the LTR. Due to the periodic nature of the system, the total density of VAl decreases between t =∼ 9 to 12 ns. While Fig. 5.19 provides some insight into the amount of diffusion that has occurred during the simulation, it does not provide any insight as to what factors caused the diffusion to occur. To answer the question of what factors are driving the diffusion in the simulations, we consider the diffusion driving forces defined by [44] ! ρi Zi e Õ Z j e di = ∇xi + (xi − yi )∇ ln p + − yj E, (5.162) p mi j mj where p = nT, E is the electric field, and ρi = mi /ni is the mass density. The atom fraction and mass fraction are given by xi = ni /n and yi = ρi, /ρ, respectively, with n = iNi ni and ρ = iNi ρi . Í Í The first term on the right-hand side of Eq. (5.162) corresponds to Fickian diffusion (diffusion due to concentration gradients), the second term corresponds to barodiffusion (diffusion due to pressure gradients) and the last term corresponds to electrodiffusion (diffusion due to electric fields). We display the spatio-temporal evolution of the diffusion driving forces with the LTR and HTR for C [panels (a) – (c)] and V [panels (d) - (f)] in Figs. 5.20 and 5.21. A comparison of Fig. 5.20(c) and Fig. 5.21(c) highlight the periodic boundary conditions and show how the HTR causes faster diffusion than does the LTR. In both for both the HTR and LTR cases, the dominant diffusive flux 167 Figure 5.20: Space-time diagrams of Fickian, electro-, and barodiffusion fluxes with the LTR. (a) − (c) show the diffusive fluxes for C where (a) is the Fickian diffusion flux, (b) is the barodiffusion flux, and (c) is the electrodiffusion flux. (d) − (f) show the diffusive fluxes for V where (d) is the Fickian diffusion flux, (e) is the barodiffusion flux, and (f) is the electrodiffusion flux. is electrodiffusion. The charge imbalance at the interface results in stron electric fields. The role that these electric fields play is that the C species in the CHO region moves away from the interface and the V species in the VAl region move toward the center of the VAl region. 5.7 Conclusions and Outlook The focus of this chapter has been on the development and numerical implementation of kinetic models for simulating strongly coupled plasmas. We have validated our numerical scheme against data for multi-species UNP temperature data and have used our simulations to elucidate various processes in UNPs and HED plasmas. Specifically, we have been able to simulate the tail filling rate in UNPs showing that the tail filling rate is sensitive to the choice of collision operator and collision frequency. The next steps are to validate our models with experimental data. We have also analyzed the role that collisions play in the expansion of single-species UNPs with a Gaussian initial condition. We find that the collisions rapidly drive the system to equilibrium as evidenced by the agreement of the Vlasov and BGK simulations. By carrying out explicit 168 Figure 5.21: Space-time diagrams of Fickian, electro-, and barodiffusion fluxes with the HTR. (a) − (c) show the diffusive fluxes for C where (a) is the Fickian diffusion flux, (b) is the barodiffusion flux, and (c) is the electrodiffusion flux. (d) − (f) show the diffusive fluxes for V where (d) is the Fickian diffusion flux, (e) is the barodiffusion flux, and (f) is the electrodiffusion flux. Ca+ -e simulations, we test various assumptions of analytic models for the electron temperature in a single-species UNP. We find that the spatial independence assumption is quite accurate and that the electrons can be treated adiabatically eliminating the need to explicitly simulate them. For UNP mixtures initialized with a Gaussian profile, we find that certain plasma conditions result in self-similar expansion. Specifically, by picking two ion species with similar mass, we find that Vlasov and BGK simulations agree to within a percent. Our numerical results suggests that for this choice of plasma conditions, the UNP mixture is time-reversible like the case of single species UNPs with a Gaussian initial condition. These results can be used to guide future experimental calculations to verify our models. Lastly, for HED plasmas relevant to experiments on the Z machine at Sandia National Labora- tories, we find that the main source of diffusive mixing is due to strong electric fields that occur in regions of a sharp interface. To mitigate the diffusive mixing, specific choices of elements can be chosen to avoid a disparate charge imbalance reducing the strength of the electric field. 169 The results in this chapter can be expanded upon in a variety of ways. First, UNPs are often spherically symmetric which lends itself to simulations in a spherical coordinate system. As such, we have provided a derivation of the Vlasov equation in spherical coordinates, showing that there are additional terms that appear due to fictitious forces. Additionally, we have developed a second- order spherical FVM stencil (see Appendix C) in r space which still needs to be implemented numerically. In general, the numerical method provided here relies on the operator splitting technique and one improvement could be to solve the “two-dimensional" FVM directly allowing for the implementation of high-order time integrators. Moreover, an implicit time integrator could be used for the multi-species BGK collision operator allowing for larger time steps when needed. 170 CHAPTER 6 CONCLUSIONS AND FUTURE WORK In this dissertation, we have presented four projects all with applications to non-ideal plasmas. The first project focused on an exhaustive benchmarking study which validated the use of simple force laws for molecular dynamics simulation of dense plasmas. The results of this work have allowed us to dismiss force laws that lack the non-parametric nature of the force-matched potentials we employed here. By using multiple metrics for comparison, we found that an agreement of time-independent properties like the radial distribution function is not sufficient for estimating the validity of force laws as compared to a Kohn-Sham MD calculation. We have also illustrated that although Kohn-Sham MD includes important physics that simpler models, results from Kohn- Sham MD can be heavily plagued by finite-size effects, introducing large amounts of statistical errors. Extensions of this work include simulations of additional elements at different densities and temperatures. This will allow for a greater understanding of the functional form of pair potentials in the warm dense matter regime. Additionally, the natural extension of this work is to apply this data-driven discovery of force laws from high-fidelity data to plasma mixtures. By obtaining pair potentials for plasma mixtures, we gain access to Kohn-Sham MD accurate pair potentials for simulating systems that are pertinent to nuclear fusion experiments. The second project focused on deriving theoretical models for interdiffusion of binary plasma mixtures. For multiple plasma conditions, our models eliminate the need to perform costly MD simulations altogether while only incurring an error on the order of a few percent. Our closed form formulae provide accurate approximations for values that are typically ignored altogether in the MD community. Additionally, we were able to dismiss and highlight the inadequacies of the so-called “Darken formula" for the interdiffusion coefficient by highlighting its inability to converge to the true result. The culmination of the work results in a simple-to-use closed form expression for the interdiffusion in a binary ionic mixture, which is reasonably accurate in the strongly coupled regime. Extensions of this work include a more comprehensive study of the efficacy of these models for the thermodynamic factor which relies on the generation of datasets that span temperature, density, 171 and atomic number. The third project focused on employing machine learning techniques for combining datasets of multiple fidelities. By applying this general machine learning framework to plasma transport coefficient data, we showed that interpolating data from multiple models ranging in fidelity results in a more accurate, confident prediction in regions absent of high-fidelity data. We highlighted the use of this framework in multiple dimensions and developed a sampling approach for high- fidelity data to reduce prediction errors. We could further extend the work done in this project by developing models which allow for more accurate extrapolation beyond the range of data based on a low-fidelity trend. Additionally, while Gaussian-process regression was the primary method used for interpolation in this project, we could explore the use of other methods such as neural networks for multi-fidelity modeling. Lastly, the fourth project focused on developing kinetic models to simulate strongly coupled plasma mixtures. We simulated UNPs and HED plasmas using 0D and 1D kinetic simulations. Our kinetic model included the LBD collision operator, the BGK collision operator and multi- species BGK operator. We used our kinetic models to study the time evolution of single-species UNPs and UNP mixtures and also the rate of entropy production in these plasmas. By varying the initial plasma conditions, we determined plasma conditions that minimize the amount of entropy being produced suggesting that the specific UNP mixture is time-reversible. In addition, we used experimental data to validate collision rate models for the BGK operator adding confidence to their use for non-ideal plasmas. We found that in our 0D simulations that we can accurately reproduce data from MD simulation for UNP mixtures decreasing the computation cost by approximately 3000 times. By carrying out an explicit ion-electron UNP simulation, we quantified the errors associated with various approximations in analytic formulae used to approximate electrons in a UNP. We found that the temperature of the electrons in a UNP well-treated by an adiabatic mode that assumes no spatial dependence. For the HED plasmas, we simulated a plasma mixture that is relevant to ongoing interface mixing experiments on the Z machine at Sandia National Laboratory. We found that the diffusive mixing in these experiments is dominated by strong electric fields that 172 originate due to charge imbalances between the species on either side of the interface. Although much of this work in this project is numerical, we have used our results to suggest experimental cases to further validate our models. 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Therefore, Poisson’s equation for these two cases is      1 d r 2 d ϕ = −4πρ , for r ≤ R,  r 2 dr    dr 0 (A.1)      r12 dr d d r 2 dr ϕ = 0, for r > R.    We solve the first equation by first expanding out the derivative using the product rule which yields d2 2 d ϕ + ϕ = −4πρ0 . (A.2) dr 2 r dr Multiplying by r 2 we get d2 d r2 2 ϕ + 2r ϕ = −4r 2 πρ0 . (A.3) dr dr We now assume a solution of the form a ϕ = r λ and determine the complementary solution from r 2 λ(λ − 1)r λ−2 + 2rλr λ−1 = 0.   (A.4) The above equation yields the auxiliary equation λ(λ + 1) = 0, (A.5) which implies a complementary solution of the form c2 ϕc (r) = c1 + . (A.6) r We note that the complementary solution is also the solution to the r > R case in Eq. (A.1). For the particular solution, we use variation of parameters. The particular solution has the form 1 ϕ p (r) = u1 ϕ1 + u2 ϕ2 = u1 + u2 . (A.7) r 199 We have the following Wronskians: 1 1 1 r 0 r 1 0 W= , W1 = , W2 = . (A.8) 0 − r12 −4πρ0 − r12 0 −4πρ0 Using the fact that u01 = W1 /W and u02 = W2 /W, we have 4 u1 = −2πρ0r 2, u2 = πρ0r 3, (A.9) 3 which gives 2πr 2 ρ0 ϕp = − . (A.10) 3 Thus, the solution to Eq. (A.1) is 2πr 2 ρ0  ϕ = c1 + c2 3 ,   − for r ≤ R,  in   r  (A.11)    ϕout = c3 + r, c4 for r > R.    Where the subscripts “in" and “out" denote the solution inside and outside the sphere of uniform charge density respectively. To determine the unknown constants c1, c2, c3 and c4 , we employ the boundary conditions d ϕin = 0, (A.12) dr r=0 d d ϕin = ϕout , (A.13) dr r=R dr r=R ϕin (R) = ϕout (R), (A.14) lim ϕout = 0. (A.15) r→∞ Applying these conditions result in c1 = 6πR2 ρ0 /3, c2 = 0, c3 = 0, and c4 = 4πR3 ρ0 /3. Eq. (A.11) becomes    r2 ϕ = 2 ,  2πρ 0 R − for r ≤ R,   in 3    (A.16)  4πR3 ρ   ϕout = 0 , for r > R.    3r 200 From Eq. (A.16) we can obtain the electric field via E = − dx d ϕ which gives  4πρ0 r E = 3 ,   for r ≤ R,  in    (A.17)  4πR3 ρ   Eout = 0 , for r > R.    3r 2 We use Eq. (A.17) to verify that our numerical scheme for solving Eq. (5.144) in Sec. 5.5.3. 201 APPENDIX B DYNAMIC STRUCTURE FACTOR IN THE RANDOM PHASE APPROXIMATION The dynamic structure factor for a Coulomb potential (OCP) can be computed by   T 1 S(k, ω) = − Im , (B.1) πuC (k)ω ε(k, ω) where uC (k) = 4π(Ze)2 /k 2 is the Coulomb potential and  2 ! kD ω ε(k, ω) = 1 + W p , (B.2) k k T/m ∫ ∞  2 1 x −x W(Z) = √ dx exp , (B.3) 2π −∞ x − Z − iη 2 where k D2 = 4πnZ 2 e2 /T. For a generic potential u(k), we re-write the dynamic structure factor using the density response function   1 1 χ(k, ω) = − 1− , (B.4) uC (k) ε(k, ω) which can be rearranged so that 1 uC (k) χ(k, ω) + 1 = . (B.5) ε(k, ω) Substituting Eq. (B.5) into Eq. (B.1) gives T S(k, ω) = − Im [ χ(k, ω)] . (B.6) πω With the ideal gas density response function, we have for some arbitrary potential u(k) that χ0 (k, ω) χ(k, ω) = . (B.7) 1 − u(k)[1 − G(k, ω)] χ0 (k, ω) Setting the dynamic local-field correction G(k, ω) = 0, we have the random phase approximation (RPA) χ0 (k, ω) χRPA (k, ω) = . (B.8) 1 − u(k) χ0 (k, ω) By plugging Eq. (B.8) into Eq. (B.6), we have T S RPA (k, ω) = − Im χRPA (k, ω) ,   (B.9) πω 202 where the imaginary part of χRPA (k, ω) is obtained from solving 2 ! ω  kD 1 − uC (k) χ0 (k, ω) = 1 + W p , (B.10) k k T/m which simplifies to !  2 1 kD ω χ0 (k, ω) = − W p . (B.11) uC (k) k k T/m Note that when ω = 0, the ideal gas response function is simply n χ0 = − . (B.12) T B.1 Fourier Transform of Interaction Potentials We begin with ∫ v(k) = d 3rv(r)eik·r . (B.13) Writing the above equation in spherical coordinates (assuming that the potential is radially sym- metric), we have ∫ 2π ∫ ∞ ∫ π u(k) = dφ dr dθ v(r)eikr cos θ r 2 sin θ 0 0 0 ∫ ∞ ∫ π = 2π dr dθ v(r)eikr cos θ r 2 sin θ 0 0 ∫ ∞ ∫ π = 2π drv(r) dθ eikr cos θ r 2 sin θ. (B.14) 0 0 Letting u = cos θ means that du = − sin θdθ and the new limits of the θ integral are u = −1 and 1 . Thus, ∫ ∞ ∫ 1 u(k) = 2π drv(r)r 2 du eikru −1 ∫0 ∞ 1  ikru  1 = 2π drv(r)r 2 e −1 0 ikr ∫ ∞ 2 = 2π drv(r)r 2 sin(kr) kr ∫0 ∞ 1 = 4π drv(r)r sin(kr). (B.15) 0 k Which yields ∫ ∞ 4π u(k) = dr rv(r) sin(kr). (B.16) k 0 203 B.2 Computing u(k) Using a Discrete Sine Transform We make use of Python’s discrete sine transform (DST-I) function to compute u(k). We begin by making the following definitions xi = a + i∆x, k j = b + j∆k. (B.17) The DST-I algorithm assumes the form N−1 π(i + 1)( j + 1) Õ   F(k j ) = 2 f (xi ) sin . (B.18) i=0 N +1 To map our problem into the above convention, we begin by defining, for some maximum distance rmax , the real space grid and the Fourier grid rmax (i + 1) π( j + 1) xi = , kj = . (B.19) N +1 rmax With a Riemann sum, we have N−1 π Õ I(k j ) = ∆x xi u(xi ) sin(k j xi ). (B.20) kj i=0 B.3 Computing the W(Z) function To compute the W(Z) function we have from Ref. [8] that π ∫ Z Z2   2 r  2 y Z W(Z) = 1 − Zexp − dy exp +i Zexp − 2 0 2 2 2 π  2  ∫ Z  2 r  Z y = 1 − Zexp − dy exp −i 2 0 2 2  2  "r ∫ Z  2 # π r Z 2 y =1−Z exp − dy exp −i 2 2 π 0 2  2 " r ∫ Z # π r  2 Z 1 2 y = 1 − iZ exp − dy exp −1 . (B.21) 2 2 i π 0 2 204 The above expression reduces further  2 " r ∫  2 # π 1 2 Z r Z y W(Z) = 1 + iZ exp − 1− dy exp 2 2 i π 0 2  2 " r # π 1 2√ −ix r ∫ Z   = 1 + iZ exp − 1− 2i dt exp −t 2 2 2 i π 0 π r  2  ∫ −ix   Z 2 = 1 + iZ exp − 1− √ dt exp −t 2 2 2 π 0 π r   = 1 + iZ 2 exp −x erfc(−ix). (B.22) 2 We can use Python’s “wofz" function to obtain π √ r W(Z) = 1 + iZ wofz(Z/ 2). (B.23) 2 205 APPENDIX C SECOND-ORDER UPWINDING STENCIL IN SPHERICAL COORDINATES The procedure shown here has been detailed in Ref. [262] for Cartesian and cylindrical coordinates; we extend the approach to spherical coordinates. Our goal is to construct a second-order upwinding stencil for the Vlasov equation in Spherical coordinates. We use a one-dimensional polynomial reconstruction with a second-order polynomial P(r) = p2r 2 + p1r + p0 .d (C.1) We can derive a second-order FVM stencil by solving a linear system for the unknown coefficients {p2, p1, p0 } in terms of the cell averaged distribution function which we denote here as h f ii . Once we have the coefficients of the polynomial P(r), we evaluate it at the cell interfaces to obtain a numerical flux function at the interface. For a second-order upwinding stencil, we require three values of our cell averaged distribution function. Specifically, we have that the linear system is given by the relations ∫ ri−3/2 h f ii−2 = P(r)4πr 2 dr, (C.2) r ∫ i−5/2 ri−1/2 h f ii−1 = P(r)4πr 2 dr, (C.3) r ∫ i−3/2 ri+1/2 h f ii = P(r)4πr 2 dr, (C.4) r ∫ i−1/2 ri+3/2 h f ii+1 = P(r)4πr 2 dr. (C.5) ri+1/2 From the above relations, we solve the linear system to obtain the coefficients {p2, p1, p0 }. For the case where the advection speed v > 0, we construct a linear system from h f ii−1, h f ii, and h f ii+1 . When v < 0 we construct a linear system from h f ii−2, h f ii−1, and h f ii . Assuming that v > 0 and 206 solving the linear system for the coefficients of P(r) and then evaulating P(r) at ri+1/2 yields  1 h f ii+1/2 = 24π∆r 3 (4 + 6` − 9` 2 − 20` 3 + 15` 4 + 30` 5 + 10` 6 ) +(24 + 96` + 144` 2 + 90` 3 + 20` 4 )h f ii−1 +(69 − 96` − 63` 2 + 90` 3 + 50` 4 )h f ii  +(−3 + 9` − 10` )h f ii+1 , 2 4 (C.6) where ` = ri+1/2 /∆r . For v < 0  1 h f ii+1/2 = 24π∆r 3 (4 − 6` − 9` 2 + 20` 3 + 15` 4 − 30` 5 + 10` 6 ) +(−3 + 9` 2 − 10` 4 )h f ii−2 +(69 + 96` − 63` 2 − 90` 3 + 50` 4 )h f ii−1  +(24 − 96` + 144` − 90` + 20` )h f ii . 2 3 4 (C.7) 207