DYNAMIC RESPONSE AND KINETIC PHENOMENA IN HIGH ENERGY DENSITY
MULTI-SPECIES PLASMAS

By

Thomas Chuna

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy
Computational Mathematics, Science, and Engineering—Dual Major

2024

ABSTRACT

High energy density (HED) science, concerned with matter at pressures in excess of 1 MBar, inves-

tigates the processes occurring inside nuclear fusion and giant planets, enhancing our understanding

of the universe’s most energetic events. This work contains three primary results. First, we in-

corporate conservation of momentum into the collisional multi-species dynamic response models.

Second, we extend the single species hybrid kinetic-fluid model of Degond et al. to multi-species

[P. Degond, S. Jin, and L. Mieussens, JCP 209.2 (2005): 665-694]. Third, we present data-driven

observations of system equilibration, which can assess the quality of machine-learned model clo-

sures in extended moment hydrodynamics. Each result uses expansions about equilibrium, but

contributes to HED science in different ways.

Measuring the material properties of HED matter is challenging since they exist for a short time

in a confined space at conditions that damage nearby equipment. Thus, experimental diagnostics

rely on scattered and emitted electromagnetic spectra to investigate material properties. Connecting

the spectra to material properties requires theoretical models of dynamic response. Typical dynamic

response models include the Mermin model, predicting Drude-like conductivity [N. D. Mermin,

PRB 1.5 (1970): 2362], and the Drude-Smith model, predicting non-Drude-like conductivity [N.

V. Smith, PRB 64.15 (2001): 155106]. However, the often used Mermin model does not satisfy the

relevant sum rules, and the Drude-Smith model lacks interpretability. In this dissertation, develop

a new interpretable dynamic dielectric function for multi-species plasmas which includes mean

field interactions as well as number and momentum conserving multi-species collisions. This

interpretable model satisfies relevant sum rules. We demonstrate the impact of each conservation

law on the predicted dynamic structure factor of a pure deuterium-tritium (DT) HED plasma as

well as a carbon contaminated DT HED plasma. Additionally, we present a new dynamic non-

Drude conductivity model that has a clear interpretation. Comparing our conductivity model to the

Drude-Smith conductivity model, we conclude that Smith’s intensely debated phenomenological

parameter violates local number conservation.

Simulations are conducted to complement and inform HED experiments. Historically, HED

scientists have used radiation–hydrodynamic codes. However, Eulerian codes assume the mean

free path in the plasma is infinitesimally small, placing the system in local equilibrium. This

assumption neglects dissipation and forces species in the same location to share a bulk velocity

and temperature. Current codes correct for dissipation, which improves predictions, but they

cannot correct for velocity and temperature separation. A fully kinetic code could account for

these phenomena, but such a code is computationally infeasible for realistic 3D simulations. In

this dissertation, we present a hybrid model which can smoothly transition between Haack et al.’s

multi-species kinetic PDE [J. R. Haack, C. Hauck, and M. S. Murillo, J. Stat. Phys. 168, 4] and

multi-species hydrodynamic PDEs. We validate the hybrid model on the Sod shock problem and

then investigate multi-species mixing in HED experiments. Within our simulation, we identify

electro-diffusion at the interfaces as well as persistent velocity and temperature separation between

species, phenomena that are missed by purely hydrodynamic codes.

As an alternative to hybrid models, extended moment hydrodynamic models can be employed

[N. M. Hoffman, et al. Physics of Plasmas 22.5 (2015)]. However, to close the hierarchy of

moments, these models often assume local equilibrium. Machine learning is an emerging approach

to the moment closure problem, which can avoid such assumptions. We construct a complex-valued,

multi-step neural network to close Grad’s extended moment equations [H. Grad, Comm. pure and

applied mathematics 2.4 (1949): 331-407]. Additionally, the quality of a closure is typically

assessed on its long time stability and ability to describe diffusion/dissipation. We conduct data

driven observations of the dissipation process. In particular, we use dimension reduction techniques

and dynamic mode decomposition to provides new metrics to assess a neural network’s ability to

inform on dissipation.

ACKNOWLEDGEMENTS

Chapter 4 of this work was supported by the U.S. Department of Energy through the Los Alamos

National Laboratory. Los Alamos National Laboratory is operated by Triad National Security,

LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No.

89233218CNA000001). Research presented in this article/presentation/report was supported by

the Laboratory Directed Research and Development program of Los Alamos National Laboratory

under project number 20190005DR. Chapter 5 of this work was performed under the auspices

of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract

DE-AC52-07NA27344 Lawrence National Security, LLC.

I acknowledge the academic advisors from Michigan State University and Wittenberg Univer-

sity. First, I acknowledge my PhD advisor Michael Murillo. I thank Michael for expanding my

presentation and publication skill set as well as my knowledge of machine learning. I recognize the

patience it took to help me express ideas succinctly and the skill it took to select relevant projects

that were within my reach. I acknowledge Alexei Bazavov, under whom I earned my MSc, I thank

Alexei for expanding the number of mathematical tools at my disposal. I acknowledge Elizabeth

George, under whom I earned my BSc, I thank Elizabeth George for fostering my curiosity and

giving me space and support to explore.

I acknowledge the scientists from Los Almos National Laboratory. Jeff Haack who guided

my understanding of near-equilibrium expansions and computational fluid dynamics and Irina

Sagert who guided the editing process on my first published paper. As well as the scientists

from Lawrence-Livermore National Laboratory. Frank Graziani who guided my understanding of

emergent phenomena and Lee Ellison who guided me through his exciting post-doctoral research.

I acknowledge the members of the MSU community who have helped my along my professional

path. To my committee members (in alphabetical order): Metin Aktulga, Wade Fisher, Michael

Murillo, Brian O’Shea, and Scott Pratt, after many committee meetings, I was not sure I would

make it, but we have arrived at a dissertation. I acknowledge Kim Crosslan, Remco Zegers, and

Steve Zeph who maintain Michigan State’s physics graduate program. To the veteran MurilloLab

iv

members David, Lucciano, and Luke your support and many sanity checks were crucial. To the

new members: Zach, Chris, Jorge, and Jannik, I wish the absolute best. Additionally to: Yannis,

Yash, and Alejandro, I wish the absolute best.

I acknowledge the many personal connections that sustained my emotional well-being. To my

family: Karen, Trevor, Kelsie, Alex, Erin, Eric, Trevor, Pam, and the cousins Sam, Keith, Gabe,

Kyle, Sarah, thank you for your unending lines of communication. To Camila Monsalve Avendaño,

I will be with you tomorrow. To my Michigan State friends: Alyssa Turcsak and Corey Cooling,

Abbie Cathcart and Isaac Yandow, Maggie Magilligan and Morgan Koetje, Tom-Erik Haugen

and Adam Anthony, Tamas Budner and Kyle Krowpman, Carl Fields, Jason Surbrook, Meredith

Wagner and Tino Burse, Roy Salinas, Jordan Owens-Fryer, Tracy Edwards, Dan Salazar, Mostafa

Ali, Camille Mikolas, and Sheng Lee thank for helping me heal. To my loved ones: Jordan, Sage,

Alex Quinn Durham, Guillermo, TK(fire!), Wam Sitman, Señor, Sam Pagliaroli, Autumn Hill and

the Hills, That1kid, and Yung Grady, Steve Stuthers, Lucas George, Daddy Chris, and Daddy

Andrew, thank you for balancing me. To Margaret Dutko and Thomas E. Chuna may you rest in

peace.

To everyone I have named here, I let my stress prevent me from cherishing our connections; I

have learned my lesson and I look forward to loving more.

v

TABLE OF CONTENTS

LIST OF TABLES .

. .

LIST OF FIGURES .

. .

.

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.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Definition of a Plasma .
1.2 Classification of a Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 High Energy Density Science
1.4 Thesis Organization .
.
.
BIBLIOGRAPHY . .

1
1
4
6
.
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

CHAPTER 2

MULTI-SCALE NATURE OF PLASMA . . . . . . . . . . . . . . . .

. 15
2.1 Classical Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Kinetic Theory . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Hydrodynamics .
.
2.4 Time and length scales associated to the various frameworks
. . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
BIBLIOGRAPHY . .

. .

.
.

.

CHAPTER 3

.
.
. .
. .

.
. .
. .

Introduction .

CONSERVATIVE DIELECTRIC RESPONSE AND ELECTRICAL
CONDUCTIVITY FROM THE MULTI-SPECIES BHATNAGAR GROSS
KROOK KINETIC EQUATION . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. 34
. 39
. 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. 56

.
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Kinetics . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
3.3 Results
3.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BIBLIOGRAPHY .
APPENDIX 3A
APPENDIX 3B

.
FOURIER TRANSFORMING OUR KINETIC EQUATION .
CALCULATING THE COLLISIONAL INVARIANT FOR
ENFORCEING MOMENTUM CONSERVATION . . . . . . . 58
EXPANDING PARTIAL FRACTIONS . . . . . . . . . . . . . . 59
IMPLEMENTING OUR MODEL . . . . . . . . . . . . . . . . 60
EXPANDING THE DIELECTRIC FUNCTION AT LONG
WAVELENGTHS . . . . . . . . . . . . . . . . . . . . . . . . . 63

APPENDIX 3C
APPENDIX 3D
APPENDIX 3E

.

.

.

CHAPTER 4

.

.

.

.

Introduction .

MULTI-SPECIES KINETIC-FLUID COUPLING FOR HIGH-ENERGY
DENSITY SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . 65
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1
. 68
4.2 The multispecies kinetic and hydrodynamic models . . . . . . . . . . . . . .
4.3 Coupling of the BGK and continuum equations for multiple species in 3D . . . 71
4.4 Example for the BGK-Navier-Stokes Coupling for Two Species . . . . . . . . . 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Numerical Results .
4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 94
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
BIBLIOGRAPHY .

.

.

.

.

vi

APPENDIX 4A

APPENDIX 4B
APPENDIX 4C

COMPUTING THE FLUID CORRECTION 𝑓 (1)
REGION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
CORRECTIONS TO THE STRESS TENSOR . . . . . . . . . . 104
CORRECTIONS TO THE HEAT FLUX . . . . . . . . . . . . . 105

𝑖𝐹 IN BUFFER

CHAPTER 5

.

. .

Introduction . .

DATA DRIVEN OBSERVATIONS OF SYSTEM EQUILIBRATION . . 106
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Grad’s Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Neural Closures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4
. 116
5.5 Summary and outlook .
BIBLIOGRAPHY . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
APPENDIX 5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
Invariant Manifolds .

UPDATE ERRORS IN BOTH MULTI-STEP NODES AND
MULTI-STEP NEURAL CLOSURES . . . . . . . . . . . . . . 120

. .

.

.

CHAPTER 6

BIBLIOGRAPHY . .

SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . 122
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

. .

.

vii

LIST OF TABLES

Table 1.1 Table identifying the density 𝜌, temperature 𝑇, ion-ion coupling parameter Γ,
and electron degeneracy parameter Θ of various plasmas. Omega and NIF
estimates indicate the density and temperature at peak capsule compression in
inertial confinement fusion (ICF) experiments [9, 10].

. . . . . . . . . . . . . .

5

Table 2.1 Tabulation of theories from fine to coarse degrees of freedom. Each theory’s
relevant dynamical equations and closure information are identified.

. . . . . . . 16

Table 3.1 Tabulated plasma parameters for a pure deuterium (D) plasma, a pure tritium
(T) plasma, a pure L plasma, and a mixed D and T plasma; all plasmas are at a
mass density of 1002 g/cc and temperature of 928 eV [36]. The effective charge
of the ion 𝑍 is computed using More’s Thomas-Fermi ionization estimate [37].
The coupling parameter is defined Γ𝑖 ≡ (𝑍𝑖)2 /𝑎𝑖𝑇 where 𝑎𝑖 = (3𝑍𝑖/4𝜋𝑛𝑒)1/3
and 𝑛𝑒 = (cid:205)𝑖 𝑍𝑖𝑛𝑖. The screening parameter is defined ˜𝜅 ≡ (𝜆𝑠 𝑘 𝐷,𝑒)−1, where
the screening length is given in Stanton and Murillo’s work [31]. Lastly, for 𝜈
we use the temperature relaxation collision rates defined in Haack et al. [28].

. . 47

Table 3.2 Tabulated plasma parameters for contaminated light species plasmas at three
different levels of contamination: 1 carbon atom per 105, 104, 103 light species
atoms. Tabulated plasma parameters are computed in the same way as in
Table. 3.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

. .

. .

.

.

Table 4.1 Material location, number densities, densities, and ionization levels in the

MARBLE preheat problem.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Table 5.1 Visualization of the complex valued neural network’s sequential input data (x)

and output data (Y).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

viii

LIST OF FIGURES

Figure 1.1 Plot mapping the ionization of hydrogen across density and temperature space.
Different plasma examples from Table 1.1 are plotted for comparison. We
see that the National Ignition Facility (NIF) and the Omega laser facility
(Omega) can generate matter within the high energy density region (> 1
MBar). This dissertation will consider plasmas at the NIF direct drive and
pre-heated MARBLE pore conditions.

. . . . . . . . . . . . . . . . . . . . .

.

6

Figure 2.1 The relations between the quantum, classical molecular dynamics, kinetic

theory, and hydrodynamics formalisms. . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.2 Plot indicating the total time elapsed and total spatial extent of various pub-
lished simulations. classical molecular dynamics simulations can reach larger
spatial scales and longer time scales than quantum methods (i.e., orbital-free
and Kohn-Sham molecular dynamics). There is strong overlap between hy-
drodynamics and kinetics simulations. Plot reprinted from Luke Stanek’s
PhD dissertation with author permission. . . . . . . . . . . . . . . . . . . . .

. 28

Figure 3.1 Plots of the long-wavelength expansion of Im{𝜀−1} in the one-component
plasma (OCP) case, for the Mermin (green), completed Mermin (CM, red),
and Atwal-Ashcroft (AA, purple) dielectric functions from Eq(3.37). We
evaluate the functions at ˜𝑘 ≈ 0.05, (𝜔 𝑝𝜏)−1 = .2, 𝜁 2 = 1, 𝜔 𝑝,𝑒 = 1.

. . . . . . . 44

Figure 3.2 Left: A plot of the frequency-sum (f-sum) rule, Eq(3.35), for the random-
phase approximation (RPA, blue), Mermin (green), completed Mermin (CM,
red), and Atwal-Ashcroft (AA, purple) dielectric functions of the Yukawa
one component plasma (YOCP). Only the Mermin dielectric function does
not integrate to −𝜋 (black), and therefore does not satisfy the f-sum rule.
Right: A plot of the screening-sum rule, Eq(3.36), for the single-species RPA
(blue), Mermin (green), and completed Mermin (red) dielectric functions of
the Yukawa one component plasma (YOCP). In both plots, wiggles arise in
the long-wavelength limit because the susceptibilities become Dirac deltas
and numerical integration becomes impossible; we have truncated our plots
before that happens.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. 45

Figure 3.3 Left: A plot of the frequency-sum rule, Eq(3.35), for the RPA (blue), Mer-
min (green), and completed Mermin (red) dielectric functions of the binary
Yukawa mixture (BYM). Only the Mermin dielectric function does not in-
tegrate to −𝜋 (black), and therefore does not satisfy the f-sum rule. Right:
A plot of the screening sum rule, Eq(3.36), for multi-species RPA (blue),
Mermin (green), and completed Mermin (red) of the binary Yukawa mixture
(BYM) case. All results converge to −𝜋 (black) in the long wavelength limit.
In both plots wiggles arise in the long wavelength limit because the suscepti-
bilities become Dirac deltas and numerical integration becomes impossible,
we have truncated our plots before that happens.

. . . . . . . . . . . . . . . . . 46

ix

Figure 3.4 All plots compare 𝑆(𝑘, 𝜔) at fixed 𝑘/𝑘 𝐷,𝑒 = .63 across different DSF models
of the binary Yukawa Mixture (BYM). Each panel contains a pure deuterium
(pure D), deuterium mixed with tritium (mixed D), a pure tritium (pure T)
and tritium mixed with deuterium (mixed T) plotted in gray. Each panel also
contains a pure light species [defined Eq(3.41)] plotted in solid line. The
plots show that all five cases have qualitative agreement across a given set
of conservation laws. However, the completed Mermin model has a stronger
plasmon peak than the RPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 3.5 All plots compare 𝑆(𝑘/𝑘 𝐷,𝑒, 𝜔) at fixed 𝑘/𝑘 𝐷,𝑒 = .63 across different DSF
models of the binary Yukawa Mixture (BYM). Each panel contains different
carbon purity levels for a given multi-species DSF. The black line indicates a
pure light species. The blue dashed line indicates 1 carbon particle per 105
light species particles. The orange dotted line indicates 1 carbon particle per
104 light species particles. The green dotted line indicates 1 carbon particle
per 103 light species particles.

. . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 3.6 We have plotted the real part of the conductivity 𝜎𝑟 to demonstrate how
number conservation violation (𝑎 < 1, plotted as green lines) and momentum
preservation (𝑏 ≠ 0, plotted as red lines) affect Chen et al.’s Drude conductiv-
ity fit (plotted as a solid black line) [Z. Chen, et al., Nature communications,
12.1, 1638, (2021)].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 4.1 Example of a buffer function ℎ(𝑥). As ℎ(𝑥) varies within the interval [0, 1],
it transitions the hybrid model between pure hydrodynamic and pure kinetic
regions. For 𝑥 ≤ 𝑎, a value of ℎ(𝑥) = 0 indicates the continuum regime. For
𝑥 ≥ 𝑏, the transition function is ℎ(𝑥) = 1 and matter is in non-equilibrium.

. . . 72

Figure 4.2 Snapshots of the density (left) and velocity (right) at time t = 1 as obtained
by the kinetic model for different values of R. We increase collisionality by
increasing the non-dimensional hard sphere radius 𝑅. As 𝑅 increases, we
recover the fluid limit. We emphasize that the Euler simulation matches
the Sod analytic solution and that the BGK simulation converges to the Sod
analytic solution in the large R (i.e. hydrodynamic) limit.

. . . . . . . . . . . . 84

Figure 4.3 Estimates of the deviation from equilibrium for different collisionalities via
the Knudsen number definitions in Eqn. Eq(4.87) and Eq(4.88), varied via the
nondimensional particle radius 𝑅, for the Sod problem. For reference, we also
plot the (scaled) density profile of the Euler solution. Left: effective Knudsen
number via integrated deviations from Maxwellian, see Eq. Eq(4.87). Right:
effective Knudsen number via moment ratio, see Eq. Eq(4.88). Both models
for the effective Knudsen number show that the deviation from a Maxwellian
is greatest near the the shock.

. . . . . . . . . . . . . . . . . . . . . . . . . .

. 85

x

Figure 4.4 Sod problem with 𝑅 = 1. Left:

the kinetic-hydro buffer region is placed
in the space interval [1, 3], i.e. away from the shock. Right:
the buffer
region is placed in [5, 7], i.e., around the shock. Since the coupled model
is derived for a system near equilibrium, this assumption is imprinted in the
buffer region. While in the left plot, the coupled model correctly follows
the kinetic and continuum solutions where appropriate, in the right plot, it
tracks the hydrodynamic solution and only transitions to the reference kinetic
solution near the edge of the buffer region.

. . . . . . . . . . . . . . . . . . . . 86

Figure 4.5 Left: Illustration of the Marble-type foam which is studied in this chapter.
Orange represents the CD foam while the purple disks represent the macro-
pores that are filled with TH gas. Note that the CD foam also contains
many smaller micro-pores; for the purpose of this study we consider the foam
region to be a homogenous CD material. Right: Initial densities used in the
1D planar MARBLE pore preheat problem (see Sec. (4.5.2)). We simulate a
200𝜇𝑚 slice of carbon-deuterium foam with a 20𝜇𝑚 hydrogen-tritium pore
located at the center.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 88

Figure 4.6 Left: Transition function ℎ(𝑥) for the MARBLE pore preheat problem (Sec.
4.5.2). The edges of the buffer regions are indicated by the dashed lines.
Right: Material densities together with the edges of the buffer regions. Due
to the sharp interfaces which define the pore, we expect kinetic effects to be
important around the pore. To ensure our assumption that the buffer region is
placed in a hydrodynamic region is satisfied, we place the buffer edges away
from where we expect shock fronts (i.e in the foam).

. . . . . . . . . . . . . . 88

Figure 4.7 Evolution of the material densities as compared across models for the MAR-
BLE pore preheat problem (Sec. 4.5.2). Each heat map has time evolution
In the hybrid
on vertical axis and spatial position on the horizontal axis.
simulation, we visualize the buffer region as dashed lines. Note that initial
conditions plotted in Figure 4.6 Right are a cross section from the heat map
at time 0ps. The top row shows the deuterium in the foam while the bottom
row gives the evolution of the tritium densities. The left to right columns
correspond to the BGK, hybrid, and Euler methods. For every model, these
heat maps show that the pore is compressed for approximately 150ps as the
foam/pore expands. For the Euler model the compression is not nearly as
great as the others. Eventually the pore is compressed enough to trigger
a rarefaction wave. This rarefaction wave looks sharpest for the BGK and
hybrid model because the compression of the pore was greatest.

. . . . . . .

. 89

Figure 4.8 Electric field profiles at early times in the MARBLE pore preheat problem.
The electric fields evolution for the kinetic model (BGK) and Euler model are
indistinguishable so only BGK is presented here. The strong electric fields
accelerate positively charged HT ions The constituents of the plasma quickly
redistribute to diminish the electric fields. (i.e. less than 10 ps)

. . . . . . . . . 90

xi

Figure 4.9 Time evolution of the velocity profiles in the MARBLE pore preheat problem.
The top row shows the velocity in the deuterium foam material, while the
bottom row gives the velocities in the pore tritium. The left to right columns
correspond to the BGK, hybrid, and Euler solutions. Since the hydrodynamic
model assumes a single velocity, the deuterium and tritium are both propelled
inward. In the kinetic and hybrid models, however, the tritium distribution
shows an additional velocity jet at early times which corresponds to a very
small amount of tritium ejected from the pore by the electric field at a large
velocity. Furthermore, tritium ions show a non-zero velocity field beyond
the edges of the pore (90𝜇m - 100𝜇m). This is due to the few ejected ions
which have been sprinkled throughout the CD foam. Note that for display
purposes, the maximum velocity in the color map is set to ±80 km/s to ensure
a representative color map on the region of interest; the velocities in the
ejection ’plumes’ typically exceed ±200 km/s.

. . . . . . . . . . . . . . . . . . 91

Figure 4.10 Comparison of effective Knudsen number Eq(4.87) across species for the
MARBLE pore preheat problem, at time 250ps. The (rescaled) density profile
is shown in the background for reference. As expected, the effective Knudsen
number is larger in carbon and deuterium where there rapid changes in density,
i.e., where the gradient scale length is small. Additionally, the tritium and
hydrogen ejections produce a large Knudsen number which travels into the
buffer region (i.e. [50, 70] and [130, 150]). However the associated densities
are negligible. Thus, the hydrodynamic model does not cause the hybrid
results to differ from the BGK results.

. . . . . . . . . . . . . . . . . . . . . . 92

Figure 4.11 Plots of the effective Knudsen number 𝐾𝑛1 for BGK, hybrid, and Euler so-
lutions of the Marble preheat problem. For reference we have added a solid
white contour which marks where the material number density is less that
1 particle per cc. The minuscule amount of particles ejected by the strong
interface electric fields carry high Knudsen values as they propagate through
the foam until they collide with each other on the periodic boundary condi-
tions. As can be seen by 𝐾𝑛1 in the buffer region for the hybrid method,
the transition to Euler in the buffer region suppresses the high speed, uncol-
lided ejected particles from penetrating further into the foam. In both cases,
the contour indicates that the high Knudsen values occur where a negligible
amount of particles exist. Thus, the Knudsen number of the ejecta does not
corrupt our hydrodynamic assumptions.

. . . . . . . . . . . . . . . . . . . . . 93

xii

Figure 4.12 Each plot contains BGK, hybrid, and Euler density curves for a different
species and time. The rows separate deuterium (top) and tritium (bottom)
density profiles. From left to right, the plots are 250ps, 375ps, and 500ps. The
top row illustrates the propagation of the rarefaction wave through the foam.
The bottom row illustrates the expansion of the pore after peak compression.
The rarefaction wave enters the buffer zone at around 500ps. We can see that a
difference emerges between the coupled model and BGK after that time. The
difference indicates that our assumption that tritium’s distribution function is
at local equilibrium around the wavefront is incorrect and we are therefore
artificially suppressing kinetic dynamics.

. . . . . . . . . . . . . . . . . . . . 94

Figure 5.1 A grid visualizing the relation between Grad’s equations (upper) and Navier-

Stokes’ equations (lower) in both spatial (left) and Fourier space (right).

. . . . 110

Figure 5.2 Left: neural closure Eq(5.5) evolved with second order multi-step neural
closure, labeled as MsNC, plotted alongside Grad’s closure Eq(5.1) evolved
with seventh order Runge-Kutta, labeled as “correct”. The difference between
the trajectories is too small to see. Right: The mean square error incurred at
each update for 20 trajectories, plotted alongside the smoothed median error,
and cumulative smoothed median error. The plot demonstrates that the neural
closure does not have exponentially increasing error. . . . . . . . . . . . . . . . 112

Figure 5.3 Plot of the reducibility of an ensemble of trajectories, as assessed by the
inverse reconstruction error [23]. The ensemble is more reducible at later
times. The ensemble is reduced using PCA into a various number of dimen-
sions (varied along the y-axis) the evolution time is plotted along the x-axis.
The yellow and orange regions indicate that reducing and un-reducing the
data destroys the ensemble’s local structure and the purple and blue regions
indicate that local structure is preserved. Both Grad’s closure and the neural
closure’s evolve to a reduced subspace, though the neural closure is not as
reducible as Grad’s closure at late time. . . . . . . . . . . . . . . . . . . . . . . 113

Figure 5.4 Visualization of DMD on a sliding window. DMD is conducted on only the
subsequence of data contained in the blue band. Left: View of the sliding
window from the global time series level. Right: View of sliding window
from the localized data level. The blue window contains 5 time steps, the first
4 comprise 𝑌 and the last 4 comprise 𝑌 ′.

. . . . . . . . . . . . . . . . . . . . . 114

Figure 5.5 Time series of DMD eigenvalues. Left: Sliding Window DMD conducted on
full Grad’s equations Eq(5.1). Right: Sliding Window DMD conducted on
Grad’s equations. For both Grad’s closure and the neural closure, the eigen-
values separate by orders of magnitude by 𝑡 = 1. This indicates both Grad’s
closure and the neural closure evolve towards a slow manifold. However,
the neural closure converges worse than Grad’s closure because the smallest
eigenvalues do not reduce to numerical zero.

. . . . . . . . . . . . . . . . . . . 115

xiii

Figure 5.6 Plots of the similarity between a DMD eigenvector at time 𝑡𝑖 and at time 𝑡 𝑗
averaged across 20 random initial conditions (similarity measured by com-
plex dot product). The eigenvector corresponding to the largest eigenvalue is
labeled “1”, while the eigenvector corresponding to the second largest eigen-
value is labeled “2”. A clear transition occurs in both Grad’s closure and
the neural closure, where the DMD eigenvectors discovered after 𝑡 = 1 are
similar to each other. However, with the eigenvectors associated to the neural
closure display weaker similarity than the eigenvectors associated to Grad’s
closure. .

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 5.7 Representation of an ensemble of simulations equilibrating according to
Grad’s eqns towards a slow manifold (image altered from [13]).

. . . . . . . . 116

Figure 5.1 Left: A sample trajectory of Grad’s full equations Eq(5.1) updated with RK7,
labeled as “correct”, alongside a multi-step neural closure trajectory, labeled
as MsNC, and a multi-step neural closure trajectory, labeled as MsNODE. The
difference between the MsNODE trajectories and the correct trajectory is large
enough to be visible. Right: The relative mean square error (MSE) incurred
at each update for 20 different trajectories, plotted alongside the smoothed
median error and cumulative smoothed median error. The plot demonstrates
that the error decreases as the order of the multi-step update increases. This
plots demonstrates that using a multi-step neural ODE to update the system
leads to a growing relative error across all orders. Alternatively, the relative
error decreases, across all orders, when conservation of mass, momentum,
and energy is enforced.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xiv

CHAPTER 1

INTRODUCTION

1.1 Definition of a Plasma

Plasmas are systems of charged particles formed through ionization, the process where electrons

are stripped from atoms, creating a mix of free electrons and ions, i.e., atoms that have lost

electrons. Traditionally, physicists are first exposed to ionization in the context of spectroscopy,

Compton scattering, or photoelectric effect. In these cases, ionization occurs in the presence of

electromagnetic radiation. However, ionization can occur if the system’s thermal energy is high

enough for electrons to be knocked loose by collisions. Additionally, ionization can occur if the

system’s pressure is high enough for electrons to be squeezed from their bound states. Examples

of plasmas can be found looking up at the night sky. Stars are massive plasma spheres undergoing

thermonuclear fusion. While thermonuclear fusion can be initiated here on earth, it is not yet an

energy source like the sun is.

Instead, diamond and chip fabrication are more reflective of the

industrial applications of plasma.

The properties of a plasma are distinct from a solid, liquid, or gas because the constituents of

the plasma carry charge. The presence of charge leads to two key properties. The first property

is “quasi-neutrality” meaning that number of positive and negative charges in a plasma tends to

balance out over large distances, resulting in an overall electrically neutral medium. The second

property is “collective behavior” meaning that the long range of the Coulomb force gives rise to

emergent phenomena, i.e., behavior that only occurs when many particles interact simultaneously.

Violations of either of these two properties (e.g, non-neutral plasmas [1]) are interesting and hence

the exceptions prove the rule.

Plasmas exhibit quasi-neutrality because of charge conservation; when a free electron is created

in a plasma, an ion is also created. However, even if the plasma is globally neutral, there will

be local fluctuations in the charge. Fluctuations refer to sub-domains that are not charge-neutral.

Assuming the plasma is in thermodynamic equilibrium, the local deviations in the average charge,

1

which arise from particle interactions, are [2, section 116]

⟨𝑒𝛿𝑛(𝑟)⟩ = 𝑒𝑛0

(cid:16)

𝑒−𝑒Φ(𝑟)/𝑇 − 1

(cid:17)

.

(1.1)

Here Φ(𝑟) is the average potential per particle and 𝑇 has units of eV.

A system’s tendency towards quasi-neutrality can be seen from a back-of-the-envelope calcu-

lation [3]. Consider a charged particle system of electrons and ions with number density 𝑛𝑒 and

𝑛𝐼 respectively. Consider a sphere of fully ionized hydrogen plasma with radius 𝑟 = 10−3 and ion

number density 𝑛𝐼 = 1020. Assume the electron number density is 1% smaller 𝑛𝑒 = 0.99𝑛𝐼, then

the total charge in the sphere is

𝑄net =

𝜋𝑟 3

4
3

𝑒𝑛𝐼
100

and the electric potential at the surface would be

Φ =

1
4𝜋𝜖0

𝑄net
𝑟

=

𝑟 2𝑒
3𝜖0

𝑛𝐼
100

(1.2)

(1.3)

where 𝜖0 is the permittivity of free space. Evaluating this expression indicates that the potential

on our plasma’s surface is approximately 6000 volts, enough to create a 2 mm electrical arc

through air (also more than double the voltage used in Nebraska’s electric chair executions) [4]. In

short, violating quasi-neutrality produces an electric potential which attempts to rectify the uneven

distribution, thus it takes a lot of work to create macroscopic charge imbalances.

Collective phenomena occur in plasmas because the long range electromagnetic force allows

a single charged particle to influence and be influenced by many charged particles. Consider a

test charge in a volume of plasma; a sub-volume of plasma located a distance 𝑟 away acts on the

test charge with a force that diminishes as 1/𝑟 2. However, for a given solid angle (that is, where

Δ𝑟/𝑟 = constant), the volume of plasma acting on the test charge increases as 𝑟 3. [5, section 1.2].

Therefore, while any one particle from the plasma has little impact, the collection of particles can

impact the test charge.

2

A quick derivation shows that background charges screen a test charge’s potential. Consider

Poisson’s equation

−

1
4𝜋

∇2Φ(𝑟) = 𝑒𝛿𝑛(𝑟) + 𝑄𝛿(𝑟)

(1.4)

where 𝑒𝛿𝑛(𝑟) is the local deviation from uniform charge density, given in Eq(1.1), and 𝑄𝛿(𝑟) is a

test charge placed at the origin. In Fourier space, this equation is given as

𝑘 2
4𝜋

Φ(𝑘) = 𝑒𝛿𝑛(𝑘) + 𝑄

(1.5)

Assume that the average potential energy is much greater than the average kinetic energy 𝑈 =

𝑒Φ(𝑟) << 𝑇, this is known as the “weakly coupled” limit. Then Eq(1.1) gives 𝛿𝑛(𝑘) =

−𝑛0𝑒Φ(𝑘)/𝑇.

Inserting this expression into the Poisson equation Eq(1.5) yields the screened-

Coulomb potential

Φ(𝑘) = 4𝜋

𝑄
𝑘 2 + 𝑘 2
𝐷

,

(1.6)

where 𝑘 𝐷,𝑒 = 4𝜋𝑛𝑒𝑒2

𝑇

is the Debye wavenumber. In short, by allowing many electrons to redistribute

themselves, test charges interact with other charges via a screened charge.

Another well known collective phenomenon is plasma oscillation or, if quantized, the plasmonic

mode. A quick derivation can expose a plasma’s ability to produce plasmonic modes. Consider a

slab of quasi-neutral plasma

𝑛𝑒 =

∑︁

𝑍𝑖𝑛𝑖,

(1.7)

𝑖
where 𝑍𝑖 is the effective ionization of the 𝑖th species and 𝑛𝑖 is uniform. If we displace the electrons

by an infinitesimal distance 𝛿𝑥, the portions of the ions and the displaced electrons that do not

overlap are effectively like two plates of a capacitor, each with a surface charge 𝜎 = ±𝑒𝑛𝑒𝛿𝑥. The

resulting electric field is uniform 𝐸 = 4𝜋𝜎 = 4𝜋𝑒𝑛𝑒𝛿𝑥. Because the mass of the ions is much

greater than the mass of the electrons we approximate the ions as stationary and consider the motion

of the electrons. The electrons feel a force

𝐹 ≡ 𝑚𝑒 (cid:165)𝛿𝑥,

𝐹 = −𝑒𝐸 (𝛿𝑥) = −4𝜋𝑒2𝑛𝛿𝑥,

3

(1.8)

(1.9)

which together describe an oscillatory equation of motion

(cid:165)𝛿𝑥 = −𝜔2

𝑝𝛿𝑥.

(1.10)

Here 𝜔 𝑝,𝑒 =

√︃ 4𝜋𝑒2𝑛𝑒
𝑚𝑒

is known as the electron plasma frequency.

Together, charge fluctuation, charge screening, and plasma oscillation form a small subset of the

rich and interconnected behaviors that are characteristic of plasmas. Charge fluctuations demon-

strate that free electrons have non-trivial local arrangements, while maintaining global charge

neutrality. Non-trivial local charge arrangements lead to charge screening, which alter the interac-

tions between particles in a plasma. Charged particle interactions give rise to collective behavior

like plasma oscillations.

Investigating the characteristic behaviors enables plasma scientists to

utilize plasmas in new technologies.

1.2 Classification of a Plasma

Plasmas exist across a wide range of number densities and temperatures.

It is useful to

classify the various types of plasma in 𝜌𝑇 space (density-temperature space) with a collection of

dimensionless parameters. Further, these dimensionless parameters are often expansion parameters

in derivation, e.g., the weakly coupled limit.

The first dimensionless plasma parameter we focus on is the coupling parameter Γ which is the

ratio of an ion’s average potential energy from ion-ion interactions with the ion’s average kinetic

energy

Γ =

4𝜋𝑒2
𝑎𝑠𝑇

.

(1.11)

Here 𝑎𝑠 = (3𝑛𝑒/4𝜋)1/3 is the Wigner-Sietz radius, the average distance between ions in a plasma.

Γ ≪ 1 indicates a weakly coupled plasma where kinetic energy dominates. This is the assumption

we used to derive screened Coulomb Eq(1.6). Whereas Γ ≫ 1, implies a strongly coupled plasma.

In contrast with the screened Coulomb interaction, a purely repulsive force, we expect that the

ion-ion interactions acquire attractive wells in the strongly coupled limit [6, chapter 10].

The dimensionless quantum degeneracy parameter Θ is the ratio of the electron’s average kinetic

4

energy at a given temperature 𝑇 to its kinetic energy at 𝑇 = 0, i.e., its Fermi energy

Θ =

𝑇
𝐸𝐹

.

(1.12)

Θ is crucial in characterizing the significance of quantum effects. When Θ ≪ 1 the electrons

interact with each other as a quantum degenerate fermi gas; the thermal de Broglie wavelength

exceeds the interatomic spacing. Conversely, Θ ≫ 1 indicates electrons follow a Maxwellian

distribution, where quantum effects can be neglected.

A collection of hydrogen (H) plasmas, along with their respective densities, temperatures, and

dimensionless parameters, are listed in Table 1.1. These example plasmas are plotted on a heat map

of H’s ionization in Figure 1.1. The Saha ionization estimates for both ideal and Van der Waal’s

equation of state (EoS) were computed using [7]

𝑍 2
1 − 𝑍

=

1
𝑛𝐻

(cid:18) 2𝜋𝑚𝑒𝑇
ℎ2

(cid:19) 3/2

𝑒𝐸𝐻 −𝐼1/𝑇

(1.13)

where 𝐸𝐻−𝐼1 = −13.6 eV the ionization energy of an electron in the Hydrogen ground state. The

Thomas-Fermi ionization heat map was generated using More’s Thomas-Fermi fit [8, Table IV].

The 1 MBar pressure line was generated using the ideal gas equation of state.

Table 1.1 Table identifying the density 𝜌, temperature 𝑇, ion-ion coupling parameter Γ, and electron
degeneracy parameter Θ of various plasmas. Omega and NIF estimates indicate the density and
temperature at peak capsule compression in inertial confinement fusion (ICF) experiments [9, 10].

Physical System
Typical Fusion Reactor [5]
Ideal Fusion Reactor [5]
Lightning [11]
Jupiter Interior [12] [13]
Omega Direct Drive [9]
NIF Direct Drive [9]
Solar Interior [14]
NIF In-direct Drive [10]
MARBLE Pre-heated Pore [15]
White Dwarf [16]

𝜌 (g/cc)
4.2 × 10−5
4.2 × 10−4
1.3 × 10−4
3.0 × 101
1.8 × 102
8.8 × 102
1.5 × 102
1.5 × 103
2.0 × 10−3
1.0 × 104

𝑇 (eV)
1.0 × 102
3.0 × 104
2.1 × 100
2.2 × 100
3.4 × 102
9.6 × 102
1.3 × 103
1.4 × 104
5.0 × 101
1.0 × 100

Γ
0.01
0.00
0.16
5.34
0.229
0.15
0.06
0.01
0.03
467.64

Θ
0.00
0.00
0.02
12.44
1.470
1.53
0.35
0.15
0.00
15,584.14

5

Figure 1.1 Plot mapping the ionization of hydrogen across density and temperature space. Different
plasma examples from Table 1.1 are plotted for comparison. We see that the National Ignition
Facility (NIF) and the Omega laser facility (Omega) can generate matter within the high energy
density region (> 1 MBar). This dissertation will consider plasmas at the NIF direct drive and
pre-heated MARBLE pore conditions.

1.3 High Energy Density Science

The upper right portion of 𝜌𝑇 space is the high energy density (HED) regime, defined as having

a pressure in excess of 1 Mbar [17]. HED can be loosely understood as the regime that is too dense

for ideal plasma theory to work [5]. A subsection of the HED regime is the warm dense matter

(WDM) regime, located in 𝜌𝑇 space between the ideal gas regime (low 𝜌 and high 𝑇) and the solid

state regime (high 𝜌 and low 𝑇). WDM is characterized as having an ion-ion coupling parameter

Γ and an electron quantum degeneracy parameter Θ of order 1 [18].

In experimental settings, HED plasmas can be produced with direct laser drive, laser driven

hohlraums (i.e. indirect laser drive), Z-pinches, ultra-fast lasers, or high energy density beams [5].

The final two methods, ultra-fast and high energy density lasers, produce pressure greater than a

6

MBar by creating hot and fast (i.e. relativistic) charged particles. Such systems are not extremely

dense. Alternatively, the first three methods (in/direct drive and Z pinch) use inertial confinement to

create a plasma that is warm and dense, where the combination of high densities and temperatures

produce pressures in excess of 1 MBar.

This dissertation centers on high energy density plasmas created through inertial confinement.

This approach is taken in the Laboratory for Laser Energetic (LLE) Omega laser system, Lawrence

Livermore’s National Laboratory’s (LLNL) National Ignition Facility (NIF), and Sandia National

Laboratory’s (SNL) Z-pinch machine. The basic idea behind inertial confinement is to compress

a small capsule that is made of high Z material (e.g. diamond/carbon) and filled with deuterium

and tritium (DT) gas. Compression is achieved via a collection of high intensity lasers which heat,

ionize, and vaporize the high Z shell. As the outer shell explodes radially outward, conservation of

momentum causes the inner shell to implode radially inward. This process is known as ablation, and

the rapid compression causes the deuterium-tritium fuel to reach extreme densities and temperatures,

see Figure 1.1 and Figure 1.1. The energies and densities can get high enough that fusion events

are possible, so inertial confinement is often referred to as inertial confinement fusion (ICF).

Experimental diagnostics rely on a collection of techniques to estimate material properties

[19]. Since HED matter exists for a short time in a confined space at conditions that damage

nearby equipment, scattered and emitted electromagnetic spectra are commonly used to infer the

dynamic conductivity of the HED plasma [20, 21] and X-ray Thompson scattering (XRTS) is used

to infer the plasma’s properties, i.e., number density, temperature, and ionization [18, 22]. Further,

bremsstrahlung radiation emitted from the plasma can be used to observe localized mixing [23, 24].

Each of these experimental diagnostics use an observed signal to fit a parameterized model and

thus extract the relevant properties. Therefore, models of a plasma’s response to electromagnetic

signals enable HED instruments.

A considerable number of semi-empirical and first-principles models have been created to de-

scribe the dynamic response of a collisionally damped charged particle system. However, known

challenges persist in established dynamic structure factors (DSF), dielectric functions, and conduc-

7

tivities. For instance, the semi-empirical Drude-Smith conductivity [25] lacks interpretability and

the first principles Mermin dielectric function [26] does not satisfy the frequency sum rule [27].

In this dissertation, we will present a new dynamic response function for multi-species plasmas as

well as a new dynamic non-Drude conductivity model.

Furthermore, extensive simulations are conducted to compliment and inform HED experiment.

Historically, HED scientists have relied on augmenting radiation–hydrodynamic (rad-hydro) codes.

For example, the Eulerian rad-hydro code RAGE [28] was upgraded to the xRAGE code to describe

ICF experiments [29]. However, Eulerian codes assume the plasma’s mean free path divided by

its characteristic length (i.e.

the Knudsen number) is very small and thus the system is in local

equilibrium. This assumption removes kinetic phenomena and leads to predictions that do not match

experiment. ICF experiments can be parsed into four stages: the early stage, the acceleration stage,

the deceleration stage, and the peak-compression/burn stage. Each stage is characterized by unique

physical conditions and processes where kinetic effects play crucial roles [10]. A review of kinetic

phenomena (i.e. non-equilibrium physics) in ICF is given by Renderknecht et al. [30]. Fully kinetic

simulations would be able to describe all this behavior, but from a computational perspective such

approaches are usually very expensive or simply not feasible. In this dissertation, we investigate

computationally efficient methods for including kinetic effects like velocity/temperature separation

and dissipative processes (i.e. electro-diffusion, baro-diffusion, thermo-diffusion, viscosity, thermal

conduction).

The separation of velocity and temperature between species is not accounted for in current

radiation-hydrodynamics codes, despite evidence of this phenomenon from experiments [31, 32, 33]

and fully kinetic simulations [34]. Eulerian codes cannot account for separation between species

because they rely on a single bulk momentum and temperature equation [28, 35, 36]. To use a

single transport equation these codes combine species’ conductivities into a single cell conductivity

[37], but investigations have found that predictions of ICF capsule performance and X-ray flux from

vacuum hohlraums are sensitive to how species’ conductivies were combined into a cell conductivity

[38, 39]. In particular, ICF capsule performance are more sensitive to how species’ conductivities

8

were combined into a cell conductivity than to the species’ conductivities [38]. Haack et al. has

shown that going beyond local equilibrium (i.e., Eulerian fluids), to a near equilibrium Navier-

Stokes assumption, leads to separated velocities, but not temperatures [40]. In this dissertation,

we present a new model hybrid model which can smoothly transition between Haack et al.’s

kinetic multi-species PDE and multi-species hydrodynamic PDEs. This hybrid model provides

computationally efficient kinetics, allowing for both velocity and temperature separation as well as

avoiding concocting mixture conductivities, but only where such are needed.

Additionally, dissipation is needed to model inertial confinement. Electro-diffusion is known

to rocket particles with a large charge to mass ratios across the interfaces [34, 41, 42]. There are

experimental observations at the Omega laser facility that suggest electro-, baro-, and thermal-

diffusive processes are more important than hydrodynamic instabilities for multi-material mixing

[43]. Additionally, experimental observations suggest that viscosity plays an important role in

stabilizing the plasmas laser driven shocks propagate through the capsule [44]. Further, studies

have shown that viscosity is needed to resolve the discrepancy between simulation and experiment

[45, 46, 47]. To account for these dissipative processes, extended moment hydrodynamic equations

are employed [48]. In this dissertation, we explore using neural networks as a tool to inform the

dissipation in extended moment hydrodynamic simulations. Further, we use dimension reduction

techniques and dynamic mode decomposition to quantitatively assess a neural network’s ability to

inform dissipation.

1.4 Thesis Organization

This thesis addresses current topics in HED physics as described in Section 1.3. Chapter

2 develops the many-body formalism necessary to describe the electromagnetic response of a

plasma and non-equilibrium phenomena. Chapter 3 details the electromagnetic response of a

collisional multi-species plasma. It derives a multi-species susceptibility from the multi-species

Bhatnagar-Gross-Krook (BGK) kinetic equation, introduces a new dynamic non-Drude conductiv-

ity model, and presents a one-to-one correspondence between the phenomenological Drude-Smith

conductivity and Mermin’s number conserving conductivity. Chapter 4 introduces a multi-species

9

kinetic-fluid coupling for high-energy density simulations, deriving a set of coupled partial differ-

ential equations (PDEs) that include both the multi-species BGK model and its limiting Euler or

Navier-Stokes hydrodynamic equations. Chapter 5 explores a data-driven approach to incorporating

dissipation and methods for assessing the quality of the dissipation. It describes training a neural

network to forecast the pressure tensor and heat flux, and employing data compression algorithms

and dynamic mode decomposition to characterize the system’s invariant manifold. Finally, chapter

6 summarizes and draws major conclusions from the work.

10

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14

CHAPTER 2

MULTI-SCALE NATURE OF PLASMA

The dynamics of plasmas are inherently multi-scale. Many-body physics provides a framework to

describe the dynamics at each scale. In this chapter, we begin by discussing classical molecular

dynamics (MD) which describes the system in terms of deterministic dynamics of individual

particles. This approach is entirely based on evolving Hamilton’s equations of motion. Then we

discuss kinetic theory, a statistical framework that describes the system in terms of distributions

of particle velocities and positions. This approach is encapsulated in evolving the Boltzmann

integro-differential equation or the Bhatnagar-Gross-Krook (BGK) partial differential equation.

Finally, we discuss fluid dynamics, which coarse-grains away the details of individual particles

and describes the system with position dependent density, velocity, and temperature fields. This

approach is exemplified by the Navier-Stokes equations. Together, these frameworks provide

insight into plasma: molecular dynamics describes the microscopic scale, kinetic theory describes

the mesoscopic scale, and fluid dynamics describes the macroscopic scale.

Each of these theories has unknowns which must be supplied to close the governing equations;

these unknowns are referred to as closure information. For fluid dynamics, the plasma’s pressure

tensor 𝑃, heat flux (cid:174)𝑞, and dielectric function 𝜀(𝑘, 𝜔) are closures that can be provided by kinetic

theories. However, kinetic theories themselves require closures. Two examples of kinetic closures

are the collision cross sections 𝜎 needed to evolve the Boltzmann equation and the effective

relaxation/collision rate 𝜈 needed to evolve the BGK kinetic equation. These closures can be

provided by molecular dynamics simulations which track individual particles. However, the closure

problem also affects molecular dynamics, where effective inter-atomic potentials (informed by

quantum mechanics) are needed to evolve Hamilton’s equations. It’s turtles all the way down! At

every stage a finer scale theory is needed to inform the necessary closure information. Table 2.1

organizes the hierarchy of closure information described in this paragraph.

The hierarchical structure of closures is no accident.

In the early 1900s Hilbert posed 23

problems, his sixth problem called for a rigorous derivation showing how macroscopic degrees of

15

Table 2.1 Tabulation of theories from fine to coarse degrees of freedom. Each theory’s relevant
dynamical equations and closure information are identified.

Theory
Classical Molecular Dynamics
Kinetic Theory
Kinetic Theory
Hydrodynamics

Example Equation
Hamilton’s Equations
Boltzmann’s Equation
BGK’s Equation
Navier-Stokes’ Equations

Closure Information
Interaction Potential 𝜙
Collision cross section 𝜎
Collision rate 𝜏
𝑃, (cid:174)𝑞, and 𝜖

Figure 2.1 The relations between the quantum, classical molecular dynamics, kinetic theory, and
hydrodynamics formalisms.

freedom (e.g. density, momentum, and energy fields in a fluid) emerge from microscopic degrees

of freedom (e.g., Hamilton’s equations). The problem proved influential, works are still being

published in an endless pursuit of rigor [1].

Typical answers to Hilbert’s sixth problem have the following form. Microscopic degrees of

freedom (DoF) are integrated over, leaving behind only macroscopic DoF. However, the coupling

between macro and micro DoF typically produces a hierarchy of equations, preventing the integra-

16

tion from producing a closed system. The “truncation” assumption cuts off the hierarchy producing

a closed system of equations. These assumptions typically hinge on the time scale separation

between the coarse and fine grain theories (n-body, kinetic, and hydrodynamic descriptions). Here

the hydrodynamic description corresponds to the longest time scale [2, 3]. A diagram organizing

the answer to Hilbert’s sixth problem according to integration and truncation steps is provided in

Figure 2.1.

An equivalent framing of Hilbert’s sixth problem uses invariant manifolds. This framework is

typically used to interpret extended moment hydrodynamics models [1]. An invariant manifold is

a subset of the system’s phase space characterized as being invariant under the system’s dynamical

equations. This implies that if the system’s state is on the invariant manifold at one point in time,

it will remain on the manifold at all future times. The slow manifold is a subset of the invariant

manifold, containing only the longest lasting modes, it forms a low-dimensional model of the

system’s long time behavior [4]. Muncaster frames the process of constructing coarse grained

theories from fine theories as a process of constructing the slow invariant manifold of the fine

grained theory [5]. This is extended by Gorban and Karlin who formulate hydrodynamics as the

problem of discovering invariant manifolds in the space of distribution functions [6]. For more

details, Gorban and Karlin give a thorough introduction to this framing of Hilbert’s sixth problem

[7].

Every major result of this thesis constitutes closure information; this chapter shows the reader a

small sample of where and how closure information arises. In this chapter, we go over the essential

steps laid out in Figure 2.1. First, we identify how classical molecular dynamics emerges from

Schrödinger’s equation. Then we distill kinetic theory from classical molecular dynamics. Finally,

we derive fluid dynamics as a limiting case of kinetic theory. The many body formalism developed

in each section will be applied to more substantive problems in later chapters.

2.1 Classical Molecular Dynamics

Molecular dynamics (MD) delves into the detailed interactions and trajectories of individual

particles. At this level, the role of quantum mechanics becomes apparent, as it treats individual

17

particles as probability waves. The Schrödinger equation, which describes the wave state evolution,

is the typical description. The multi-scale nature of molecular dynamics is exemplified by Born-

Oppenheimer MD, which uses quantum mechanics to calculate the forces on classical nuclei. In

this section, we integrate out the quantum electronic DoF and truncate the system of equations to

arrive at classical MD [8, Chapter 10].

Consider the partition function Z of a system of ions and electrons

Z = N Tr

(cid:110)

𝑒−𝛽(𝐻𝐼 +𝐻𝑒+𝑈𝐼𝑒)(cid:111)

(2.1)

Here N is a normalization constant, 𝐻𝐼 is the ion Hamiltonian, 𝐻𝑒 is the electron Hamiltonian,

and 𝑈𝐼𝑒 is the ion-electron interaction energy. "For brevity, we neglect the one-body potential 𝑈ext

arising from the external field, but will introduce it into the final expression Eq(2.14). 𝐻𝐼 commutes

with 𝐻𝑒 and if we assume that the ions are classical, then 𝐻𝐼 also commutes with 𝑈𝐼𝑒. By the

Baker-Campbell-Hausdorff identity, this produces

Z = N

∫ 𝑁
(cid:214)

𝑖=𝑠

𝑑𝑟𝑖𝑑𝑝𝑖𝑒−𝛽𝐻𝐼 Tr

(cid:110)

𝑒−𝛽(𝐻𝑒+𝑈𝐼𝑒)(cid:111)

Integrating over electronic DoF alters the classical exponent, producing

Z = N

𝑁
(cid:214)

𝑖=𝑠

𝑑𝑟𝑖𝑑𝑝𝑖𝑒−𝛽(𝐻𝐼 +𝐹𝐼𝑒).

(2.2)

(2.3)

Here we have inserted the free energy 𝐹𝐼𝑒 ≡ −𝑇 ln 𝑍𝐼𝑒, where 𝑍𝐼𝑒 = Tr (cid:8)𝑒−𝛽(𝐻𝑒+𝑈𝐼𝑒)(cid:9). From

Eq(2.3), we identify the effective Hamiltonian governing the ion equations of motion

𝐻eff

𝐼 = 𝐻𝐼 + 𝐹𝐼𝑒.

(2.4)

Notice the coupling 𝑈𝐼𝑒 between ion and electron DoF persists within 𝐹𝐼𝑒, so for this effective

Hamiltonian the ion equations of motions would be coupled to the electron equations of motion.

Let us examine the functional form of 𝐻eff
𝐼

. Assuming a standard coulomb interaction, the

Fourier representation of the ion Hamiltonian is [8, chapter 10]

𝐻𝐼 =

𝑝2
𝑖
2𝑚𝑖

+

1
2𝐿3

∑︁

𝑖

∑︁

(cid:16)

𝑣 𝐼 𝐼 (𝑘)

𝑛𝐼 (𝑘)𝑛𝐼 (−𝑘) − 𝑁𝐼

(cid:17)

.

(2.5)

𝑘

18

where 𝑣 𝐼 𝐼 (𝑘) = 4𝜋𝑍 2

𝐼 𝑒2/𝑘 2. Next, formulate 𝐹𝐼𝑒 in terms of tractable quantities. We proceed
by treating the ion-electron interaction as weak and associating a coupling constant 𝜆 with the

interaction. In this approximation, the effective free energy is [9]

𝐹𝐼𝑒 = 𝐹𝑒 +

∫ 1

0

𝑑𝜆 ⟨𝑈𝐼𝑒⟩𝜆

(2.6)

where ⟨·⟩𝜆 indicates an ensemble average with weight exp[−𝛽(𝐻𝑒 + 𝜆𝐻𝐼𝑒)]. Assuming periodic

boundary conditions of length 𝐿, then the Hartree interaction, in momentum space, is given as

𝑈𝐼𝑒 (𝑘, 𝜆) =

1
𝐿3

∑︁

𝑘≠0

𝑣 𝐼𝑒 (𝑘)𝑛𝐼 (𝑘)𝑛𝑒 (𝑘),

(2.7)

up to some constant 𝑈0. The simplest assumption is the Coulomb approximation 𝑣 𝐼𝑒 (𝑘) =

4𝜋𝑍𝑖𝑒2/𝑘 2.

In some cases, a hard-core ion-electron interaction model is used [10]. At 𝜆 = 0,

the electron density is unaltered by the presence of ions ⟨𝑛𝐼 (𝑘)𝑛𝑒 (−𝑘)⟩𝜆=0 = 0. However, at the

next order the electron positions are altered by the presence of ions. We write the Eq(2.6) as

𝐹𝐼𝑒 = 𝐹𝑒 +

1
𝐿3

∑︁

𝑘≠0

𝑣 𝐼𝑒 (𝑘)𝑛𝐼 (𝑘)

∫ 1

0

𝑑𝜆 ⟨𝛿𝑛𝑒 (𝑘)⟩𝜆.

(2.8)

Here ⟨𝛿𝑛𝑒 (𝑘)⟩𝜆 captures the magnitude by which the ions alter the electron from its uniform

distribution.

Next we need to evaluate the integral in Eq(2.8). Assuming that the density perturbations of

electrons are time-independent and can be described by linear response theory,

𝛿𝑛𝑒 (𝑘) = 𝐶𝑒,0(𝑘) (𝑣𝑒𝑒 (𝑘)𝛿𝑛𝑒 (𝑘) + 𝜆𝑣 𝐼𝑒 (𝑘)𝑛𝐼 (𝑘))

(2.9)

where 𝐶𝑒,0(𝑘), sometimes denoted 𝜒, is the static susceptibility of the ideal electron gas. Rear-

ranging this equation yields

𝛿𝑛𝑒 (𝑘) = 𝜆

(cid:18)

𝐶𝑒,0(𝑘)
1 − 𝑣𝑒𝑒 (𝑘)𝐶𝑒,0(𝑘)

(cid:19)

𝑣 𝐼𝑒 (𝑘)𝑛𝐼 (𝑘).

(2.10)

Inserting Eq(2.10) into Eq(2.8) yields our final result

𝐹𝐼𝑒 = 𝐹𝑒 +

1
2𝐿3

∑︁

𝑘

𝐶𝑒 (𝑘)|𝑣 𝐼𝑒 (𝑘)|2𝑛𝐼 (𝑘)𝑛𝐼 (−𝑘),

(2.11)

19

where

𝐶𝑒 (𝑘) ≡

𝐶𝑒,0(𝑘)
1 − 𝑣𝑒𝑒 (𝑘)𝐶𝑒,0(𝑘)

.

(2.12)

We have now formulated 𝐹𝐼𝑒 in terms of ion density and electron susceptibility, cleanly separating

the electron and ion DoFs.

Our ultimate goal, deriving classical equations of motion for the ions, is within reach. Inserting

Eq(2.5) and Eq(2.11) into Eq(2.4) yields

𝐻eff

𝐼 =

𝑝2
𝑖
2𝑚𝑖

+

1
2𝐿3

∑︁

𝑖

∑︁

(cid:16)

𝑣 𝐼 𝐼 (𝑘) + 𝐶𝑒 (𝑘)|𝑣 𝐼𝑒 (𝑘)|2(cid:17) (cid:16)

𝑛𝐼 (𝑘)𝑛𝐼 (−𝑘) − 𝑁𝐼

(cid:17)

+ 𝐹′
𝑒,

(2.13)

𝑘

where 𝐹′

𝑒 = 𝐹𝑒 + 𝑁𝐼
2𝐿3

(cid:205)𝑘 𝐶𝑒 (𝑘)|𝑣 𝐼𝑒 (𝑘)|2. In Eq(2.13), the electron and ion DoF are separated and

the associated Hamilton’s equations are,

(cid:164)𝑟𝑖 = 𝑝𝑖/𝑚,
(cid:32)
𝑈ext(𝑟𝑖) +

(cid:164)𝑝𝑖 = −∇𝑟𝑖

(cid:33)

𝑈eff

𝐼 𝐼 (|𝑟𝑖 − 𝑟 𝑗 |)

.

∑︁

𝑗≠𝑖

(2.14a)

(2.14b)

In these equations, {𝑟𝑖, 𝑝𝑖} represents the phase space position of the 𝑖𝑡ℎ ion. The effective 2 body

interaction potential is defined

𝑈eff

𝐼 𝐼 (𝑟) ≡

∫

1
2𝐿3

𝑑3𝑘 𝑒𝑖𝑘𝑟 (cid:16)

𝑣 𝐼 𝐼 (𝑘) + 𝐶𝑒 (𝑘)|𝑣 𝐼𝑒 (𝑘)|2(cid:17) (cid:16)

𝑛𝐼 (𝑘)𝑛𝐼 (−𝑘) − 𝑁𝐼

(cid:17)

.

(2.15)

Thus, at linear order, an effective two-body interaction is the only required closure needed for

classical molecular dynamics. Beyond linear response 3 through 𝑁 body forces emerge [10].1 This

concludes our derivation of MD closure information using linear response theory. Eq(2.14) forms

the foundation of kinetic models and Eq(2.15) demonstrate one of many uses for the susceptibility

(which will be computed in Chapter 3).

2.2 Kinetic Theory

Kinetic theory, sometimes called non-equilibrium statistical mechanics, serves as a bridge

connecting the motion and interactions of individual particles to observable properties of matter,

1In the high temperature limit, Eq(2.15) easily produces the earlier derived Yukawa interaction potential. Other

closures have been computed using Eq(2.15) [11, 12, 10].

20

such as density, bulk velocity, temperature, and pressure. The multi-scale nature of kinetic theory

is exemplified by the classical Liouville equation, which describes the temporal evolution of the

𝑁 particle system. The equation is instrumental in linking the detailed, microscopic behavior of

particles with the emergent, macroscopic properties of the system because it describes the evolution

in terms of a probability distribution function.

Consider a system of 𝑁 charged particles, whose motion is described by the classical MD

equations previously derived in Eq(2.14). The Hamiltonian is

𝐻 =

𝑁
∑︁

𝑖=1

𝐻 (1)

𝑖 +

1
2

∑︁

𝑖≠ 𝑗

𝑈eff

𝐼 𝐼 (|𝑟𝑖 − 𝑟 𝑗 |)

where 𝐻 (1) is the one-body Hamiltonian

𝐻 (1)

𝑖 ≡

𝑝2
𝑖
2𝑚𝑖

+ 𝑈ext
𝑖

(𝑟𝑖)

(2.16)

(2.17)

The mean field (MF) approximation is used to incorporate the collective behavior of our plasma

system through 𝑈ext(𝑟𝑖). In the mean field approximation, 𝑈ext(𝑟𝑖) is related to the electric potential

Φ(𝑟), which is produced by the total system of charged particles and taken as an external field for

any given particle. In the electro-static approximation, the Poisson equation defines the electric

potential as

𝑘 2
4𝜋

Φ(𝑘) =

∑︁

𝛼

𝑞𝛼𝑛𝛼 (𝑘)𝜖 −1

𝛼 (𝑘).

(2.18)

Here 𝛼 and 𝛽 indexes the ion species and 𝜖 −1

𝛼 (𝑘) is the position-dependent dielectric function

defined in terms of the static susceptibility 𝐶𝛼,𝛽 (𝑘),

𝜖 −1
𝛼 (𝑘) ≡ 1 +

∑︁

𝛽

𝛼𝛽 (𝑘)𝐶𝛽 (𝑘).
𝑣eff

(2.19)

A given particle couples to this potential by its charge. If we include time dependence, then further

Maxwell equations are needed.

Now let us proceed to the objective, reformulate the evolution of 𝑁 particles as an evolution of

a 𝑁 body probability distribution. First, define the function 𝑓 (𝑁) ({𝑟𝑖}, {𝑝𝑖}, 𝑡), with 𝑖 = 1, . . . , 𝑁,

21

which is the local density of particles in 𝑁 body phase space. Since 𝑓 (𝑁) is a probability distribution

it is normalized such that

∫ (cid:32)

(cid:214)

(cid:33)

𝑑𝑟𝑖𝑑𝑝𝑖

1 =

𝑖

𝑓 (𝑁) ({𝑟𝑖}, {𝑝𝑖}, 𝑡).

Given that probability is locally conserved, 𝑓 (𝑁) must satisfy the continuity equation,

𝜕𝑡 𝑓 (𝑁) + ∇𝑟,𝑝 · u𝑟,𝑝 𝑓 (𝑁) = 0,

(2.20)

(2.21)

The phase space velocity is u𝑟,𝑝 = ( (cid:164)𝑟1, . . . , (cid:164)𝑟𝑁 , (cid:164)𝑝1, . . . , (cid:164)𝑝𝑁 ). Next, we simplify this equation using

the product rule and Hamilton’s equations Eq(2.14) to arrive at the material derivative formulation

of the continuity equation

𝜕𝑡 𝑓 (𝑁) + u𝑟,𝑝 · ∇𝑟,𝑝 𝑓 (𝑁) = 0.

(2.22)

Finally, we recognize u𝑟,𝑝 · ∇𝑟,𝑝 as the Poisson bracket and we arrive at the classical Liouville

equation

𝜕𝑡 𝑓 (𝑁) + { 𝑓 (𝑁), 𝐻} = 0.

(2.23)

The characteristic curves of the Liouville equation Eq(2.23) are defined by Eq(2.16)’s associated

Hamilton equations [3]. Thus, the Liouville equation is the crucial step between the particle de-

scription and the statistical description. From here on we can describe our system with probabilistic

approaches.

We will now determine the dynamics of the 1 body distribution. The 𝑠 body marginal distribution

is defined by integrating the 𝑁 body distribution over the phase space coordinates of 𝑁 − 𝑠 particles
∫ (cid:32) 𝑁
(cid:214)

(cid:33)

𝑑𝑟𝑖𝑑𝑝𝑖

𝑓 (𝑁) (𝑟𝑖, 𝑝𝑖, 𝑡).

(2.24)

𝑓 (𝑠) (𝑅1, 𝑝1, 𝑡) =

𝑁!
(𝑁 − 𝑠)!

𝑖=𝑠

The dynamics of the one-body distribution, denoted 𝑓 instead of 𝑓 (1) for brevity, can be computed

explicitly from Eq(2.23) by integrating over the coordinates of 𝑁 − 1 particles. The result is [13,

Chapter 2]

𝜕𝑡 𝑓 (𝑟1, 𝑝1, 𝑡) =

(cid:110)

𝑓 (𝑟1, 𝑝1, 𝑡), 𝐻 (1)(cid:111)

+ 𝑄

(2.25)

22

where 𝑄, sometimes denoted as

𝛿 𝑓
𝛿𝑡

(cid:12)
(cid:12)
(cid:12)𝑐

, is the collision operator

∫

𝑄 ≡

𝑑𝑟2𝑑𝑝2∇𝑟1

𝐼 𝐼 (𝑟1 − 𝑟2) · ∇𝑝1
𝑉 eff

𝑓 (2) (𝑟1, 𝑟2, 𝑝1, 𝑝2, 𝑡).

(2.26)

Eq(2.25) is the first equation in the BBGKY hierarchy. We see the evolution of the 1 body evolution

depends on the evolution of the 2 body distribution. This persists in higher 𝑛 body dynamics which

depend on 𝑛 + 1 body dynamics. Thus, even though Eq(2.25) is formally exact, it is not a closed

equation. The unknown quantity 𝑄 arises from the interactions between DoFs which were integrated

and DoFs that were not integrated. We will spend the remainder of the section reformulating 𝑄 to

close the equation.

Many approaches have been developed to express the collision operator 𝑄 solely in terms of 𝑓

[14], and new operators geared towards plasmas continue to be proposed [15]. However, in keeping

with Figure 2.1, we aim to compute the Boltzmann equation. To do so we would integrate Eq(2.23)

to determine the dynamics of 𝑓 (2), assume the three-body collision operator is zero (i.e., 𝑓 (3) = 0)

and reduce the two equations.However, this process is laborious and not relevant to subsequent

chapters. Interested readers are referred to Tong’s kinetic theory notes [13, Chapter 2] or Liboff’s

introductory textbook [3, Chapter 3].

Instead, we show how to derive the BGK collision operator from the Boltzmann collision

operator. The well-known Boltzmann collision operator is given

∫

𝑄 =

𝑑Ω 𝑑3 𝑝2 𝑔𝜎 (cid:0) 𝑓 (𝑟, 𝑝′
1

, 𝑡) 𝑓 (𝑟, 𝑝′
2

, 𝑡) − 𝑓 (𝑟, 𝑝1, 𝑡) 𝑓 (𝑟, 𝑝2, 𝑡)(cid:1) .

(2.27)

Here 𝑝1 is the momentum of particle 1 pre-collision, 𝑝′

1 is the momentum of particle 1 post-
collision, 𝑔 = |p1 − p2| is the relative pre-collision momentum, and 𝜎(𝑔, Ω) is the differential cross

section. Suppose that the velocity distribution is in equilibrium after collisions, then we represent

the post-collision distribution function as a Maxwellian 𝑀,

∫

𝑄 =

𝑑Ω 𝑑3 𝑝2 𝑔𝜎 (𝑀 (𝑟, 𝑝1, 𝑡) 𝑀 (𝑟, 𝑝2, 𝑡) − 𝑓 (𝑟, 𝑝1, 𝑡) 𝑓 (𝑟, 𝑝2, 𝑡)) .

(2.28)

Suppose 𝑓 (𝑟, 𝑝2, 𝑡) is represented well by a Maxwellian. This approximation is not motivated by a

near-equilibrium assumption. Rather, 𝑀 (𝑟, 𝑝2, 𝑡) is defined such that its 1, 𝑣, 𝑣2 moments produce

23

the moments of 𝑓 (𝑟, 𝑝2, 𝑡). The final result is the BGK collision operator

𝑄 =

1
𝜏

(𝑀 (𝑟, 𝑝, 𝑡) − 𝑓 (𝑟, 𝑝, 𝑡)) .

where we have relabeled 𝑝1 → 𝑝 and the collision frequency is defined

∫

= 𝜈 ≡

1
𝜏

𝑑Ω 𝑑3 𝑝2 𝑔𝜎 𝑓 (𝑟, 𝑝2, 𝑡).

(2.29)

(2.30)

Dimensionally, 𝜏 represents a rate, and from Eq(2.25) we can see that this is the rate at which the

one-body distribution relaxs towards a Maxwellian.

We have detailed the closures Eq(2.30) and Eq(2.19) needed to evolve the BGK kinetic equation

Eq(2.25). Insert Eq(2.18) and Eq(2.26) into Eq(2.25) and expand out the Poisson Bracket to arrive

at an expression for the BGK kinetic equation

𝜕𝑡 𝑓 + v · ∇x 𝑓 + F · ∇p 𝑓 =

1
𝜏

(𝑀 (𝑟, 𝑝, 𝑡) − 𝑓 (𝑟, 𝑝, 𝑡)) .

(2.31)

The force is defined as 𝑚a = −∇𝑈ext. This concludes our derivation of kinetic equations from

N-body MD. We have completed the step from microscopic DoF to intermediate mesoscopic DoF.

We see that the effective ion-ion interaction, the dielectric function 𝜖 (related to susceptibility via

Eq(2.19)), and the collision rate 𝜈 are necessary closures for the BGK kinetic equation.

2.3 Hydrodynamics

As the coarsest theory, fluid dynamics formulates the system entirely in terms of observable

thermodynamic quantities, it describes the bulk motion of momentum, mass, and energy in a fluid.

Each equation (momentum, mass, and energy) corresponds to the dynamics of a moment of the

one-body distribution function derived in the previous section. In effect, Navier-Stokes’ equations

only track the evolution of the lowest moments (i.e. 1, v, 𝑣2). Certain models, known as extended

moment hydrodynamics, track the evolution of more than 3 moments. For example, one approach

preserves 6000+ moments [16].

In this section, we derive fluid dynamics from the BGK kinetic equation. We will begin

by macroscopic averaging, i.e., taking 𝑀 moments of the 1 body kinetic equation. A hierarchy

of moments will emerge, where the 𝑀-th moment depends on the 𝑀 + 1 moment. To form

24

a closed system, the 𝑀 + 1 moment must be expressed in terms of the previous 𝑀 moments.

Many approaches to close this hierarchy have emerged, typically truncating it by expanding about

equilibrium. Including deviations from equilibrium in the expansion manifests non-local transport

phenomena, e.g., viscosity.

We conduct macroscopic averaging on our kinetic equation to arrive at fluid transport equations.

Consider the BGK equation Eq(2.31) derived in the previous section. The BGK collision operator

Eq(2.29) satisfies local mass, momentum, and energy conservation [17]











Now, take the moment with respect to one of these collisional invariants, denoted by 𝐴(𝑣),


0




0



0

























𝑄 =

𝑑3𝑣

𝑣2

∫

1

v

.

(2.32)

∫

𝑑3𝑣 𝐴(𝑣) (𝜕𝑡 𝑓 + v · ∇x 𝑓 + a · ∇v 𝑓 ) = 0.

(2.33)

This equation can be simplified with calculus identities to

𝜕𝑡𝑛⟨𝐴⟩ + ∇x · 𝑛⟨𝑣 𝐴⟩ − 𝑛⟨∇x 𝐴𝑣⟩ − 𝑛⟨a · ∇v 𝐴⟩ = 0.

(2.34)

where

𝑛(𝑥, 𝑡) ≡

∫

𝑑3𝑣 𝑓 ,

(2.35)

From Eq(2.34) we can generate equations for mass, momentum, and energy transport, 𝐴 =
𝑚, 𝑚v, 𝑚

2 (𝑣 − 𝑢)2 respectively. These transport equations, in their conservative form, are

𝜕𝑡 𝜌 = −∇x · (𝜌u),

𝜕𝑡 𝜌u = −∇x · (𝑃 + 𝜌u ⊗ u − 𝜌a) ,

𝜕𝑡

3𝑝
2

= −∇x · (q + 𝑝u) + ∇xu : P.

(2.36a)

(2.36b)

(2.36c)

25

where

𝜌(𝑥, 𝑡) ≡ 𝑚

∫

𝑑3𝑣 𝑓 ,

∫

∫

u(𝑥, 𝑡) ≡

𝑃(𝑥, 𝑡) ≡

𝑑3𝑣 v 𝑓 ,

𝑑3𝑣 (v − u) (v − u) 𝑓 ,

𝑝(𝑥, 𝑡) ≡

q(𝑥, 𝑡) ≡

1
3
𝑚

2

∫

Tr{𝑃} =

∫

1
3

𝑑3𝑣 (𝑣 − 𝑢)2 𝑓 ,

𝑑3𝑣 (v − u) (𝑣 − 𝑢)2 𝑓 .

(2.37a)

(2.37b)

(2.37c)

(2.37d)

(2.37e)

(2.37f)

While exact, these equations do not form a closed set. The four variables 𝜌, u, 𝑃, q are the 0𝑡ℎ, 1𝑠𝑡,

2𝑛𝑑, 3𝑟𝑑 moments respectively. However, there are only three equations; the 𝑞 is connecting the

higher moments to the lower moments.

Based on Eq(2.36), the heat flux q and the off-diagonal components of the pressure tensor 𝑃

constitute the closure information needed for these three macroscopic equations. There are many

known approaches to derive q and 𝑃 [18]. In fact, new methods are still being suggested [19, 20]!

We will follow the Chapman-Enskog expansion procedure to inform 𝑞 and 𝑃.

Based on the definitions of 𝑃 and q from Eq(2.37), fixing 𝑓 (𝑥, 𝑣, 𝑡) determines the closure.

To determine 𝑓 (𝑥, 𝑣, 𝑡), expand the BGK kinetic equation Eq(2.31) about local Maxwellian as

𝑓 (𝑥, 𝑣, 𝑡) ≈ 𝑀 (𝑥, 𝑣, 𝑡) + 𝛿 𝑓 (𝑥, 𝑣, 𝑡). At first order, the equation yields

𝜕𝑡 𝑀 + v · ∇x𝑀 + a · ∇v𝑀 = 𝜖 𝜈𝛿 𝑓 .

(2.38)

We can solve this equation for 𝛿 𝑓 and then insert 𝑓 = 𝑀 + 𝛿 𝑓 into Eq(2.37d) to compute the

pressure

P = 𝑛𝑇

(cid:18)

𝐼 −

1
𝜈

(∇xu + (∇xu)𝑇 )

(cid:19)

and Eq(2.37f) to compute the heat flux

q = −

𝑛𝑇
𝑚𝜈

5
2

∇x𝑇 .

26

(2.39)

(2.40)

The external force 𝑚a has dropped out, but collision frequency 𝜈 has been carried into the hydrody-

namic closures. In the equilibrium case (i.e., 𝑓 (𝑥, 𝑣, 𝑡) ≈ 𝑀 (𝑥, 𝑣, 𝑡)), Eq(2.39) reduces to P = 𝑛𝑇I

where 𝐼 is the identity matrix and q = 0 2.

We have completed our task, Eq(2.39) and Eq(2.40) provide closure for Eq(2.36). This con-

cludes the final step from mesoscopic DoF to macroscopic DoF. We have truncated the hydrody-

namic hierarchy by expressing the two-body distribution function 𝑓 in terms of the lowest moments

𝑛, u, and 𝑝 and showed what constitutes closure information for hydrodynamic equations.

2.4 Time and length scales associated to the various frameworks

In applications ranging from controlled nuclear fusion to stellar evolution, physicists wish to

predict how plasmas will evolve. Accurately modeling the dynamics of a large number of inter-

acting particles in three dimensions via Born-Oppenheimer MD requires immense computational

resources, often beyond the capacity of modern clusters. Thus all three frameworks presented in

this chapter, molecular dynamics, kinetic equations, and fluid dynamics, are used in practice. Due

to the current state of computation resources each framework has a limited domain of temporal

and spatial extents in physical three-dimensional systems. A sampling of real word simulations

illustrates the time and length scales of actual simulations.

2Using higher order expansions yields higher order gradients, i.e., 𝑓 = 𝑀 + 𝜖 𝛿 𝑓 + 𝜖 2𝛿 𝑓 ′ produces ∇3 terms [21].
In effect fluid dynamics is “rectangular”, in the sense that the equation’s “width” is determined by the highest order
spatial derivative and the height is determined by the number of moments you evolve.

27

Figure 2.2 Plot indicating the total time elapsed and total spatial extent of various published
simulations. classical molecular dynamics simulations can reach larger spatial scales and longer
time scales than quantum methods (i.e., orbital-free and Kohn-Sham molecular dynamics). There is
strong overlap between hydrodynamics and kinetics simulations. Plot reprinted from Luke Stanek’s
PhD dissertation with author permission.

28

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[3] Richard L Liboff. Kinetic theory: classical, quantum, and relativistic descriptions. Springer

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Jean-Pierre Hansen and Ian Ranald McDonald. Theory of simple liquids: with applications
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[12] Tobias Dornheim, Panagiotis Tolias, Zhandos A Moldabekov, Attila Cangi, and Jan Vorberger.
Effective electronic forces and potentials from ab initio path integral monte carlo simulations.
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[13] David Tong. Kinetic theory. https://www.damtp.cam.ac.uk/user/tong/kintheory/, 2012. Ac-

cessed: 2024-01-24.

[14] Setsuo Ichimaru. Statistical Plasma Physics, Vol1: Basic Principles. CRC Press, 1991.

[15] Scott D Baalrud and Jérôme Daligault. Mean force kinetic theory: A convergent kinetic

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[16] Jeong-Young Ji, Gunsu S Yun, Yong-Su Na, and Eric D Held. Electron parallel transport for

arbitrary collisionality. Physics of Plasmas, 24(11), 2017.

[17] P. L. Bhatnagar, E. P. Gross, and K. Krook. Uniformly stable numerical schemes for the
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94:511–524, 1954.

[18] Sydney Chapman and Thomas George Cowling. The mathematical theory of non-uniform
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system. Communications on pure and applied mathematics, 67(3):464–518, 2014.

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30

CHAPTER 3

CONSERVATIVE DIELECTRIC RESPONSE AND ELECTRICAL CONDUCTIVITY
FROM THE MULTI-SPECIES BHATNAGAR GROSS KROOK KINETIC EQUATION

3.1

Introduction

Many scientific investigations make use of dynamic response models, e.g., for the dynamic

structure factor (DSF), dielectric function, and conductivity. Only a small subset are mentioned

here. One source of demand for such models is X-ray Thompson scattering (XRTS) diagnostics,

which use a DSF model to infer the plasma’s properties, e.g., number density, temperature, and

ionization [1, 2]. Another source of demand arises in optical conductivity experiments, which use a

conductivity model to extrapolate a material’s DC conductivity from optical-regime measurements

[3]. Similarly, in density functional theory (DFT) calculations of the electrical conductivity, a

conductivity model is needed to extrapolate a material’s DC conductivity from Kubo-Greenwood

estimates [4]. The final source of demand we list is radiation hydrodynamics (rad-hydro) simula-

tions, which use dielectric functions to estimate a plasma’s stopping power (e.g., SRIM [5]) and

DSF models to compute Bremsstrahlung emissions rates [6]. In conjunction with such simulations,

experimentally observed Bremsstrahlung emissions can be used to quantify plasma mixing [7].

This work provides conservative models of the dielectric function, dynamic structure factor (DSF),

and conductivity to be used in these applications and more.

These sources of demand are met by many different models. We term one grouping of models

“semi-empirical”; models in this grouping have free parameters tuned to match data. Semi-

empirical models can require laborious fitting procedures or can lack clear interpretations. The labor

intensity is exemplified by Cochrane et al.’s work, which fit the Lee-More-Desjairlais semi-empirical

conductivity model to DFT estimates [8]. The lack of clear interpretations is exemplified by non-

Drude conductivity models, which are used by both high energy density (HED) and nanomaterials

scientists [9, 10, 11]. When non-Drude behavior is observed, the primary alternative is the Drude-

Smith conductivity model [12]. However, Smith’s model has a phenomenological parameter of

unknown meaning [13, 14]. Smith maintains that his modification to the Drude model includes

31

charge carrier back scattering [15], while Cocker et al. contest Smith’s claim, having derived a

similar modification by assuming localized charge carriers [16]. Thus, interpretable alternatives to

semi-empirical models are needed.

We call the alternative to semi-empirical models “first-principles” models. These models

have calculated inputs rather than tuned parameters and, as a result, have clearer interpretations.

We will consider Mermin’s collision-corrected dielectric function [17], which requires a calculated

dynamical collision frequency. To improve the Mermin model’s predictions, research has improved

collision-frequency estimates [18, 19, 20, 21]. Yet, regardless of collision-frequency choice, the

Mermin model predicts nonphysical behavior. The Mermin model’s nonphysical behavior has

been tied to its lack of momentum conservation. Atwal and Ashcroft showed that Mermin’s

dielectric function does not have an infinitesimal plasmon width and thus does not satisfy the

frequency sum rule [22]. Further, they show that a dielectric function that includes momentum

and energy conservation has an infinitesimal plasmon width and thus does satisfy the frequency

sum rule. Morawetz and Fuhrmann established that Mermin’s dielectric function scales incorrectly

in the high-frequency limit, but the inclusion of momentum conservation corrects this scaling

[23]. However, both works also demonstrate that if a single-species dielectric function conserves

momentum, then it also predicts an infinite conductivity. This implies that only multi-species

dielectric functions can conserve momentum without predicting infinite conductivity. Thus, a

momentum-conserving multi-species first-principles model is well motivated.

In this work, we derive our momentum-conserving multi-species first-principles model from

a kinetic equation in the relaxation-time approximation; this is a typical approach that has been

used previously [24, 22]. In these previous works, the distribution function exponentially decays

toward a target function, which is characterized by local perturbations in the chemical potential,

velocity, and temperature. These perturbations are constrained to enforce number, momentum, and

energy conservation and are substituted into the linearized kinetic equation to produce dynamical

response functions. In this way, Selchow and Morawetz [25] derive the Mermin’s single-species

dielectric function from the classical Bhatnagar, Gross, and Krook’s (BGK) kinetic equation

32

[26] and the classical Fokker-Plank kinetic equation. Additionally, Atwal and Ashcroft derive

a number-, momentum-, and energy-conserving single-species dielectric function from the BGK

kinetic equation [22]. Currently, a multi-species Mermin dielectric function exists [27], but it

has not been extended to include momentum conservation. We produce the first number- and

momentum-conserving multi-species susceptibility from Haack et al.’s multi-species BGK kinetic

equation [28] and recover, as a limit, the known multi-species Mermin susceptibility. We call our

result the completed Mermin dielectric function because it satisfies the frequency sum rule.

Our multi-species completed Mermin susceptibility offers a new DSF model to inform a plasma’s

ionic structure. Both XRTS and bremmstrahlung emission models require estimates of the ions’

DSFs [2] [6]. In inertial confinement fusion (ICF) experiments, hydrodynamic instabilities inject

the ICF capsule’s ablator material, e.g., carbon, into the deuterium-tritium hot-spot [29]. However,

the carbon contaminants create an inherently multi-species system, which is tractable with our multi-

species susceptibility. We apply the completed Mermin model to understand how conservation

laws impact carbon-contaminated DSFs.

The single-species limit of our completed Mermin model offers a new non-Drude dynamical

conductivity model.

In dynamical conductivity estimates, the measured DC conductivity can

be suppressed while the measured optical conductivity is enhanced relative to the Drude model.

Thus, Chen et al. recommend the use of non-Drude conductivities, e.g., Drude-Smith when fitting

optical conductivity measurements [9]. We apply the completed Mermin model to understand

how conservation laws impact the conductivity. We show that partial number and momentum

conservation can also suppress the DC conductivity and enhance optical conductivity. We compare

our completed Mermin model to the established Drude-Smith model and find that Smith’s parameter

violates conservation laws.

The structure of the paper is as follows. In Sec. 3.2, we introduce the multi-species BGK kinetic

equation and show how to conserve local number, momentum, and energy using local variations of

the chemical potential, drift velocity and temperature. In Sec. 3.3, we develop applications of the

conserving relaxation-time approximation. First, we derive the multi-species completed Mermin

33

susceptibility and show that it satisfies sum rules that the multi-species Mermin does not. Next,

we investigate the DSF of mixtures at NIF hot-spot conditions. We quantify the impact of the

light-species approximation and demonstrate that momentum conservation has a qualitative impact

on the shape of the DSF. Then we use the light-species approximation to observe the impact of

carbon contaminants on the light-species DSF. Finally, we derive a dynamical conductivity from

our single-species completed Mermin susceptibility and demonstrate how number- and momentum-

conservation parameters impact the model. We compare this new conductivity model to both the

Drude and Drude-Smith models to demonstrate that Smith’s phenomenological parameter violates

number conservation. Details required to implement our new completed Mermin susceptibility can

be found in the appendices.

3.2 Kinetics

3.2.1 Describing the system

We consider a classical system containing 𝑁 different charged-particle species. Many-body

particle interactions and externally applied fields govern the dynamics of a multi-species system of

charged particles. We account for the many-body interactions with an effective one-body description

for each of the 𝑁 species. To this end, two terms govern the dynamics of a single particle: an

effective single-particle Hamiltonian (i.e., a mean-field interaction) and an effective single-body

inter- and intra-species collisional operator.

3.2.2 Formulating the system

The dynamics of the single-particle distribution function are governed by

(cid:0)𝜕𝑡 + v · ∇r + atot,i · ∇v

(cid:1) 𝑓𝑖 (r, v, 𝑡) =

∑︁

𝑄𝑖 𝑗 .

𝑗

(3.1)

𝑄𝑖 𝑗 denotes the effective one-body description of the intra-species 𝑖 = 𝑗 collisions and inter-species

𝑖 ≠ 𝑗 collisions. atot,i denotes the total acceleration and is defined by

𝑚𝑖atot,i = −∇r𝑈ext − ∇r𝑈ind

𝑖

,

(3.2)

where the external potential 𝑈ext is inherently a one-body potential, and the induced potential 𝑈ind

𝑖

is an effective one-body potential that describes the electrostatic energy.

In Fourier space, our

34

induced potential is given by the Hartree potential

𝑈ind

𝑖 ≡

∑︁

𝑗

𝑣𝑖 𝑗 (𝑘)𝛿𝑛 𝑗 (k, 𝜔),

(3.3)

which expresses that interactions 𝑣𝑖 𝑗 (𝑘) between species 𝑖 and 𝑗 cause density fluctuations 𝛿𝑛 𝑗 (k, 𝜔)

in species 𝑗 to affect the electrostatic potential 𝑈ind

𝑖

experienced by species 𝑖. This formulation

facilitates species dependent ion-ion interactions which include the effects of electron screening,

e.g., screened Coulomb potential or force matched potentials [30]. In the case of a single species

of electrons, we will use a Coulomb potential.

In Eq(3.1), we assume Haack et al.’s multi-species relaxation to equilibrium [28] as our colli-

sional operator:

𝑄𝑖 𝑗 ≡

1
𝜏𝑖 𝑗

(cid:0)𝑀𝑖 𝑗 (r, v, 𝑡) − 𝑓𝑖 (r, v, 𝑡)(cid:1) .

The Maxwellian target distribution 𝑀𝑖 𝑗 is defined by

(cid:18)

𝑚𝑖
2𝜋𝑇𝑖 𝑗 (r, 𝑡)

(cid:19) 3/2

( 𝜀𝑖 𝑗 (r,𝑡 ) −𝜇𝑖 (r,𝑡 ) )
𝑇𝑖 𝑗 (r,𝑡 )

,

𝑒−

(𝑣 − 𝑢𝑖 𝑗 (r, 𝑡))2,

𝑀𝑖 𝑗 ≡

𝜀𝑖 𝑗 ≡

1
𝜏𝑖

≡

𝑔𝑖
𝜆3
𝑖,𝑡ℎ
𝑚𝑖
2
∑︁

𝑗

1
𝜏𝑖 𝑗

,

(3.4)

(3.5)

(3.6)

(3.7)

where 𝜆𝑖 (r, 𝑡) ≡

𝑔𝑖
𝜆3

𝑖,𝑡 ℎ

𝑒𝜇𝑖 (r,𝑡)/𝑇𝑖 𝑗 (r,𝑡) is the local fugacity. The target velocity 𝑢𝑖 𝑗 and target temperature

𝑇𝑖 𝑗 are defined so that the collision operator satisfies the H-theorem and number-, momentum-, and

energy-conservation laws. However, in this work, we will only use 𝑢𝑖 𝑗 and 𝑇𝑖 𝑗 as expansion

parameters.

The standard interpretation of the relaxation approximation is that Vlasov dynamics govern

the particle’s phase-space dynamics and that for every infinitesimal time interval 𝑑𝑡, a fraction

𝑑𝑡/𝜏𝑖 𝑗 of the particles experiences a collision event that sets their velocity distribution to the target

distribution 𝑀𝑖 𝑗 . For ion-ion collisions, Haack et al. present different choices for the ion-ion

relaxation times 𝜏𝑖 𝑗 [28]. These relaxation times rely on the Stanton-Murillo transport (SMT)

model [31] which accounts for electron screening, but suppresses the frequency dependence in

35

𝜏𝑖 𝑗 . Thus, if the energy levels involved in the collision process lie above the Fermi energy, then a

frequency dependent relaxation time needs to be chosen [21].

3.2.3 Linearizing the multi-species BGK equation

To linearize the kinetic equation Eq(3.1), we assume that the 𝑖th species’ distribution function

𝑓𝑖 (r, v, 𝑡) and the target Maxwellian 𝑀𝑖 𝑗 (r, v, 𝑡) have small deviations from a global (0) mixture

(M) equilibrium distribution

𝑓𝑖 (r, v, 𝑡) = 𝜆𝑖 𝑓𝑖𝐺 (𝑣) + 𝜆𝑖𝛿 𝑓𝑖𝐺 (r, v, 𝑡),

𝑀𝑖 𝑗 (r, v, 𝑡) = 𝜆𝑖 𝑓𝑖𝐺 (𝑣) + 𝜆𝑖𝛿𝑀𝑖 𝑗 (r, v, 𝑡),

(3.8)

(3.9)

where the global fugacity 𝜆𝑖 ≡

𝑔𝑖
𝜆3

𝑖,𝑡 ℎ

𝑒𝜇𝑖/𝑇 has been factored out from each term, and 𝑓𝑖𝐺 (𝑣) is the

global mixture equilibrium distribution, defined as

𝑓𝑖𝐺 (𝑣) ≡

u ≡

(cid:17) 3/2

(cid:16) 𝑚𝑖
2𝜋𝑇
(cid:16)∑︁ 𝑚𝑖𝑛𝑖u𝑖

𝑚𝑖
2𝑇
𝑚𝑖𝑛𝑖,

∑︁

(cid:17)

/

(cid:16)

−

exp

(𝑣 − 𝑢)2(cid:17)

,

𝑖

𝑇 ≡

(cid:16)∑︁ 𝑛𝑖𝑇𝑖

(cid:17)

/

∑︁

𝑛𝑖.

𝑖

Lastly, the deviations from global equilibrium define the density fluctuations, with

∫

𝛿𝑛 𝑗 (r, 𝑡) ≡

𝑑v 𝛿 𝑓 𝑗 (r, v, 𝑡).

(3.10)

(3.11)

(3.12)

(3.13)

We insert expansions Eq(3.8) and Eq(3.9) into Eq(3.1), cancel the fugacity factors, and then

Fourier transform the resulting equation to arrive at the following:

(cid:0)v · k − 𝜔𝜏𝑖

(cid:1) 𝛿 𝑓𝑖𝐺 (k, v, 𝜔) +

𝑖
𝜏𝑖 𝑗

∑︁

𝑗

𝛿𝑀𝑖 𝑗 (r, v, 𝑡) = 𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺 𝑈𝑡𝑜𝑡 .

(3.14)

The Fourier conventions and relevant steps are described in Appendix 3A. We have grouped the

𝛿 𝑓𝑖𝐺 terms, and thus, the frequency has been shifted by 𝜔𝜏𝑗 ≡ 𝜔 + 𝑖/𝜏𝑗 .

In Eq(3.14), 𝛿𝑀𝑖 𝑗 (r, v, 𝑡) is an unknown term. The target Maxwellian 𝑀𝑖 𝑗 , from Eq(3.5),

recovers the global mixture equilibrium distribution 𝑓𝑖𝐺, from Eq(3.10), when the target velocity

u𝑖 𝑗 (r, 𝑡) reduces to the bulk velocity u and the target temperature 𝑇𝑖 𝑗 (r, 𝑡) reduces to the bulk

36

temperature 𝑇. This implies that, at zeroth order, u𝑖 𝑗 (r, 𝑡) ≈ u and 𝑇𝑖 𝑗 (r, 𝑡) ≈ 𝑇. Therefore,

𝛿𝑀𝑖 𝑗 (r, v, 𝑡) contains the local equilibrium’s local deviations in chemical potential, velocity, and

temperature. We expand 𝛿𝑀𝑖 𝑗 (r, v, 𝑡) to linear order, obtaining

𝛿𝑀𝑖 𝑗 =

(cid:18) 𝜕M𝑖 𝑗
𝜕𝜇𝑖

(cid:12)
(cid:12)
(cid:12)M𝑖 𝑗 = 𝑓𝑖𝐺

(cid:19)

𝛿𝜇𝑖 +

(cid:18) 𝜕M𝑖 𝑗
𝜕u𝑖 𝑗

(cid:12)
(cid:12)
(cid:12)M𝑖 𝑗 = 𝑓𝑖𝐺

(cid:19)

𝛿u𝑖 𝑗 +

(cid:18) 𝜕M𝑖 𝑗
𝜕𝑇𝑖 𝑗

(cid:12)
(cid:12)
(cid:12)M𝑖 𝑗 = 𝑓𝑖𝐺

(cid:19)

𝛿𝑇𝑖 𝑗 .

(3.15)

We factor out −

𝜕 𝑓𝑖𝐺
𝜕𝜖 from every term and use the chain rule to reformulate

𝜕 𝑓𝑖𝐺
𝜕𝜖 as

= (cid:169)
(cid:173)
(cid:173)
(cid:171)
This produces our final equation for 𝛿𝑀𝑖 𝑗 :

𝜕 𝑓 𝐺𝐶
𝑖,0
𝜕𝜖𝑖

𝜕 𝑓 𝐺𝐶
𝑖,0
𝜕v

k ·

−1

.

k ·

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:171)

𝜕𝜖𝑖
𝜕v

(cid:170)
(cid:174)
(cid:174)
(cid:172)

𝛿𝑀𝑖 𝑗 = −

(cid:32)

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺
k · v

𝛿𝜇𝑖 + p𝑖 · 𝛿u𝑖 𝑗 +

(cid:32) 𝑝2
𝑖
2𝑚𝑖

− 𝜇𝑖

(cid:33)

.

(cid:33) 𝛿𝑇𝑖 𝑗
𝑇𝑖 𝑗

Inserting Eq(3.17) into Eq(3.14) yields

(3.16)

(3.17)

𝛿 𝑓𝑖𝐺 (k, v, 𝜔) =

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺 (𝑣)
v · k − 𝜔𝜏𝑖
𝑖
v · k

(cid:32)
𝑈𝑡𝑜𝑡 +

𝑗

∑︁

×

(cid:32)

1
𝜏𝑖 𝑗

𝛿𝜇𝑖 + p𝑖 · 𝛿u𝑖 𝑗 +

(cid:32) 𝑝2
𝑖
2𝑚𝑖

− 𝜇𝑖

(cid:33) 𝛿𝑇𝑖 𝑗
𝑇

(cid:33)(cid:33)

,

(3.18)

which indicates that the external potential, the induced potential, as well as local deviations in

chemical potential, velocity, and temperature can all cause perturbations from global equilibrium.

3.2.4

Incorporating conservation laws using linear perturbations

Previous single-species models violated the momentum-conservation law to account for scatter-

ing events with the other species implicitly contained in the system. For instance, this violation was

necessary to predict that a gas of electrons, implicitly contained in a metal, has finite conductivity.

Because we are accounting for all 𝑁 species in the system, it is physical to include momentum

conservation.

In Eq(3.18), 𝛿𝜇𝑖, 𝛿u𝑖 𝑗 , and 𝛿𝑇𝑖 𝑗 are unknowns. We intend to enforce conservation laws by

constraining these three unknowns with three collisional invariants, i.e., number, momentum and

37

energy. Our collisional invariants are formulated as follows:

∫

∫

∫

𝑑v 𝑄𝑖 𝑗 = 0,

𝑑v 𝑚𝑖v𝑄𝑖 𝑗 +

∫

𝑑v 𝑚 𝑗 v𝑄 𝑗𝑖 = 0,

𝑑v

𝑚𝑖
2

𝑣2𝑄𝑖 𝑗 +

∫

𝑑v

𝑚 𝑗
2

𝑣2𝑄 𝑗𝑖 = 0.

(3.19a)

(3.19b)

(3.19c)

For example, satisfying Eq(3.19a) ensures that the same number of members of species 𝑖 are present

before and after 𝑖’s collisions with species 𝑗.

It may spuriously appear that our collisional invariants uniquely determine the values of local

perturbations, i.e., 𝛿𝜇𝑖, 𝛿u𝑖 𝑗 , and 𝛿𝑇𝑖 𝑗 . However, notice that 𝛿𝜇𝑖 and 𝛿𝑇𝑖 𝑗 are accompanied by even

moments of momentum in Eq(3.17). Thus, the mass collisional invariant Eq(3.19a) and energy

collisional invariant Eq(3.19c) will both preserve terms with 𝛿𝜇𝑖 and 𝛿𝑇𝑖 𝑗 , coupling these two

constraints together. Because 𝛿𝜇𝑖 depends on a single index and 𝛿𝑇𝑖 𝑗 depends on two indices, there

remains an ambiguity about how to select 𝑗 for the number-density constraint Eq(3.19a). This issue

could be resolved by changing 𝛿𝜇𝑖 to 𝛿𝜇𝑖 𝑗 and Eq(3.19a) to

∫

∫

𝑑v 𝑚𝑖𝑄𝑖 𝑗 +

𝑑v 𝑚 𝑗 𝑄 𝑗𝑖 = 0.

(3.20)

However, this mass-conservation constraint would allow the system to convert species 𝑖 into species

𝑗 to reach chemical equilibrium, which is unphysical. Therefore, no attempt is made to conserve

number, momentum and energy simultaneously. Instead, we limit ourselves to the iso-thermal 𝛿𝑇𝑖 𝑗

case, conserving only number and momentum. We explore the error introduced by the iso-thermal

approximation for a single-species dielectric in section 3.3.4. In the single-species limit, we find

that energy conservation corrections enter at order 𝑘 2.

We can conserve momentum even though 𝛿u𝑖 𝑗 also depends on 𝑖, 𝑗, because 𝛿𝜇𝑖 and 𝛿u𝑖 𝑗

are accompanied by even and odd powers of momentum, respectively, in Eq(3.17). Therefore,

𝛿u𝑖 𝑗 does not appear in the mass collisional invariant Eq(3.19a), and 𝛿𝜇𝑖 does not appear in the

momentum collisional invariant Eq(3.19b). Hence, these equations are decoupled. Evaluating

38

constraints Eq(3.19a) and Eq(3.19b) produces

where

𝛿𝜇𝑖 = −𝛿𝑛𝑖/𝐵𝑀
𝑖,0

,

k · 𝛿u𝑖 𝑗 = 𝜔

(cid:18) 𝑚𝑖𝛿𝑛𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝛿𝑛 𝑗 𝜏𝑖 𝑗
𝑚𝑖𝑛0,𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝑛0, 𝑗 𝜏𝑖 𝑗

(cid:19)

,

𝐶 𝑀

𝑖,𝑛 (k, 𝜔) ≡

∫

𝑑v|p|𝑛

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺
v · k − 𝜔

,

𝑖,𝑛 ≡ 𝐶 𝑀
𝐵𝑀

𝑖,𝑛 (k, 0).

(3.21a)

(3.21b)

(3.22a)

(3.22b)

The momentum integration is conducted in Appendix 3B. Equations Eq(3.21a) and Eq(3.21b)

constrain the unknowns in our system’s dynamic response.

3.3 Results

3.3.1 Susceptibilities

We begin our results section by examining the susceptibility, from which other quantities,

e.g., the dielectric function and the dynamic structure factor, will be produced. The susceptibility

quantifies an external potential’s ability to cause density fluctuations 𝛿𝑛 (see Eq(3.13)) about the

global equilibrium:

𝛿𝑛𝑖 (𝑘, 𝜔) = 𝜒𝑖 (𝑘, 𝜔)𝑈ext(𝑘, 𝜔).

(3.23)

The 𝑖 index runs over all species (𝑖 = 1, . . . , 𝑁). We formulate the system’s dynamical response by

integrating Eq(3.18) over velocity; the steps are shown in Appendix 3C. Our final result is

𝛿𝑛𝑖 = 𝐶 𝑀

𝑖,0(𝑈ext +

∑︁

𝑣𝑖 𝑗 𝛿𝑛 𝑗 ) +

(cid:16)

𝑖
𝜔𝜏𝑖 𝜏𝑖

𝐶 𝑀
𝑖,0 − 𝐵𝑀
𝑖,0

(cid:17)

𝛿𝜇𝑖 (k)

+ 𝐶 𝑀
𝑖,0

𝑚𝑖
𝑘 2

∑︁

𝑗

𝑗

𝑖
𝜏𝑖 𝑗

k · 𝛿u𝑖 𝑗 +

𝑖
𝜏𝑖 𝑗

(cid:19)

(cid:18)(cid:18) 𝐶𝑖,2 − 𝐵𝑖,2
2𝑚𝑖

∑︁

𝑗

− 𝜇𝑖 (𝐶𝑖,0 − 𝐵𝑖,0)

(cid:19) 𝛿𝑇𝑖 𝑗
𝑇𝑖 𝑗

.

(3.24)

39

The 𝑗 index runs over all species ( 𝑗 = 1, . . . , 𝑁). Substituting expressions 𝛿𝜇𝑖 from Eq(3.21a), 𝛿u𝑖 𝑗

from Eq(3.21b), and 𝛿𝑇𝑖 𝑗 = 0 yields

𝛿𝑛𝑖 = 𝐶 𝑀

𝑖,0(𝑈ext +

∑︁

𝑗

𝑣𝑖 𝑗 (𝑘)𝛿𝑛 𝑗 ) −

(cid:16)

𝑖
𝜔𝜏𝑖 𝜏𝑖

𝑖,0 − 𝐵𝑀
𝐶 𝑀
𝑖,0

(cid:17)

𝛿𝑛𝑖/𝐵𝑀
𝑖,0

+𝑖𝐶 𝑀
𝑖,0

𝑚𝑖𝜔
𝑘 2

∑︁

𝑗

1
𝜏𝑖 𝑗

(cid:18) 𝑚𝑖𝛿𝑛𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝛿𝑛 𝑗 𝜏𝑖 𝑗
𝑚𝑖𝑛0,𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝑛0, 𝑗 𝜏𝑖 𝑗

(cid:19)

.

(3.25)

Grouping the 𝛿𝑛𝑖 terms yields

𝐶 𝑀
𝑖,0

𝑈ext =𝛿𝑛𝑖 − 𝑣𝑖𝑖 (𝑘)𝐶 𝑀
𝑖,0

𝛿𝑛𝑖 −

(cid:32)

1 −

𝑖
𝜔𝜏𝑖 𝜏𝑖

− 𝐶 𝑀
𝑖,0

∑︁

𝑗≠𝑖

𝑣𝑖 𝑗 (𝑘)𝛿𝑛 𝑗 −

𝑖𝜔𝑚𝑖
𝑘 2

𝐶 𝑀
𝑖,0

𝐶 𝑀
𝑖,0
𝐵𝑀
𝑖,0
∑︁

𝑗≠𝑖

(cid:33)

𝛿𝑛𝑖 −

𝑖𝜔𝑚𝑖
𝑘 2𝑛𝑖,0𝜏𝑖𝑖

𝐶 𝑀
𝑖,0

𝛿𝑛𝑖

1
𝜏𝑖 𝑗

(cid:18) 𝑚𝑖𝛿𝑛𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝛿𝑛 𝑗 𝜏𝑖 𝑗
𝑚𝑖𝑛0,𝑖𝜏𝑗𝑖 + 𝑚 𝑗 𝑛0, 𝑗 𝜏𝑖 𝑗

(cid:19)

.

(3.26)

Next, we limit ourselves to two species and solve for the susceptibility using the 2x2 matrix-

inversion formula. The result is

𝜒1,𝐶 𝑀 =

𝜀∗
1

𝜀∗
2

− 𝐶1,0𝐶2,0

(cid:16)

(cid:16)

𝐶 𝑀
𝜀∗
2 +
1,0
𝑣12(𝑘) + 𝑖 𝜔
𝑘 2

𝑣12(𝑘) + 𝑖 𝜔
𝑘 2
𝑚1𝑚2
𝑚1𝑛0,1𝜏21+𝑚2𝑛0,2𝜏12

𝑚1𝑚2
𝑚1𝑛0,1𝜏21+𝑚2𝑛0,2𝜏12
(cid:17) (cid:16)

𝐶 𝑀
2,0
𝑣21(𝑘) + 𝑖 𝜔
𝑘 2

(cid:16)

(cid:17)

(cid:17)

𝑚1𝑚2
𝑚1𝑛0,1𝜏21+𝑚2𝑛0,2𝜏12

,

(cid:17)

𝜒2,𝐶 𝑀 =

𝜀∗
1

𝜀∗
2

− 𝐶1,0𝐶2,0

(cid:16)

(cid:16)

(cid:16)

𝐶 𝑀
2,0

𝜀∗
1 +
𝑣12(𝑘) + 𝑖 𝜔
𝑘 2

𝑣21(𝑘) + 𝑖 𝜔
𝑘 2
𝑚1𝑚2
𝑚1𝑛0,1𝜏21+𝑚2𝑛0,2𝜏12

𝑚1𝑚2
𝑚1𝑛1𝜏21+𝑚2𝑛2𝜏12
(cid:17) (cid:16)

𝐶 𝑀
1,0
𝑣21(𝑘) + 𝑖 𝜔
𝑘 2

(cid:17)

(3.27a)

(cid:17)

𝑚1𝑚2
𝑚1𝑛0,1𝜏21+𝑚2𝑛0,2𝜏12

,

(cid:17)

𝜀∗
1 = 𝜀1 −

𝜀∗
2 = 𝜀2 −

𝑖
𝜔𝜏1
𝑖
𝜔𝜏2

𝜏1

𝜏2

(cid:32)

(cid:32)

1 −

1 −

(cid:33)

(cid:33)

𝐶 𝑀
1,0
𝐵𝑀
1,0
𝐶 𝑀
2,0
𝐵𝑀
2,0

− 𝑖

− 𝑖

𝜔
𝑘 2

𝜔
𝑘 2

(cid:32) 𝑚1
𝑛1,0𝜏11
(cid:32) 𝑚2
𝑛2,0𝜏22

+

+

𝜏21
𝜏12

𝜏12
𝜏21

𝑚2
1
𝑚1𝑛0,1𝜏21 + 𝑚2𝑛0,2𝜏12
𝑚2
2
𝑚1𝑛0,1𝜏21 + 𝑚2𝑛0,2𝜏12

(cid:33)

(cid:33)

𝐶 𝑀
1,0

,

𝐶 𝑀
2,0

.

(3.27b)

(3.27c)

(3.27d)

These equations are the primary result of this paper; we refer to them as the multi-species completed

Mermin susceptibility. For ease of use, this result is broken down into dimensionless, numerically

implementable equations in Appendix 3D.

We recover the multi-species Mermin-like susceptibility [32, 33] from Eq(3.27) by neglecting

40

the terms with a factor of 𝜔/𝑘 2. The result is

𝜒1,𝑀 ≡ 𝐶 𝑀
1,0

𝜒2,𝑀 ≡ 𝐶 𝑀
2,0

𝜀∗
1

𝜀∗
1

2 + 𝑣12(𝑘)𝐶 𝑀
𝜀∗
2,0
𝜀∗
− 𝐶1,0𝐶2,0𝑣2
2
12
1 + 𝑣12(𝑘)𝐶 𝑀
𝜀∗
1,0
− 𝐶1,0𝐶2,0𝑣2
12
(cid:33)
(cid:32)

𝜀∗
2

𝑖
𝜔𝜏𝑗 𝜏𝑗

1 −

𝐶 𝑀
𝑗,0
𝐵𝑀
𝑗,0

,

,

(𝑘)

(𝑘)

(3.28a)

(3.28b)

(3.28c)

.

𝜀∗
𝑗 = 𝜀 𝑗 −

If we also let the relaxation time 𝜏 go to infinity, we recover the random-phase approximation (RPA).

Notice that in both the Mermin and the RPA susceptibilities, the interaction potential 𝑣12(𝑘) in

Eq(3.28) is the only term coupling species 1’s susceptibility to species 2’s susceptibility, whereas,

in the completed Mermin susceptibility Eq(3.27), the conservation of momentum also couples the

susceptibilities.

We recover the single-species limit of the completed Mermin model by neglecting the coupling

terms in Eq(3.27),

𝜒𝐶 𝑀 ≡

𝐶0

𝜀 −

𝑖

𝜔𝜏𝜏

𝐶0
𝐵0

(cid:170)
(cid:174)
(cid:174)
(cid:172)

1 −

(cid:169)
(cid:173)
(cid:173)
(cid:171)

,

− 𝑖𝑚𝜔
𝑘 2𝑛0𝜏

𝐶0

whereas the single species Mermin is given by

𝜒𝑀 ≡

𝐶0

.

𝜀 −

𝑖

𝜔𝜏𝜏

𝐶0
𝐵0

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
1 −
(cid:173)
(cid:173)
(cid:171)

(3.29)

(3.30)

Comparing Eq(3.30) with Eq(3.29), the completed Mermin susceptibility includes the momentum-

conservation correction

𝑖𝑚𝜔
𝑘 2𝑛𝑖,0𝜏

𝐶𝑖,0.

(3.31)

The form of this momentum-conservation correction Eq(3.31) matches Morawetz and Fuhrmann’s

single-species local field correction [23]. The correction Eq(3.31) arises from the single-species

41

version of the momentum constraint Eq(3.21b), with

k · 𝑛0𝛿u = 𝜔𝛿𝑛.

(3.32)

Comparatively, Mermin produced his number-conservation constraint by enforcing k·j = 𝜔𝛿𝑛 [17].

Our momentum constraint, Eq(3.21b), differs only in that j = 𝑛0 𝛿u. This suggests that enforcing

j = 𝑛0 𝛿u in the Mermin continuity equation and varying local equilibrium with respect to velocity

leads to momentum conservation. Thus, we refer to our susceptibility as the “completed Mermin”

susceptibility.

3.3.2 Dielectric functions

The dielectric function is defined as

𝑈ind
𝑖

(𝑘, 𝜔) ≡

(cid:18)

1
𝜀𝑖 (𝑘, 𝜔)

(cid:19)

− 1

𝑈ext(𝑘, 𝜔).

(3.33)

Inserting the multi-species Hartree potential Eq(3.3) and comparing to the definition of the suscep-

tibility Eq(3.23) yields

1
𝜀𝑖 (𝑘, 𝜔)

= 1 +

∑︁

𝑗

𝑣𝑖 𝑗 (𝑘) 𝜒 𝑗 (𝑘, 𝜔).

(3.34)

For the single-species mean-field corrected susceptibility 𝜒 = 𝐶0/(1 − 𝑣(𝑘)𝐶0), this expression

produces the common expression 𝜀(𝑘, 𝜔) = 1 − 𝑣(𝑘)𝐶0(𝑘, 𝜔).

An essential property of every dielectric function is the fulfillment of the sum rules, which

determines the quality of the dielectric function moment by moment [34]. The frequency sum

(f-sum) rule expresses whether the local continuity equation is satisfied; it is expressed as

∫ ∞

−∞

𝑑𝜔 𝜔Im{𝜀−1

𝑖 } = −𝜋 ∑︁

𝑗

𝜔2

𝑝,𝑒

𝑘 2

𝐷,𝑒

𝑚𝑒
𝑚 𝑗

𝑛𝑖
𝑇

𝑣𝑖 𝑗 (𝑘)𝑘 2.

(3.35)

For an isolated species of electrons in the unscreened limit, the RHS of Eq(3.35) reduces to the

familiar −𝜋𝜔2

𝑝. Another sum rule is the perfect-screening sum rule, which is valid when there is no

appreciable 𝑘 dependence, that is, when the relation between the induced density and the external

potential is a purely local one [35]; it is expressed as

lim
𝑘→0

lim
˜𝜅→0

∫ ∞

−∞

𝑑𝜔
𝜔 Im{𝜀−1} = −𝜋.

(3.36)

42

For the screening sum rule, the no-screening limit (i.e., ˜𝜅 → 0) is taken first, and the long-wavelength

limit (i.e., 𝑘 → 0) is taken second.

For the one-component plasma (OCP), we produce analytic expressions of lim
k→0

Im{𝜀−1} for

each known collisional single-species case: Mermin (M) Eq(3.30), our completed Mermin (CM)

Eq(3.29), and Atwal-Ashcroft (AA), which locally conserves number, momentum, and energy [22].

In terms of dimensionless parameters, the functional forms are as follows:

(cid:27)

(cid:26) 1
𝜀M
(cid:26) 1
𝜀CM

Im

lim
k→0

Im

lim
k→0

(cid:27)

= −

= −

𝜁 2( ˜𝜔/ ˜𝜏)
𝜁 4( ˜𝜔/ ˜𝜏)2 + (𝜁 2 ˜𝜔2 − 1)2
˜𝑘 2 ˜𝜏 ˜𝜔
−2 ˜𝑘 2(𝜁 2 ˜𝜔2 − 1) + (1 + ˜𝜏2 ˜𝜔2) (𝜁 2 ˜𝜔2 − 1)2

,

,

(cid:27)

(cid:26) 1
𝜀AA

= −

Im

lim
k→0

˜𝑘 2 ˜𝜏 ˜𝜔

−2 ˜𝑘 2 (cid:16) 1+3 ˜𝜏2 ˜𝜔2−2𝜁 −2 ˜𝜏2

1+ ˜𝜏2 ˜𝜔2

(cid:17)

(𝜁 2 ˜𝜔2 − 1) + 1

3 (1 + ˜𝜏2 ˜𝜔2) (𝜁 2 ˜𝜔2 − 1)2

(3.37a)

(3.37b)

.

(3.37c)

We have used electronic quantities to render the functions dimensionless: ˜𝜏 = 𝜏𝜔 𝑝,𝑒, ˜𝜔 = 𝜔/𝜔 𝑝,𝑒,
˜𝑘 = 𝑘/𝑘 𝐷,𝑒, and 𝜁𝑖 ≡ √︁𝑚𝑖/𝑚𝑒. The steps taken to compute these dimensionless expressions are

shown in Appendix 3E.

All three functional forms in Eq(3.37) are Lorentzian. Mermin’s width parameter is 1/ ˜𝜏, while

both the completed Mermin and the Atwal-Ashcroft width parameters are ˜𝑘. As 𝑘 → 0, the

completed Mermin and Atwal-Ashcroft dielectric functions become Dirac deltas. We plot the

long-wavelength expansions from Eq(3.37) for wave number ˜𝑘 ≈ 0.05 in Fig. 3.1. The completed

Mermin and Atwal-Ashcroft functions have narrower widths than the Mermin because Mermin’s

width parameter 1/ ˜𝜏 remains finite in the long-wavelength limit.

When the width becomes infinitesimally small, we recover a Dirac delta function:

Im{𝜀−1} = −𝜋𝜔 𝛿(𝜔 − 𝜔 𝑝).

(3.38)

By substitution, we see that the Dirac delta satisfies both sum rules Eq(3.35) and Eq(3.36). There-

fore, the completed Mermin and the Atwal-Ashcroft dielectric functions satisfy both the f-sum rule

and the screening-sum rule. Because the Mermin dielectric function does not converge to a Dirac

delta, it will not satisfy the f-sum rule; however, the Mermin dielectric function will satisfy the

43

Figure 3.1 Plots of the long-wavelength expansion of Im{𝜀−1} in the one-component plasma (OCP)
case, for the Mermin (green), completed Mermin (CM, red), and Atwal-Ashcroft (AA, purple)
dielectric functions from Eq(3.37). We evaluate the functions at ˜𝑘 ≈ 0.05, (𝜔 𝑝𝜏)−1 = .2, 𝜁 2 = 1,
𝜔 𝑝,𝑒 = 1.

screening-sum rule because it has no 𝑘 dependence. In the RPA limit (i.e., 𝜏 → ∞), the Mermin

model also becomes a Dirac delta function and satisfies the f-sum rule.

We can also assess whether the sum rules hold outside the OCP long-wavelength limit. The

sum rules are computed numerically at a finite 𝑘, for the Yukawa one component plasma (YOCP)

and the results are shown in Fig. 3.2. For the f-sum rule, only the Mermin dielectric function does

not integrate to −𝜋, which matches the behavior seen in the long-wavelength limit of the OCP.

For the screening-sum rule, all results converge to −𝜋 in the long wavelength limit. Additionally,

Fig. 3.3 shows that for the binary Yukawa mixture (BYM) the multi-species dielectric functions

exhibit the same sum rule behavior as their YOCP counterparts.

3.3.3 Dynamic structure factors

The DSF is an essential input into HED experimental diagnostics, often determining the quality

of the experimental inferences. The classical fluctuation dissipation theorem (FDT) defines the

DSF as

𝑆(𝑘, 𝜔) =

−2𝑇
𝑛𝑖 (𝜔/𝜔 𝑝,𝑒)

(cid:18)

𝑍 ∗
𝑖

(cid:19)

√︂ 𝑛𝑖
𝑛𝑒

𝑚𝑒
𝑚𝑖

𝜒′′(𝑘, 𝜔),

(3.39)

44

Figure 3.2 Left: A plot of the frequency-sum (f-sum) rule, Eq(3.35), for the random-phase ap-
proximation (RPA, blue), Mermin (green), completed Mermin (CM, red), and Atwal-Ashcroft
(AA, purple) dielectric functions of the Yukawa one component plasma (YOCP). Only the Mermin
dielectric function does not integrate to −𝜋 (black), and therefore does not satisfy the f-sum rule.
Right: A plot of the screening-sum rule, Eq(3.36), for the single-species RPA (blue), Mermin
(green), and completed Mermin (red) dielectric functions of the Yukawa one component plasma
(YOCP). In both plots, wiggles arise in the long-wavelength limit because the susceptibilities be-
come Dirac deltas and numerical integration becomes impossible; we have truncated our plots
before that happens.

where 𝜒′′

𝑖 is the imaginary part of the susceptibility, and 𝑍 ∗

𝑖 is the effective charge of the 𝑖𝑡ℎ species.

We examine the effects of high-Z contaminants on deuterium’s (D) and tritium’s (T) DSFs in an

inertial confinement fusion (ICF) hot-spot. Based on Hu et al.’s work on National Ignition Facility

(NIF) direct drive, we assume that the hot-spot is at a mass density of 1002 g/cc and temperature

of 928 eV [36]. We consider two binary Yukawa mixtures. In the first mixture, we neglect carbon

(C) contaminants and consider an uncontaminated D and T plasma; we explore the impact of the

light-species approximation, whereby D and T are treated as a single species. In the second mixture,

we include C contaminants in the hot-spot and use the light-species approximation to reduce the

contaminated D, T, and C plasma to a mixture of only light species and C. Including electrons as

a third species would reduce to the two species case, due the mass scale separation [35, Section

10.9].

First, neglect C contaminants. Given mass density 𝜌 the number density of the individual

45

Figure 3.3 Left: A plot of the frequency-sum rule, Eq(3.35), for the RPA (blue), Mermin (green),
and completed Mermin (red) dielectric functions of the binary Yukawa mixture (BYM). Only the
Mermin dielectric function does not integrate to −𝜋 (black), and therefore does not satisfy the
f-sum rule. Right: A plot of the screening sum rule, Eq(3.36), for multi-species RPA (blue),
Mermin (green), and completed Mermin (red) of the binary Yukawa mixture (BYM) case. All
results converge to −𝜋 (black) in the long wavelength limit.
In both plots wiggles arise in the
long wavelength limit because the susceptibilities become Dirac deltas and numerical integration
becomes impossible, we have truncated our plots before that happens.

components is determined by

𝜌 = 𝑚𝐷𝑛𝐷 + 𝑚𝑇 𝑛𝑇 .

(3.40)

We consider three different combinations of 𝑛𝑇 and 𝑛𝐷: pure D (i.e., 𝑛𝑇 = 0), pure T (i.e., 𝑛𝐷 = 0),

and equal parts D and T (i.e., 𝑛𝐷 = 𝑛𝑇 ) Additionally, we consider the light species approximation

and treat D and T as indistinguishable components of a single species plasma. The relative number

density of the tritium and deuterium sets the mass of this light (L) species. This is formulated

𝑚 𝐿 =

𝑛𝐷𝑚𝐷 + 𝑛𝑇 𝑚𝑇
𝑛𝐷 + 𝑛𝑇

,

𝑛𝐿 = 𝑛𝐷 + 𝑛𝑇 ,

(3.41a)

(3.41b)

which implies that 𝜌 = 𝑛𝐿𝑚 𝐿. From Eq(3.40) and Eq(3.41), we compute the number density and

the other relevant plasma parameters, which we list in Table 3.1.

For the various plasmas, we plot the DSF at a fixed 𝑘 in Fig. 3.4. The single light species

DSF is in qualitative agreement with all cases. This suggests that the light species approximation

46

Table 3.1 Tabulated plasma parameters for a pure deuterium (D) plasma, a pure tritium (T) plasma,
a pure L plasma, and a mixed D and T plasma; all plasmas are at a mass density of 1002 g/cc
and temperature of 928 eV [36]. The effective charge of the ion 𝑍 is computed using More’s
Thomas-Fermi ionization estimate [37]. The coupling parameter is defined Γ𝑖 ≡ (𝑍𝑖)2 /𝑎𝑖𝑇 where
𝑎𝑖 = (3𝑍𝑖/4𝜋𝑛𝑒)1/3 and 𝑛𝑒 = (cid:205)𝑖 𝑍𝑖𝑛𝑖. The screening parameter is defined ˜𝜅 ≡ (𝜆𝑠 𝑘 𝐷,𝑒)−1, where
the screening length is given in Stanton and Murillo’s work [31]. Lastly, for 𝜈 we use the temperature
relaxation collision rates defined in Haack et al. [28].

Uncontaminated deuterium-tritium plasmas

n (1/cc)
species
2.40e26
pure L
2.00e26
pure D
pure T
3.00e26
mixed D 1.19e+26
mixed T 1.19e+26

𝑍
0.966
0.966
0.966
0.965
0.965

Γ

˜𝜅

0.147 1.176
1.198
0.138
0.158 1.146
0.116 1.196
1.196
0.116

𝜈𝑖= 𝑗 (1/s)
6.30e-03
5.42e-03
7.56e-03
3.47e-03
2.84e-03

𝜈𝑖≠ 𝑗 (1/s)
N/A
N/A
N/A
3.04e-03
3.04e-03

No mix
No mix
No mix

𝑛𝑇 = 𝑛𝐷

Figure 3.4 All plots compare 𝑆(𝑘, 𝜔) at fixed 𝑘/𝑘 𝐷,𝑒 = .63 across different DSF models of the
binary Yukawa Mixture (BYM). Each panel contains a pure deuterium (pure D), deuterium mixed
with tritium (mixed D), a pure tritium (pure T) and tritium mixed with deuterium (mixed T) plotted
in gray. Each panel also contains a pure light species [defined Eq(3.41)] plotted in solid line. The
plots show that all five cases have qualitative agreement across a given set of conservation laws.
However, the completed Mermin model has a stronger plasmon peak than the RPA.

is reasonable for the DSF of a D and T plasma at 1002 g/cc and temperature of 928 eV. While

the light species approximation works in a given model, the models themselves have qualitative

disagreements. By including momentum conservation into the collisions a peak emerges near the

ion plasmon frequency.

In the second case, we use the light species approximation for D and T and introduce C; this is

formulated as

𝜌 = 𝑚 𝐿𝑛𝐿 + 𝑚𝐶𝑛𝐶 .

(3.42)

47

Table 3.2 Tabulated plasma parameters for contaminated light species plasmas at three different
levels of contamination: 1 carbon atom per 105, 104, 103 light species atoms. Tabulated plasma
parameters are computed in the same way as in Table. 3.1.

Carbon contaminated light species plasma

𝜂

105

104

103

species
L
C
L
C
L
C

n (1/cc)
4.08e+25
4.08e+20
4.08e+25
4.08e+21
4.06e+25
4.06e+22

𝑍
0.941
4.880
0.941
4.880
0.941
4.880

Γ
0.223
0.075
0.223
0.161
0.222
0.345

˜𝜅
1.151
1.151
1.152
1.152
1.155
1.155

𝜈𝑖= 𝑗 (1/s)
8.48e-03
3.20e-02
8.48e-03
3.20e-02
8.43e-03
3.18e-02

𝜈𝑖≠ 𝑗 (1/s)
3.20e-07
1.71e-06
3.20e-06
1.71e-05
3.18e-05
1.69e-04

We assume equal amounts of deuterium and tritium and consider a 1 part carbon per 𝜂 parts light

(i.e., 𝑛𝐶 = 𝑛𝐿/𝜂) which allow us to express the density as

𝜌 = (𝑚 𝐿 + 𝑚𝐶/𝜂)𝑛𝐿.

(3.43)

From this expression, we compute the number density and the other relevant contaminated plasma

parameters, which we list in Table 3.2.

In Fig. 3.5, the contaminated DT DSF are plotted. The pure light species, indicated by a black

line, matches the pure light species from Fig. 3.4. The colored lines indicate increases in the ratio of

carbon atoms to light species atoms. Notice, for the Mermin-like model that the carbon impurities

drive the DSF to zero at 𝜔 = 0. This is qualitatively different from the RPA and the completed

Mermin models.

3.3.4 Conductivities

Both optical conductivity experiments and Kubo-Greenwood conductivity estimates need a

dynamical model to estimate the DC conductivity. When the Drude conductivity model fails to

fit the conductivity estimates, there are few other conductivity models to use. To produce a new

model, we apply the single species limit of the completed Mermin model to a system of electrons.

The dynamical conductivity can be related to the single species dielectric function models via

𝜎(𝑘, 𝜔) =

𝑖𝜔
4𝜋

(1 − 𝜀(𝑘, 𝜔)) .

(3.44)

where 𝑣(𝑘) is a Coulomb interaction for electrons.

48

Figure 3.5 All plots compare 𝑆(𝑘/𝑘 𝐷,𝑒, 𝜔) at fixed 𝑘/𝑘 𝐷,𝑒 = .63 across different DSF models of
the binary Yukawa Mixture (BYM). Each panel contains different carbon purity levels for a given
multi-species DSF. The black line indicates a pure light species. The blue dashed line indicates 1
carbon particle per 105 light species particles. The orange dotted line indicates 1 carbon particle per
104 light species particles. The green dotted line indicates 1 carbon particle per 103 light species
particles.

Eq(3.44) requires an estimate of the dielectric function 𝜀(𝑘, 𝜔). Using Eq(3.29) and Eq(3.34),

the long wavelength expansion of the completed Mermin dielectric function, with a Coulomb

interaction, yields

𝜀CM(𝜔|𝑎, 𝑏) = 1 −

(cid:18)

𝜔2
𝑝
𝜔2
𝜏

1 − 𝑖𝑎 1
𝜔𝜏𝜏

+ 𝑖𝑏

(cid:19) −1

.

𝜔𝜏−1
𝜔2
𝜏

(3.45)

Mermin’s number conservation correction has been modified by 𝑎 ∈ [0, 1] to smoothly turn

off/on the correction. While, the momentum conservation correction has been modified by 𝑏 ∈

[−1, 1] to either fully reverse momentum (𝑏 = −1) or fully conserve momentum (𝑏 = 1) in

a collision event. The energy conservation corrections enter at order 𝑘 2, thus our completed

Mermin and the Atwal-Ashcroft model predict the same expansion. It is reasonable to make these

conservation laws variable for a single species because we expect number conservation violating

electron recombination events and momentum and energy conservation violating phonon scattering

events.

Using Eq(3.45) and Eq(3.44) we produce a new first principles conductivity model. In Fig. 3.6,

we revisit Chen et al.’s Drude fit (𝑎 = 1, 𝑏 = 0) to 300K gold data [9] and study the effects of

parameters 𝑎 and 𝑏 on the shape of the conductivity model. The central black line expresses their

Drude fit. If particle number is not conserved (𝑎 < 1), then, relative to the Drude model, the DC

49

Figure 3.6 We have plotted the real part of the conductivity 𝜎𝑟 to demonstrate how number
conservation violation (𝑎 < 1, plotted as green lines) and momentum preservation (𝑏 ≠ 0, plotted
as red lines) affect Chen et al.’s Drude conductivity fit (plotted as a solid black line) [Z. Chen, et
al., Nature communications, 12.1, 1638, (2021)].

conductivity is suppressed and the optical conductivity is enhanced. Thus, the number conservation

term primarily affects the model’s slope at lower frequencies. Whereas, if particle momentum is

partially conserved (𝑏 = 1/2), then, relative to the Drude model, the DC conductivity is enhanced

and the optical conductivity is suppressed. Oppositely, if the collision reverses the momentum

of a particle (𝑏 = −1/2), then the DC conductivity is suppressed and the optical conductivity is

enhanced. In total, the momentum conservation term primarily enhances or suppresses the DC

conductivity, but does not change the slope in the low frequency region.

Lastly, our first principles model can be compared to the primary alternative to the Drude model.

The Drude-Smith (DS) conductivity model is defined

𝜎(𝜔) =

𝜎0
1 − 𝑖𝜔𝜏

(cid:18)

1 +

𝛽1
1 − 𝑖𝜔𝜏

(cid:19)

.

Using Eq(3.44), the Drude-Smith (DS) dielectric function is given
(cid:18) 𝜔𝜏 + 𝑖𝛽1/𝜏
𝜔

𝜀DS(𝜔|𝛽1) = 1 −

(cid:19)

𝜔2
𝑝
𝜔2
𝜏

(3.46)

(3.47)

.

Comparing Smith’s model Eq(3.47) to our model Eq(3.45), we see that for 𝑏 = 0 (i.e., momen-

tum conservation is turned off), then there is a direct relationship between our number conservation,

50

𝑎, and Smith’s 𝛽1. This relation is expressed as,

(cid:19) −1

(cid:18) 𝜔𝜏 − 𝑖𝑎/𝜏
𝜔𝜏

(cid:18) 𝜔𝜏 + 𝑖𝛽1/𝜏
𝜔

(cid:19)

.

=

(3.48)

Notice, when number is conserved (i.e., 𝑎 = 1) and Smith’s parameter is off (i.e., 𝛽1 = 0) Mermin’s

model and Smith model are equivalent. Further, when number is not conserved (i.e., 𝑎 = 0) and

Smith parameter is on (i.e., 𝛽1 = −1) Mermin’s model and Smith’s model are equivalent. This

suggests Smith’s model breaks local number conservation to achieve DC conductivity suppression.

3.4 Summary and Outlook

In this chapter, we introduce a new first-principles dynamical response model to address de-

mands in XRTS diagnostics, conductivity estimates, and radiation hydrodynamics codes. This

model, named the completed Mermin susceptibility, is the first to conserve both number and mo-

mentum locally across multiple species. This extends the work of Selchow et al. [25] and Atwal and

Ashcroft [22], expanding their single-species BGK approach to encompass multi-species systems.

Moreover, as a validation of our approach, we recover the multi-species Mermin susceptibility [27]

in a specific limiting case.

We showed numerical and analytic calculations of the f-sum rule and screening sum rule, which

emphasize our completed Mermin satisfies both sum rules. The momentum conservation correction

results in an infinitesimal plasmon peak at the long wavelength limit, indicating plasmons as the

sole energy loss mechanism. We argue that the violation of Mermin’s f-sum rule stems from his

continuity equation not assuming 𝛿j = 𝑛0𝛿u. Comparatively, the momentum collisional invariant

enforces this assumption.

In the single species limit, our momentum conservation correction

matches the local field correction derived in Morawetz and Fuhrmann [23].

We produced the two-component completed Mermin ion-ion DSF and applied the model to a

plasma of deuterium and tritium at ICF hot spot conditions. Comparing the completed Mermin

DSF to the Mermin and RPA DSFs, we showed that conservation of momentum produces a peak

near the ion plasma frequency not present in the Mermin and RPA models. This suggests that

ion-phonon scattering is an important energy loss mechanism in the warm dense matter regime.

51

We also demonstrate that the light species approximation qualitatively agrees with both mixed and

unmixed deuterium and tritium plasmas. Using the light species approximation, we discovered

that carbon contamination affects the Mermin DT DSF in a qualitatively different way than either

the RPA and the completed Mermin DSF. To make direct comparisons to experiments, we must

next develop appropriate dynamical inter-species collision frequencies 𝜏𝑖 𝑗 (𝜔) and effective ion-ion

interaction potentials 𝑣𝑖 𝑗 (𝑘).

Finally, we applied our completed Mermin conductivity model to dynamical gold conductivity

measurements. We show that both number and momentum conservation suppress DC conductivity,

albeit through distinct mechanisms. Therefore, our completed Mermin conductivity model provides

a first-principles alternative to the semi-empirical Drude-Smith conductivity model. By comparing

the two models, we showed that the Drude-Smith model violates local number conservation to

suppress DC conductivity. We interpret the violation of local number conservation as the removal

of free charge carriers. IIn this regard, our results support Cocker et al. [16], who demonstrated

that charge carrier confinement can account for Smith’s phenomenological correction. Our new

interpretation of Smith’s phenomenological parameter 𝛽1 contradicts Smith’s own claim that 𝛽1

accounts for back scattering. Considering the relevant conservation laws, we conclude that the

completed Mermin conductivity model [Eq(3.45)] can incorporate back scattering by setting pa-

rameters 𝑎 = 1 (activating number conservation) and 𝑏 ∈ [−1, 0] (indicating momentum reversal

during collisions).

52

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55

APPENDIX 3A

FOURIER TRANSFORMING OUR KINETIC EQUATION

In this appendix, we expand, linearize, and Fourier transform our multi-species BGK kinetic

equation. First, we list our chosen Fourier conventions. We choose

𝛿 𝑓𝑖𝐺 (r, v, 𝜔) =

∫

𝑑𝑡 𝑒𝑖𝜔𝑡𝛿 𝑓𝑖𝐺 (r, v, 𝑡)

as the temporal convention, so that

∫

𝑑𝑡 𝑒𝑖𝜔𝑡 𝜕𝑡 𝛿 𝑓𝑖𝐺 (r, v, 𝑡) = −𝑖𝜔 𝛿 𝑓 (r, v, 𝜔)

and

𝛿 𝑓 (k, v, 𝑡) =

∫

𝑑r 𝑒−𝑖k·r𝛿 𝑓𝑖𝐺 (r, v, 𝑡)

as our spatial convention, so that

∫

𝑑r 𝑒−𝑖k·r ∇r𝛿 𝑓𝑖𝐺 (r, v, 𝑡) = 𝑖k 𝛿 𝑓𝑖𝐺 (k, v, 𝑡).

This means that the usual expression for force

results in

𝑚𝑖atot
𝑖

(r, 𝑡) = −∇r𝑈tot
𝑖

(r, 𝑡)

atot
𝑖

(k, 𝜔) = −𝑚−1

𝑖

𝑖k𝑈tot
𝑖

(k, 𝜔).

(3A.1)

(3A.2)

(3A.3)

(3A.4)

(3A.5)

We now proceed to derive Eq(3.14) from Eq(3.1). For convenience, we reproduce Eq(3.1)

(cid:0)𝜕𝑡 + v · ∇r + atot
𝑖

· ∇v

(cid:1) 𝑓𝑖 (r, v, 𝑡) =

∑︁

𝑄𝑖 𝑗 .

𝑗

(3.1’)

We insert expansions Eq(3.8) and Eq(3.9), and then Fourier transform using the specified conven-

tions. The result is

(v · 𝑘 − 𝜔) 𝜆𝛿 𝑓𝑖𝐺 (k, v, 𝜔)

∫

− 𝜆

𝑑k′𝑑𝜔′ 𝑚−1

𝑖

(k − k′) · ∇v ( 𝑓𝑖𝐺 (v)𝛿(𝑘′)𝛿(𝜔′) + 𝛿 𝑓𝑖𝐺 (k, v, 𝜔)) 𝑈tot
𝑖

(k, 𝜔)

∑︁

=

𝑗

𝑖𝜆
𝜏𝑖 𝑗

𝛿 𝑓𝑖𝐺 (k, v, 𝜔) −

𝑖𝜆
𝜏𝑖 𝑗

∑︁

𝑗

𝛿𝑀𝑖 𝑗 (r, v, 𝑡).

(3A.6)

56

Note the second term on the LHS of Eq(3A.6) is a convolution, this term arises from atot

𝑖

·∇v 𝑓𝑖 (r, v, 𝑡)

since both terms are position and time dependent. As part of the linearization process, we replaced

𝑓𝑖 (r, v, 𝑡) with its expansion about global equilibrium 𝑓𝑖𝐺 (𝑣). Since 𝑓𝑖𝐺 (𝑣) does not have position

and time dependence, it is treated as a constant and its Fourier transform acquires Dirac delta

functions. From Eq(3A.6), we drop 𝛿 𝑓 𝑀

𝑖𝐹 𝑈tot
𝑖

since it is second order and evaluate the convolution.

The simplified expression is

(v · 𝑘 − 𝜔) 𝜆𝑖𝛿 𝑓𝑖𝐺 (k, v, 𝜔) − 𝜆𝑖 𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺 (v) 𝑈tot
𝑖

(k, 𝜔)

∑︁

=

𝑗

𝑖𝜆𝑖
𝜏𝑖 𝑗

𝛿 𝑓𝑖𝐺 (k, v, 𝜔) −

𝑖𝜆𝑖
𝜏𝑖 𝑗

∑︁

𝑗

𝛿𝑀𝑖 𝑗 (r, v, 𝑡).

(3A.7)

Canceling the fugacity factors and grouping the 𝛿 𝑓𝑖𝐺 (k, v, 𝜔) terms, we arrive at our desired result

Eq(3.14).

57

APPENDIX 3B

CALCULATING THE COLLISIONAL INVARIANT FOR ENFORCEING MOMENTUM
CONSERVATION

In this appendix, we show how to produce the momentum constraint, Eq(3.21b), from the first

moment of the collision operator, Eq(3.19b). For convenience we reproduce Eq(3.19b)

∫

𝑑v 𝑚𝑖v𝑄𝑖 𝑗 +

∫

𝑑v 𝑚 𝑗 v𝑄 𝑗𝑖 = 0.

(3.19b’)

Using the definition of 𝑄𝑖 𝑗 from Eq(3.4) and the expansions from Eq(3.8) and Eq(3.9), we obtain

the expanded form of the collision operator as

𝑄𝑖 𝑗 ≈

𝜆𝑖
𝜏𝑖 𝑗

(cid:0)𝛿𝑀𝑖 𝑗 − 𝛿 𝑓𝑖𝐺 (cid:1) .

(3B.1)

After inserting Eq(3B.1) into the momentum constraint, canceling the fugacities, and grouping

terms, we produce

∫

𝑑v v

(cid:18)

𝑚𝑖

𝛿𝑀𝑖 𝑗
𝜏𝑖 𝑗

+ 𝑚 𝑗

(cid:19)

𝛿𝑀 𝑗𝑖
𝜏𝑗𝑖

∫

=

𝑑v v

(cid:18)

𝑚𝑖

𝛿 𝑓𝑖𝐺
𝜏𝑖 𝑗

+ 𝑚 𝑗

(cid:19)

.

𝛿 𝑓 𝑗𝐺
𝜏𝑗𝑖

(3B.2)

We insert the linear expansion for 𝛿𝑀𝑖 𝑗 , Eq(3.17), on the LHS and omit terms that are odd powers

of v (Gaussian integrals evaluate to zero). The momentum constraint becomes

∫

𝑚𝑖
𝜏𝑖 𝑗

𝑑v v

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺
v · k

p𝑖 · 𝛿u +

∫

𝑚 𝑗
𝜏𝑗𝑖

𝑑v v

𝑚−1

𝑗 k · ∇v 𝑓 𝑗𝐺
v · k

p 𝑗 · 𝛿u =

𝑚𝑖𝛿j𝑖
𝜏𝑖 𝑗

+

𝑚 𝑗 𝛿j 𝑗
𝜏𝑗𝑖

.

(3B.3)

To simplify the RHS, we used 𝛿j𝑖 ≡ ∫ 𝑑v v𝛿 𝑓𝑖𝐺. Evaluating the LHS integrals yields Eq(3.21b).

58

APPENDIX 3C

EXPANDING PARTIAL FRACTIONS

In this appendix, we integrate over velocity to produce the system’s linearized dynamical response.

We reproduce Eq(3.18) for the reader’s convenience,

𝛿 𝑓𝑖𝐺 (k, v, 𝜔) =

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺 (𝑣)
v · k − 𝜔𝜏𝑖
𝑖
v · k

(cid:32)
𝑈𝑡𝑜𝑡 +

𝑗

∑︁

×

(cid:32)

1
𝜏𝑖 𝑗

𝛿𝜇𝑖 + p𝑖 · 𝛿u𝑖 𝑗 +

(cid:32) 𝑝2
𝑖
2𝑚𝑖

− 𝜇𝑖

(cid:33) 𝛿𝑇𝑖 𝑗
𝑇

(cid:33)(cid:33)

.

(3.18’)

The zeroth moment of Eq(3.18) is is given by

𝛿𝑛 = 𝑈tot𝑃 [v · k] +

𝑖
𝜏𝑖

𝑃 [1] 𝛿𝜇𝑖 +

𝑖
𝜏𝑖 𝑗

∑︁

𝑗

𝑃 [p𝑖 · 𝛿u] +

𝑖
𝜏𝑖 𝑗

(cid:32)

𝑃

(cid:35)

(cid:34) 𝑝2
𝑖
2𝑚𝑖

∑︁

𝑗

− 𝜇𝑖𝑃 [1]

(cid:33) 𝛿𝑇𝑖 𝑗
𝑇

. (3C.1)

The linear functional 𝑃 is defined

𝑃 [𝑔] ≡

∫

𝑑v

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺
v · k (cid:0)v · k − 𝜔𝜏𝑖

(cid:1)

𝑔.

This results in the following functionals

𝑃 (cid:2)𝑝2𝑛

𝑖 (v · k)(cid:3) = 𝐶 𝑀
𝑖,2𝑛,

𝑃 (cid:2)𝑝2𝑛

𝑖 (v · k − 𝜔𝜏)(cid:3) = 𝐵𝑀

𝑖,2𝑛,

𝑃

(cid:34)(cid:32) 𝑝2
𝑖
2𝑚𝑖

(cid:33) 𝑛(cid:35)

=

𝑃 [p𝑖 · 𝛿u] =

1
𝜔𝜏
𝑚𝑖
𝑘 2

(cid:32) 𝐶 𝑀
2𝑛
(2𝑚𝑖)𝑛 −

𝐵𝑀
2𝑛
(2𝑚𝑖)𝑛

(cid:33)

,

𝐶 𝑀
𝑖,0 k · 𝛿u.

(3C.2)

(3C.3a)

(3C.3b)

(3C.3c)

(3C.3d)

Note that, for 𝑛 = 0, Eq(3C.3c) simplifies to 𝑃 [1]. By substituting these expressions of the

functional 𝑃 into Eq(3C.1), we obtain Eq(3.24), as claimed.

59

APPENDIX 3D

IMPLEMENTING OUR MODEL

In this appendix, we collect the information necessary to compute the multi-species completed

Mermin susceptibilty.

Expressing 𝐶𝑖,2𝑛 with Dimensional Quantities

The completed Mermin susceptibilty has been expressed in terms of 𝐶2𝑛 (k, 𝜔). Thus, our first

step is create a dimensional version of this expression for 𝑛 = 0, 2, 4. We reproduce Eq(3.22a) here

for the reader’s convenience

𝐶 𝑀

𝑖,𝑛 (k, 𝜔) ≡

∫

𝑑v|p|𝑛

𝑚−1

𝑖 k · ∇v 𝑓𝑖𝐺
v · k − 𝜔

.

(3.22a’)

In the absence of an external magnetic field, the x axis is arbitrarily chosen as the direction of the

wave vector (i.e. v · v = 𝑣2

𝑥 and v · k = 𝑣𝑥 𝑘𝑥 ). We insert ∇v 𝑓𝑖𝐺 and evaluate the 𝑣 𝑦, 𝑣𝑧 integrals.

This recasts our function as

𝐶𝑖,2𝑛 = −

𝑛𝑖
𝑇

(𝑚𝑖𝑇)𝑛

(cid:18) 1
2𝜋

(cid:19) 1/2 ∫

C

𝑑 ˜𝑣 ˜𝑣2𝑛

˜𝑣𝑒− ˜𝑣2/2
√︃ 𝑚𝑖
𝑇

˜𝑣 −

𝜔𝜏
𝑘 𝑥

.

(3D.1)

Here ˜𝑣 =

√︃ 𝑚𝑖

𝑇 𝑣𝑥 is the dimensionless velocity. We can express the term in the denominator in

relation to the electron’s dimensionless phase velocity

√︂ 𝑚𝑖
𝑇

𝜔𝜏
𝑘𝑥

= 𝜁𝑖 ˜𝑣 𝑝.

(3D.2)

This equality follows from the definitions of the electron plasma frequency 𝜔2

the electron Debye wavenumber 𝑘 2

dimensionless phase velocity ˜𝑣 𝑝 ≡

𝑝,𝑒 ≡ 4𝜋𝑛𝑒 𝑒2/𝑚𝑒,
𝐷,𝑒 ≡ 4𝜋𝑛𝑒 𝑒2/𝑇, the mass fraction 𝜁𝑖 ≡ √︁𝑚𝑖/𝑚𝑒, and the
𝜔𝜏/𝜔 𝑝,𝑒
𝑘 𝑥/𝑘 𝐷,𝑒 . We use the electron based normalization to make the

multi-species calculations simpler. The final dimensionless expression of 𝐶𝑖,2𝑛 is,

𝐶𝑖,2𝑛 (𝑘, 𝜔𝜏) = −

𝑛𝑖
𝑇

(𝑚𝑖𝑇)𝑛𝐹2𝑛 (𝜁𝑖 ˜𝑣 𝑝),

𝐵𝑖,2𝑛 ≡ 𝐶𝑖,0(𝑘, 0)
(cid:19) 1/2 ∫
(cid:18) 1
2𝜋

𝐹2𝑛 (𝑧) =

C

𝑑 ˜𝑣 ˜𝑣2𝑛

˜𝑣
˜𝑣 − 𝑧

.

60

(3D.3a)

(3D.3b)

(3D.3c)

For the reader’s convenience, we note that 𝐵𝑖,0 = − 𝑛𝑖
can be computed considering the 𝑧 → 0 limit.

𝑇 , 𝐵𝑖,2 = − 𝑛𝑖

𝑇 𝑚𝑖𝑇, and 𝐵𝑖,4 = −3

𝑛𝑖
𝑇 (𝑚𝑖𝑇)2; this

Expressing 𝜒 with Dimensionless Quantities

Our next step is to propagate these expressions for 𝐶𝑖,2𝑛 and 𝐵𝑖,2𝑛 into the completed Mermin

susceptibility Eq(3.27). Consider the modified dielectric function

1,0 −

1 = 1 − 𝑣11(𝑘)𝐶 𝑀
𝜀∗
(cid:32) 𝑚1
𝑛1,0𝜏11

𝜔
𝑘 2

− 𝑖

+

𝑖
𝜔𝜏1
𝜏21
𝜏12

(cid:32)

(cid:33)

𝜏1

1 −

𝐶 𝑀
1,0
𝐵𝑀
1,0
𝑚2
1
𝑚1𝑛0,1𝜏21 + 𝑚2𝑛0,2𝜏12

(cid:33)

𝐶 𝑀
1,0

.

(3D.4)

Substituting 𝐶𝑖,2𝑛 and 𝐵𝑖,2𝑛 from Eq(3D.3) and 𝑚𝑖/𝑇 = 𝜁 2

𝑖 𝑘 2

𝐷,𝑒/𝜔2

𝐷,𝑒, we can recast the modified

dielectric function as

1,0 −

1 = 1 − 𝑣11(𝑘)𝐶 𝑀
𝜀∗
𝑘 2
𝜁 2
𝜔
1
𝑘 2𝜏11

𝜔2

+

(cid:32)

𝑖

𝐷,𝑒

𝑝,𝑒

+ 𝑖

𝑖
𝜔𝜏1
𝜔
𝑘 2

𝜏1
𝜏21
𝜏12

(cid:0)1 − 𝐹0(𝜁1 ˜𝑣 𝑝)(cid:1)

𝜓2
𝜁 4
1
1
𝜏21 + 𝜁 2
2

𝜁 2
1

𝜓2
1

𝜓2
2

𝜏12

(cid:33)

𝑘 2

𝐷,𝑒

𝜔2

𝑝,𝑒

𝐹0(𝜁1 ˜𝑣 𝑝).

(3D.5)

We normalized our parameters to electronic scales ˜𝑘 = 𝑘/𝑘 𝐷,𝑒, ˜𝜏 = 𝜏𝜔 𝑝,𝑒,

˜𝜔 = 𝜔/𝜔 𝑝,𝑒, and

˜𝜔𝜏1 = 𝜔𝜏1/𝜔 𝑝,𝑒. The result is

1 = 1 − 𝑣11(𝑘)𝐶 𝑀
𝜀∗

1,0 −

𝑖
˜𝜔𝜏1 ˜𝜏1

(1 − 𝐹0) +

(cid:32)

𝑖

1 ˜𝜔
𝜁 2
˜𝑘 2 ˜𝜏11

+ 𝑖 ˜𝜔
˜𝑘 2

˜𝜏21
˜𝜏12

𝜓2
𝜁 4
1
1
𝜓2
1 ˜𝜏21 + 𝜁 2

2

𝜁 2
1

𝜓2
2 ˜𝜏12

(cid:33)

𝐹0,

(3D.6)

where we have suppressed the arguments in 𝐹0(𝜁𝑖 ˜𝑣 𝑝) for brevity.

Next, we non-dimensionalize 𝑣11(𝑘)𝐶 𝑀

1,0, in Eq(3D.6). To do this, we assume that 𝑣𝑖 𝑗 (𝑘) is the

screeened Coulomb interaction

𝑣𝑖 𝑗 (𝑟) =

𝑍𝑖 𝑍 𝑗 𝑒2
𝑟

𝑒−𝑟/𝜆𝑠 .

(3D.7)

We group terms into a dimensionless parameters ˜𝑟 = 𝑘 𝐷,𝑒𝑟, ˜𝜅 = (𝜆𝑠 𝑘 𝐷,𝑒)−1 and Fourier transform

over ˜𝑟 to arrive at

𝑣𝑖 𝑗 (𝑘) =

𝑇
𝑛𝑒

(cid:18) 𝑍𝑖 𝑍 𝑗
˜𝑘 2 + ˜𝜅2

(cid:19)

.

61

(3D.8)

Notice that if we let 𝜆−1

𝑠 = 𝑘 𝐷,𝑒, as is the case in Thomas-Fermi theory, then ˜𝜅 = 1. The factor of

𝑇/𝑛e will multiply −𝑛𝑖/𝑇 from the susceptibility and leave behind −𝑛𝑖/𝑛e = −𝜓2

𝑖 . Therefore the

product is expressed

𝑣𝑖 𝑗 (𝑘)𝐶 𝑀

𝑖,0 = −𝜓2
𝑖

(cid:19)

(cid:18) 𝑍𝑖 𝑍 𝑗
˜𝑘 2 + ˜𝜅2

𝐹0.

(3D.9)

Inserting Eq(3D.9) into Eq(3D.6), yields the final expression for the modified dielectric function

𝜀∗
1 = 1 + 𝜓2
𝑖

𝐹0(𝜁1 ˜𝑣 𝑝) −

(cid:33)

(cid:32) 𝑍 2
1
˜𝑘 2 + ˜𝜅2
+ 𝑖 ˜𝜔
˜𝑘 2

𝜁 2
1 ˜𝜔
˜𝑘 2 ˜𝜏11

˜𝜏21
˜𝜏12

(cid:32)

𝑖

+

𝑖
˜𝜔𝜏1 ˜𝜏1
𝜓2
𝜁 4
1
1
1 ˜𝜏21 + 𝜁 2
𝜓2

(cid:0)1 − 𝐹0(𝜁1 ˜𝑣 𝑝)(cid:1)

(cid:33)

𝐹0(𝜁1 ˜𝑣 𝑝).

𝜁 2
1
The only remaining expression in Eq(3.27) is the momentum conservation coupling term. Following

2 ˜𝜏12
𝜓2

2

(3D.10)

the same steps as above yields

(cid:18)

𝑣12(𝑘) + 𝑖

(cid:19)

𝜔
𝑘 2
(cid:32)(cid:18) 𝑍1𝑍2
˜𝑘 2 + ˜𝜅2

𝑚1𝑚2
𝑚1𝑛1𝜏21 + 𝑚2𝑛2𝜏12
𝜁 2
𝜁 2
1
2
1 ˜𝜏21 + 𝜁 2
𝜓2

+ 𝑖 ˜𝜔
𝑘 2

𝜁 2
1

(cid:19)

2

𝐶1,0

= −

(cid:33)

2 ˜𝜏12
𝜓2

𝜓2
1

𝐹0(𝜁2 ˜𝑣 𝑝).

(3D.11)

Together Eq(3D.9), Eq(3D.10), and Eq(3D.11) provide a complete dimensionless representation of

the completed Mermin susceptibilty Eq(3.27).

Representing 𝐹2𝑛 with Special Functions

To evaluate Eq(3D.9), Eq(3D.10), and Eq(3D.11), we need numerical implementations of 𝐹0(𝑧).

Ichimaru Ch.4, expresses 𝐹0(𝑧) in terms of the special function 𝑊 (𝑍) [6]. We extended Ichimaru’s

procedure to calculate 𝐹2(𝑧) and 𝐹4(𝑧). The resulting expressions are

𝐹0(𝑧) = 1 + 𝑧

(cid:18)
𝑖

WofZ(𝑧/

√

(cid:19)

,

2)

𝐹2(𝑧) = 1 + 𝑧2 + 𝑧3

𝑖

WofZ(𝑧/

√

2)

(cid:19)

,

√︂ 𝜋
2
(cid:18)

√︂ 𝜋
2
(cid:18)
𝑖

𝐹4(𝑧) = 3 + 𝑧2 + 𝑧4 + 𝑧5

WofZ(

) ≡ 𝑖

√︂

2
𝜋

𝑧
√
2

𝑒−𝑧2/2

√︂ 𝜋
2
∫ 𝑧

0

WofZ(𝑧/

√

2)

(cid:19)

,

𝑑 ˜𝑣𝑒 ˜𝑣2/2 + 𝑒−𝑧2/2.

(3D.12a)

(3D.12b)

(3D.12c)

(3D.12d)

We denote 𝑊 (𝑧) as WofZ(𝑧) to align with scipy’s notation. The only remaining unknowns are the

parameters which describe our plasma system: 𝜓𝑖, 𝜁𝑖, 𝑍𝑖, ˜𝜅, 𝜈𝑖 𝑗 .

62

APPENDIX 3E

EXPANDING THE DIELECTRIC FUNCTION AT LONG WAVELENGTHS

In this appendix, we compute the long wave expansions presented in Eq(3.37). For single species,

Eq(3.34) informs us that

1
𝜀(𝑘, 𝜔)

= 𝑣(𝑘) 𝜒(𝑘, 𝜔).

(3E.1)

Here 𝜒 is the mean field corrected susceptibility. We rewrite Eq(3.34) in terms of susceptibilities

without mean field corrections (i.e., 𝜒 = 𝜒0/(1 − 𝑣(𝑘) 𝜒0)) as

1
𝜀(𝑘, 𝜔)

=

𝑣(𝑘) 𝜒0
1 − 𝑣(𝑘) 𝜒0

.

(3E.2)

We will use Eq(3E.2) to compute the long wave expansions presented in Eq(3.37).

Starting from Mermin Eq(3.30), completed Mermin Eq(3.29), and Atwal-Ashcroft [22], we

follow the same steps as Appendix 3D to produce non-dimensional representations of the single

species susceptibilities without mean field correction. These expressions are

˜𝜒𝑀
0 ≡

−𝐹0
+ 𝑖
˜𝜏 ˜𝜔𝜏

˜𝜔
˜𝜔𝜏

,

𝐹0
−𝐹0

˜𝜒𝐶 𝑀
0 ≡

˜𝜔
˜𝜔𝜏

+ 𝑖

(cid:16) 1
˜𝜏 ˜𝜔𝜏

+

(cid:17)

𝑖 ˜𝜔
𝜁 2
˜𝑘 2 ˜𝜏

𝐹0

,

˜𝜒 𝐴𝐴
0 ≡

2 ˜𝜏 ˜𝜔 (𝐹2𝐹2 − 𝐹0𝐹4)
2 ˜𝜏2 ˜𝑘 2 ) (𝐹2𝐹2 − 𝐹0𝐹4) + 𝑖
2 ˜𝜏 ˜𝜔𝜏
where we have normalized our parameters to electronic scales: 𝜁 2

−𝐹0 + 𝑖
𝜁 2
𝑖

1
2 ˜𝜔 ˜𝜔𝜏 ˜𝜏2 +

𝐹0 + (

𝑖𝜁 2
𝑖 ˜𝜔
𝑘 2 ˜𝜏

˜𝜔
˜𝜔𝜏

+

(3E.3a)

(3E.3b)

,

(3E.3c)

(3𝐹0 − 2𝐹2 + 𝐹4)

𝑖 = 𝑚𝑖/𝑚𝑒, ˜𝑘 = 𝑘/𝑘 𝐷,𝑒, ˜𝜏 = 𝜏𝜔 𝑝,𝑒,

˜𝜔 = 𝜔/𝜔 𝑝,𝑒, and ˜𝜔𝜏 = 𝜔𝜏/𝜔 𝑝,𝑒. Additionally, we have suppressed the dependence of 𝐹0(𝜁𝑖 ˜𝑣 𝑝),

𝐹2(𝜁𝑖 ˜𝑣 𝑝), and 𝐹4(𝜁𝑖 ˜𝑣 𝑝).

As shown in Appendix 3D, 𝐹0(𝑧), 𝐹2(𝑧), and 𝐹4(𝑧) can be expressed in terms of the WofZ

function, which has known 𝑧 → ∞ expansions. Expanding 𝐹0(𝑧), 𝐹2(𝑧), and 𝐹4(𝑧) at large 𝑧 yields

63

the following analytic expressions:

𝐹0(𝑧) = −1/𝑧2 + O [𝑧−4],

𝐹2(𝑧) = −3/𝑧2 + O [𝑧−4],

𝐹4(𝑧) = −15/𝑧2 + O [𝑧−4].

lim
𝑧→∞

lim
𝑧→∞

lim
𝑧→∞

(3E.4a)

(3E.4b)

(3E.4c)

To construct Eq(3.37), we first substitute Eq(3E.4) into the susceptibilities Eq(3E.3) and then we

substitute the susceptibilities into Eq(3E.2).

64

CHAPTER 4

MULTI-SPECIES KINETIC-FLUID COUPLING FOR HIGH-ENERGY DENSITY
SIMULATIONS

4.1

Introduction

Material flows with long mean free paths (i.e. high Knudsen numbers), also known as kinetic

flows, occur in plasmas, neutron and radiation transport, and dilute gases. Their evolution is

often described by equations which are prohibitively difficult to solve numerically. However, if

the Knudsen number is small enough, moment-based, i.e. hydrodynamic, models are applicable.

These contain fewer degrees of freedom and thereby greatly reduce the computational cost. For

many applications, such as interfaces and shocks, the spatial region over which the material is

kinetic is proportionately small, suggesting the usage of a fluid-kinetic hybrid model that locally

utilizes both kinetic and hydrodynamic approaches. Such a model has been proposed, for example,

by Degond et al.

[1]. Their work introduced the concept of a buffer region, in which a convex

combination of both models is computed. Subsequent works extended this method to include time-

dependent buffer zones [2], as well as applying a micro-macro framework for coupling the equations

[3, 4, 5, 6]. A more recent study [7] proposes an infinitely thin buffer region and the transitioning

from one cell to another using flux matching. Degond’s original method was developed for single

charge-neutral particle species. Here, we generalize it to the case of mixtures with electric fields.

While this approach can be used in a wide variety of applications, in this chapter, we apply it to a

high-energy density physics (HEDP) problem, specifically an experiment in intertial confinement

fusion (ICF) to study preheat mixing in separated reactants.

ICF experiments span a wide range of plasma conditions, with densities and temperatures that

can vary over many orders of magnitude. In addition, materials can be composed of light and/or

heavy ions (e.g. deuterium (D), tritium (T) and gold (Au), respectively) and change between

strong to weak collisionality. These experiments are typically designed and analyzed by radiation

hydrodynamics simulations which attempt to capture the multiphysics nature of HEDP matter.

However, significant differences between simulations and experiments remain, with a number of

65

possible sources for these discrepancies. For example, in near-vacuum hohlraums, multi-material

mixing beyond what is predicted by hydrodynamic instabilities has been attributed to kinetic effects

[8], i.e., a breakdown in the underlying assumption that collision times and mean free paths are

sufficiently small for a fluid description to apply. Some phenomena attributed to kinetic effects

may be attributable to other radiation hydrodynamics features, e.g., a study by Pape et al.

[9]

that shows that a small amount of helium fill in the hohlraum can suppress interpenetration of the

ablator and wall plasmas, and cross-beam energy transfer is the main driver of discrepancies with

past modeling mismatches.

A kinetic description of multi-material mixing will impact predictions for several important

processes in ICF experiments [8]. For example, a mixing layer can occur at material interfaces

in the capsule, such as between the ablator and the DT ice as well as the DT ice and vapor.

Material interdiffusion due to preheating may result in significantly different configurations when

the main driving shock(s) arrive [10, 11], causing unwanted material injection into the hot spot.

The sharpness of interfaces in ICF capsules implies that physical phenomena of large particle

mean free paths might be relevant. Indeed, experiments on an Au-CDH interface on the Trident

laser [12] have shown a superdiffusive evolution of the interface which is in line with kinetic

trajectories of fast particles. Another mixing layer occurs when plasma that is ejected from the

hohlraum wall intersects with ablated material from the capsule in the region crossed by the inner

laser beams [13]. A kinetic description of this mixing layer would predict that the wall and

capsule plasmas interpenetrate. However, this phenomenon cannot be captured in a single-species

hydrodynamic code and the resulting density spike interferes with the propagation of the inner

beams. Intermediate moment models [14] have shown that a mixing layer may be described by

a hydrodynamic description in some configurations, and in fact laser-plasma interactions (e.g.

CBET) are the culprit for asymmetric drive. This hybrid model or future adaptive versions could

also be used to study the suppression of interpenetration. Finally, due to the relatively low density

of the capsule fill gas in some ICF experiments, kinetic effects manifest themselves via species

separation between the D and T atoms in the fusion gas. This results in DD and DT yield ratios

66

that differ greatly from hydrodynamic prediction [15, 16].

These mixing issues are just a few examples of kinetic effects; other non-equilibrium physics

such as laser-plasma interaction, self-generated electromagnetic fields, Knudsen layers, and detailed

shock structure may also have a significant impact on predictive modeling of ICF [8]. Due to the

integrated nature of ICF experiments, it is difficult to study any of these effects in isolation. The

MARBLE campaign fielded a unique separated reactants experiment in an attempt to measure

mixing rates in ICF implosions via changes in the capsule DT yield [17, 18]. It was conjectured

that atomic mixing and ion temperature separation played significant roles in the resulting DT/DD

fusion yields [19]. The impending BOSQUE campaign is a direct drive follow-up to MARBLE

campaign that uses larger capsules (∼2X). The larger burn volumes are expected to achieve triple

the temperatures and much higher thermonuclear yields (∼100X to ∼1000X) [20]. It is reasonable

to expect kinetic effects like atomic mixing and temperature seperation will remain significant at

higher hot spot temperatures.

Accurate predictions of ICF experiments benefit from the ability to describe temperature sep-

aration, velocity separation, viscosity, conduction, and diffusion in mixed material regions. Due

to the dimensionality and time resolution constraints required by non-equilibrium models, a fully

kinetic simulation of an ICF experiment is currently not feasible. Most approaches are limited to

short time scales and small regions in space. With that, we propose a hybrid, coupled, multispecies

fluid-kinetic approach which ensures that the added expense of kinetic modeling is only applied in

regions where it is necessary, while the relatively less expensive hydrodynamic equations are solved

in the remainder of the computational domain. This chapter builds on and extends the original

kinetic-continuum domain decomposition by Degond et al. [1] to consider multiple particle species.

Throughout the chapter, we will refer to this method as the kinetic-hydro decomposition or hybrid

method. For the kinetic and hydrodynamic models we use an entropic, conservative Bhatnagar-

Gross-Krook (BGK) description [21] and its associated multi-species fluid equations, respectively.

In this chapter, the Chapman-Enskog expansion is used to obtain the associated multi-species fluid

equations in both the Euler and Naver-Stokes limits in three dimensions (3D).

67

The chapter is organized as follows. In Section 2, we introduce the multispecies kinetic equation

and the associated Euler and Navier-Stokes hydrodynamic limits. In Section 3, we describe the

hybrid method which couples the kinetic and hydrodynamic models via a buffer region, providing

a continuous transition between both. In Section 4, we present proof of concept 1D simulations

which demonstrate the method for a Sod problem and a preheated pore inspired by the pores in a

MARBLE capsule [17]. We present summary and outlook in Section 5. Additional details of the

analytic derivation of the hybrid method, especially for the Navier-Stokes hydrodynamic model,

are provided in the Appendix.

4.2 The multispecies kinetic and hydrodynamic models

In this section, we give a brief description of the BGK model and its limiting hydrodynamic

equations. Both limits will be coupled via the hybrid approach in Section 4.3. For a single particle

species, the BGK collision operator is a nonlinear relaxation operator of the form

𝑄 𝐵𝐺𝐾 [ 𝑓 ] = 𝜈 (𝑀 [ 𝑓 ] − 𝑓 ),

(4.1)

where 𝜈 is a collision frequency, 𝑓 = 𝑓 (x, c, 𝑡) is the phase-space density function depending on

particle position x, velocity c, and time 𝑡. The Maxwellian 𝑀 [ 𝑓 ] is the local equilibrium state

based on the moments of 𝑓 :

𝑀 [ 𝑓 ] = 𝑛

(cid:16) 𝑚
2𝜋𝑇

(cid:17) 3/2

exp

(cid:18)

−

𝑚(c − v)2
2𝑇

(cid:19)

.

(4.2)

Here 𝑚 is the mass of the species, while the particle number density 𝑛, bulk velocity v, and

temperature 𝑇 are defined by

∫

𝑛 ≡

𝑓 𝑑c,

v ≡

∫

1
𝑛

c 𝑓 𝑑c,

𝑇 ≡

∫

𝑚
3𝑛

(c − v)2 𝑓 𝑑c.

(4.3)

The BGK model is the simplest kinetic approach which captures the most fundamental properties of

the Boltzmann collision model, namely, that it locally conserves mass, momentum, and energy and

satisfies Boltzmann’s H-Theorem. The multispecies BGK model is an analogue of the description

for a single species. It is a set of nonlinear relaxation operators that conserve the species masses,

68

pairwise momentum, and pairwise kinetic energy as well as satisfies the multispecies H-Theorem

[21]. In a nutshell, the model is defined as

𝑄BGK
𝑖

=

∑︁

𝑗

𝑄BGK
𝑖 𝑗

[ 𝑓𝑖, 𝑓 𝑗 ],

where 𝑄BGK

𝑖

is the collision operator for species 𝑖,

𝑄 𝐵𝐺𝐾
𝑖 𝑗

[ 𝑓𝑖, 𝑓 𝑗 ] ≡ 𝜈𝑖 𝑗 (𝑀𝑖 𝑗 [ 𝑓𝑖, 𝑓 𝑗 ] − 𝑓𝑖)

(4.4)

(4.5)

is the multi-species collision operator for the interaction between species 𝑖 and 𝑗 with the corre-

sponding phase-space density functions 𝑓𝑖 and 𝑓 𝑗 . 𝜈𝑖 𝑗 is the frequency which the 𝑖 species collides

with the 𝑗 species. Furthermore, the inter-species Maxwellians are

𝑀𝑖 𝑗 [ 𝑓𝑖, 𝑓 𝑗 ] ≡ 𝑛𝑖

(cid:19) 3/2

(cid:18) 𝑚𝑖
2𝜋𝑇𝑖 𝑗

(cid:32)

exp

−

(cid:33)

𝑚𝑖 (c − v𝑖 𝑗 )2
2𝑇𝑖 𝑗

(4.6)

To define 𝑣𝑖 𝑗 and 𝑇𝑖 𝑗 , we enforce conservation of mass, momentum, and energy [21] which gives

the following algebraic relations:

v𝑖 𝑗 =

𝜌𝑖𝜈𝑖 𝑗 v𝑖 + 𝜌 𝑗 𝜈 𝑗𝑖v 𝑗
𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖

,

𝑇𝑖 𝑗 =

𝑛𝑖𝜈𝑖 𝑗𝑇𝑖 + 𝑛 𝑗 𝜈 𝑗𝑖𝑇𝑗
𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖

+

𝜌𝑖𝜈𝑖 𝑗 (𝑣2

𝑖 𝑗 ) + 𝜌 𝑗 𝜈 𝑗𝑖 (𝑣2

𝑖 − 𝑣2
3 (cid:0)𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖(cid:1)

𝑗 − 𝑣2
𝑖 𝑗 )

.

(4.7)

Equivalent to Eq.Eq(4.3), the moments of each species are given by

∫

𝑛𝑖 ≡

𝑓𝑖𝑑c,

v𝑖 ≡

∫

1
𝑛𝑖

c 𝑓𝑖𝑑c,

𝑇𝑖 ≡

∫

𝑚𝑖
3𝑛𝑖

(c − v𝑖)2 𝑓𝑖𝑑c.

(4.8)

The multi-species collision operator forms a key part of the BGK equation,

𝜕𝑡 𝑓𝑖 + c · ∇𝑥 𝑓𝑖 + a𝑖 · ∇c 𝑓𝑖 =

∑︁

𝑗

𝑄BGK
𝑖 𝑗

[ 𝑓𝑖, 𝑓 𝑗 ],

(4.9)

where a𝑖 is an acceleration term, for example for charged particles in the presence of an electric

field.

69

4.2.1 The multi-species Navier-Stokes model

The Navier-Stokes equations for the conservation of mass, momentum and energy, associated

to the multispecies BGK model are [21, 22]

𝜕𝑡 𝜌𝑖 + ∇x · (𝜌𝑖v) + ∇x · (𝜌𝑖V𝑖) = 0,
∑︁

𝜕𝑡 (𝜌v) + ∇x · (𝜌v ⊗ v) + ∇x · P =

𝜌𝑖 a𝑖,

3
2

(𝜕𝑡 (𝑛𝑇) + ∇x · (𝑛𝑇v)) + ∇x · q + P : ∇𝑥v =

𝑖
∑︁

𝑖

V𝑖 · a𝑖.

(4.10)

(4.11)

(4.12)

Here, 𝑛 = (cid:205)𝑖 𝑛𝑖 is the total particle number density, 𝜌𝑖 = 𝑚𝑖𝑛𝑖 is the species mass density with the

total density 𝜌 = (cid:205)𝑖 𝜌𝑖. The velocity of the mixture is given by v = 1
𝜌

(cid:205)𝑖 𝜌𝑖v𝑖 while V𝑖 = v𝑖 − v is

the (macroscopic) diffusion velocity for species 𝑖. Finally, the pressure tensor1 is P, 𝑇 is the mixture

temperature, and q is the heat flux. These thermodynamic quantities can be given in terms of the

distributions 𝑓𝑖 by

P =

∑︁

𝑖

P𝑖 ≡

𝑇 ≡

𝑖
2
3𝑛

∑︁

q𝑖 ≡

q =

∑︁

𝑖

∫

∑︁

𝑚𝑖C ⊗ C 𝑓𝑖𝑑c,

∑︁

∫ 𝑚𝑖
2
𝑖
∫ 𝑚1
2

𝐶2 𝑓𝑖𝑑c,

C𝐶2 𝑓𝑖𝑑c,

(4.13)

(4.14)

(4.15)

where C = c − v is the microscopic diffusion velocity. The macroscopic diffusion velocity, pressure

tensor, and heat flux are written in terms of the hydrodynamic variables as

V𝑖 =

∑︁

𝑗

𝐷𝑖 𝑗 d 𝑗 ,

P = 𝑛𝑇I − 𝜂

(cid:18)

∇xv + (∇xv)𝑇 −

(cid:19)

,

(∇x · v)I

2
3

q = −𝜅∇x𝑇 +

∑︁

𝑖

5𝑇
2𝑚𝑖

𝜌𝑖V𝑖,

(4.16)

(4.17)

(4.18)

1Technically P is a mean-field pressure tensor, and additional contributions from the various force terms a𝑖 can
be grouped with the pressure in the resultant hydrodynamic equations, e.g., electron pressure terms. For other EOS
types, e.g. polyatomic gases with additional degrees of freedom, matching an EOS requires introducing additional
phase space variables; see [Munafo et al., JCP, 2014].

70

with the diffusion driving force term d𝑖 defined in 4A. The coefficients 𝐷𝑖 𝑗 , 𝜈, 𝜅 for interdiffusion,

viscosity, and thermal conductivity, respectively, are the transport coefficients of the hydrodynamic

model. Their formulations are directly related to the underlying BGK kinetic system [21]. Note

that in the Euler equations, these transport coefficients are zero.

4.3 Coupling of the BGK and continuum equations for multiple species in 3D

In this section, we derive the analytical description for the kinetic-hydro decomposition and

the connecting buffer region, both for the multispecies BGK, as well as the associated Euler and

Navier-Stokes equations. For simplicity, we drop the acceleration term a𝑖 in the remainder of the

chapter. As will be shown in section 4.5.2, it can be reintroduced to the fluid limit without major

changes to the resulting equations. We begin with the multi-species BGK formulation.

As in [21], we introduce a scaling parameter 𝜖 to the collision operators that is analogous to

the Knudsen number.

Its role is to emphasize the highly collisional regime and assist with the

approximation process. The corresponding BGK equations are:

𝜕𝑡 𝑓𝑖 + c · ∇𝑥 𝑓𝑖 =

1
𝜖

𝑄𝑖 [ 𝑓 ],

𝑄𝑖 [ 𝑓 ] =

∑︁

𝑗

𝑄𝑖 𝑗 [ 𝑓𝑖, 𝑓 𝑗 ],

(4.19)

where we remove the BGK superscript for better readability. We assume that the kinetic effects

are only important in relatively small, localized regions of the computational domain, for example

around an interface or shock. The domain is decomposed into fluid regions, where we expect

the usual hydrodynamic limit to hold locally, and kinetic regions, where non-equilibrium physics

has a significant effect. However, simply inserting an interface between the two models requires

devising compatible boundary conditions for each model. Following [1] we therefore introduce

a buffer region between the kinetic and hydrodynamic areas that provides a smooth transition

between the two models. In this buffer region, both the fluid and kinetic models are solved with

some modifications. The solution to the hybrid model is recovered as the weighted sum of the

coupled kinetic and hydrodynamic descriptions. At the edge of the buffer region, the modified

fluid or kinetic model becomes degenerate and the hybrid model smoothly transitions into only

71

Figure 4.1 Example of a buffer function ℎ(𝑥). As ℎ(𝑥) varies within the interval [0, 1], it transitions
the hybrid model between pure hydrodynamic and pure kinetic regions. For 𝑥 ≤ 𝑎, a value of
ℎ(𝑥) = 0 indicates the continuum regime. For 𝑥 ≥ 𝑏, the transition function is ℎ(𝑥) = 1 and matter
is in non-equilibrium.

solving the respective kinetic or fluid equations. Thus no special boundary conditions are required.

However, as we will show below, one must ensure that the buffer region is placed in an area where

a hydrodynamic limit is reasonable.

In practice, the spatial location of the kinetic region may

dynamically change, and a moving buffer region may be required. As is shown in the single species

context [2], the addition of a time-dependent buffer region adds many extra terms to the hybrid

equations. For simplicity of presentation, we focus on a fixed buffer region in this manuscript,

and the extension to moving buffer regions in the multispecies context will be the subject of future

work.

The buffer region is characterized by a continuous transition function ℎ(𝑥) which is defined to

be 1 and 0 in the kinetic and fluid regimes, respectively. For a buffer region in an interval [𝑎, 𝑏],

the simplest choice is a linear dependence on 𝑥, i.e.,

ℎ(𝑥) =






1,

0,

𝑥 ≤ 𝑎,

𝑥 ≥ 𝑏,

0 ≤ (𝑥 − 𝑏)/(𝑎 − 𝑏) ≤ 1,

𝑥 ∈ [𝑎, 𝑏].

(4.20)

Figure 4.1 gives a graphical example of a computational domain that is divided in hydrodynamic,

buffer, and kinetic regions, together with the corresponding transition function.

In order to focus on the coupled equations in the buffer region, we assume that the transition

function is fixed in time. Dynamic buffer regions, as e.g. derived for the single species hybrid

model [2], will be the subject of future work. For simplicity, we will also express the transition

72

function as ℎ. We use it to create a coupled system of kinetic and hydrodynamic equations by

splitting the full distribution function 𝑓𝑖 into kinetic and a fluid parts,

𝑓𝑖 = 𝑓𝑖𝐾 + 𝑓𝑖𝐹,

𝑓𝑖𝐹 ≡ (1 − ℎ) 𝑓𝑖,

𝑓𝑖𝐾 ≡ ℎ 𝑓𝑖.

(4.21)

(4.22)

Multiplying Eq. (4.19) with ℎ and (1 − ℎ) and using the definitions of 𝑓𝑖𝐾 and 𝑓𝑖𝐹, we can rewrite

the transport equation into a system of two coupled equations:

𝜕𝑡 𝑓𝑖𝐾 + ℎ c · ∇𝑥 𝑓𝑖𝐾 + ℎ c · ∇𝑥 𝑓𝑖𝐹 =

ℎ
𝜖

𝑄𝑖 [ 𝑓 ]

𝜕𝑡 𝑓𝑖𝐹 + (1 − ℎ) c · ∇𝑥 𝑓𝑖𝐹 + (1 − ℎ) c · ∇𝑥 𝑓𝑖𝐾 =

(1 − ℎ)
𝜖

𝑄𝑖 [ 𝑓 ].

In the following, we will take moments of the equation for 𝑓𝑖𝐹 with respect to

(cid:16)

m =

𝑚𝑖, 𝑚𝑖c,

𝑐2(cid:17)

𝑚𝑖
2

(4.23)

(4.24)

(4.25)

to obtain the equations for the mass, momentum, and energy contained in the fluid piece of the

decomposition. This results in the continuum equations

⟨𝜕𝑡m 𝑓𝑖𝐹⟩ + (1 − ℎ) ⟨mc · ∇𝑥 𝑓𝑖𝐹⟩ + (1 − ℎ) ⟨mc · ∇𝑥 𝑓𝑖𝐾⟩ =

(cid:42)

1 − ℎ
𝜖

∑︁

m

𝑗

(cid:43)

𝑄𝑖 𝑗 [ 𝑓𝑖, 𝑓 𝑗 ]

.

(4.26)

Here and for the remainder of this chapter, ⟨⟩ denotes

∫

⟨𝜙⟩ =

𝜙 dc.

(4.27)

When ℎ = 1, the system simply reduces to the original kinetic transport equation for 𝑓𝑖𝐾 as given in

Eq. Eq(4.9). For ℎ = 0 and assuming the standard Euler (or Navier-Stokes) closure, Eq. Eq(4.26)

can be written as a set of hydrodynamic equations for 𝑛𝑖, v, and 𝑇, as given in Eqs. Eq(4.10) -

Eq(4.12). Finally, when 0 < ℎ < 1, Eq. Eq(4.23) requires 𝑓𝑖𝐹 for its streaming and collision updates

while Eq. Eq(4.26) uses 𝑓𝑖𝐾 for one of its flux terms. Furthermore, we note that in the buffer region,

the moment variables 𝑛𝑖, v, and 𝑇 in Eq. Eq(4.26) correspond to the moments of 𝑓𝑖𝐹 and not the

moments of the total distribution function 𝑓 = 𝑓𝑖𝐾 + 𝑓𝑖𝐹. The main remaining task is to determine

the Euler and Navier-Stokes closures of the new fluid system in Eq. Eq(4.26), which differs from

the standard approach due to the presence of ℎ and the kinetic contribution 𝑓𝑖𝐾.

73

4.3.1 Multi-species kinetic-Euler system in the buffer region

We begin by deriving the kinetic-hydro coupling for an Euler closure. The coupling between

BGK and the Navier-Stokes equations will be discussed in the next section and naturally builds on

top of this foundation. For the Euler derivation, we assume that 𝜖 ≪ 1 in Eq. Eq(4.24) and order

terms by its powers. Recalling that the collision operator depends of the full distribution function

𝑓𝑖, the leading order term is

0 =

∑︁

𝑗

𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹 + 𝑓𝑖𝐾, 𝑓 𝑗 𝐹 + 𝑓 𝑗 𝐾 (cid:3) .

(4.28)

We can rewrite the right-hand side into

0 =

∑︁

𝑗

𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹, 𝑓 𝑗 𝐹 (cid:3) + (cid:0)𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹 + 𝑓𝑖𝐾, 𝑓 𝑗 𝐹 + 𝑓 𝑗 𝐾 (cid:3) − 𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹, 𝑓 𝑗 𝐹 (cid:3) (cid:1) .

(4.29)

Since 𝜖 ≪ 1, that is, the buffer region is in a regime where the hydrodynamic limit applies, we

assume that 𝑓 − 𝑓𝐹 ≈ 𝑂 (𝜖) and thus the difference term in Eq. Eq(4.29) satisfies

(cid:0)𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹 + 𝑓𝑖𝐾, 𝑓 𝑗 𝐹 + 𝑓 𝑗 𝐾 (cid:3) − 𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹, 𝑓 𝑗 𝐹 (cid:3) (cid:1) ≈ 𝑂 (𝜖).

(4.30)

∑︁

𝑗

As a result, at leading order, the collision operator is

𝑄𝑖 𝑗 (cid:2) 𝑓𝑖𝐹, 𝑓 𝑗 𝐹 (cid:3) = 0.

∑︁

𝑗

(4.31)

By standard H-Theorem arguments [22], 𝑓𝑖𝐹 must therefore be a Maxwellian distribution given by

Eq. Eq(4.2). With this approximation to 𝑓𝑖𝐹, we derive the resulting Euler equations. To simplify

the moment calculations we note that any reasonable collision operator, including the multi-species

BGK operator, should conserve pairwise mass, momentum, and energy. For any given distribution

functions, 𝜓 and 𝜙, these conservation properties can be written as [21]

(cid:10)𝑚𝑖𝑄𝑖 𝑗 [𝜓, 𝜙](cid:11) = 0

(cid:10)𝑚𝑖 c 𝑄𝑖 𝑗 [𝜓, 𝜙](cid:11) + (cid:10)𝑚 𝑗 c 𝑄 𝑗𝑖 [𝜓, 𝜙](cid:11) = 0

(cid:10)𝑚𝑖 𝑐2 𝑄𝑖 𝑗 [𝜓, 𝜙](cid:11) + (cid:10)𝑚 𝑗 𝑐2 𝑄 𝑗𝑖 [𝜓, 𝜙](cid:11) = 0.

(4.32)

74

For compactness in the following derivations, we define the 1𝑠𝑡, 2𝑛𝑑, and 3𝑟𝑑 moments of the kinetic

distribution function as

K1,𝑖 ≡ ⟨𝑚𝑖 c 𝑓𝑖𝐾⟩ , K2,𝑖 ≡ ⟨𝑚𝑖 (c ⊗ c) 𝑓𝑖𝐾⟩ , K3,𝑖 ≡

(cid:68) 𝑚𝑖
2

c 𝑐2 𝑓𝑖𝐾

(cid:69)

.

The total 𝑛𝑡ℎ kinetic moment across all species 𝑖 is given by

K𝑛 ≡

∑︁

K𝑛,𝑖.

𝑖

(4.33)

(4.34)

To obtain the equation for the conservation of mass, we compute

⟨𝑚𝑖 𝜕𝑡 𝑓𝑖𝐹⟩ + (1 − ℎ) ⟨𝑚𝑖 c · ∇𝑥 𝑓𝑖𝐹⟩ + (1 − ℎ) ⟨𝑚𝑖 c · ∇𝑥 𝑓𝑖𝐾⟩ =

(1 − ℎ)
𝜖

⟨𝑚𝑖 𝑄𝑖 [ 𝑓 ]⟩ ,

(4.35)

which results in

𝜕𝑡 𝜌𝑖 + (1 − ℎ) ∇𝑥 · (𝜌𝑖v) + (1 − ℎ) ∇𝑥 · ⟨𝑚𝑖 c 𝑓𝑖𝐾⟩ = 0.

(4.36)

This can be rewritten as an equation for the conservation of the number density. Defining ˜K1,𝑖 ≡

K1,𝑖/𝑚𝑖, we obtain

𝜕𝑡𝑛𝑖 + (1 − ℎ)∇𝑥 · (𝑛𝑖v) + (1 − ℎ) ∇𝑥 · ˜K1,𝑖 = 0.

(4.37)

We also note that this can be trivially re-formulated into an expression for the conservation of mass

fraction 𝑌𝑖 = 𝜌𝑖/𝜌

𝜕𝑡 (𝜌𝑌𝑖) + (1 − ℎ)∇𝑥 · (𝜌v𝑌𝑖) + (1 − ℎ) ∇𝑥 · K1 = 0.

(4.38)

Summing Eq. Eq(4.36) over all species 𝑖, we arrive at the conservation equation for the total mass

D𝑡 𝜌 + (1 − ℎ) 𝜌∇𝑥 · v + (1 − ℎ) ∇𝑥 · K1 = 0,

(4.39)

where the time derivative is given by D𝑡 = 𝜕𝑡 + (1 − ℎ) v · ∇𝑥. The equation for conservation of total

momentum can be obtained by multiplying Eq. (4.24) by 𝑚𝑖c, integrating over c, and summing the

result over all species. This gives

𝜕𝑡 (𝜌v) + (1 − ℎ)∇x · (𝜌v ⊗ v) + (1 − ℎ)∇x · P + (1 − ℎ)∇x · K2 = 0

(4.40)

75

where P = (cid:205)𝑖 P𝑖 = 𝑛𝑇I. For convenience, we reformulate this expression in terms of the primitive

variable v by using Eq. Eq(4.39). We obtain

𝜌 D𝑡v + (1 − ℎ) ∇𝑥 · P − (1 − ℎ) v (∇𝑥 · K1) + (1 − ℎ) ∇𝑥 · K2 = 0.

(4.41)

Next we define 𝐽𝑖𝐾 and 𝐽𝐾

J𝐾 =

∑︁

𝑖

J𝑖𝐾 ≡

∑︁

𝑖

∇𝑥 · K2,𝑖 − v (cid:0)∇𝑥 · K1,𝑖(cid:1) ,

(4.42)

where the subscript 𝐾 marks the fact that these contributions come from the kinetic distributions

𝑓𝑖𝐾. With this definition, we can rewrite Eq. Eq(4.41) in a more compact form as

𝜌 D𝑡v + (1 − ℎ) ∇𝑥 · P + (1 − ℎ) J𝐾 = 0.

(4.43)

Finally, for the conservation of energy, we multiply Eq. (4.24) by

𝑚𝑖
2

𝐶2, integrate again over c, and

sum over all species. This gives

𝜕𝑡

(cid:19)

𝑛𝑇

(cid:18) 3
2

+ (1 − ℎ) ∇𝑥 ·

(cid:19)

𝑛𝑇v

(cid:18) 3
2

+ (1 − ℎ) P : ∇𝑥v + (1 − ℎ) 𝐻𝐾 = 0,

(4.44)

where 𝐻𝐾 gathers moments of 𝑓𝑖𝐾 and is defined by

𝐻𝐾 =

∑︁

𝑖

𝐻𝑖𝐾 ≡

∑︁

𝑖

(cid:68) 𝑚𝑖
2

𝐶2 (c · ∇𝑥 𝑓𝑖𝐾)

(cid:69)

.

(4.45)

Using the fact that 𝑓𝑖𝐹 is a Maxwellian, we have P : ∇𝑥v = 𝑛𝑇 (∇𝑥 · v) and q = 0. We can also

reformulate this in terms of the primitive variable 𝑇. Using Eq. (4.37), we obtain

D𝑡𝑇 + (1 − ℎ)

2
3

𝑇 (∇𝑥 · v) + (1 − ℎ)

2
3

1
𝑛

𝐻𝐾 − (1 − ℎ)

1
𝑛

𝑇 ∇𝑥 · ˜K1 = 0.

(4.46)

Equations (4.23), (4.36), (4.43), and (4.46) form the closed set of hybrid kinetic-Euler equations

that describe the flow dynamics in the buffer region.

4.3.2 Multi-species kinetic-Navier-Stokes system in the buffer region

As the Euler equations assume inviscid fluids, they are often not sufficient to describe physical

flows in the continuum region.

Instead, capturing near-equilibrium effects such as viscosity or

atomic interdiffusion requires a further extension of the hydrodynamics model to the Navier-Stokes

76

closure. To describe near-equilibrium effects in the hydrodynamic model, we therefore expand 𝑓𝑖𝐹

in terms of 𝜖

𝑓𝑖𝐹 = 𝑓 (0)

𝑖𝐹 + 𝜖 𝑓 (1)
𝑖𝐹 .

(4.47)

Note that the term 𝑓 (1)

𝑖𝐹 describes the deviation from equilibrium but is different from the kinetic

distribution 𝑓𝑖𝐾. Instead, this term captures the Navier-Stokes closure in terms of the moments
of 𝑓𝑖𝐹. In a pure fluid region where ℎ = 0, the term 𝑓 (1)

𝑖𝐹 is simply the standard Navier-Stokes

correction [22, 21]
(cid:34) (cid:18) 𝑚𝑖
2𝑇

𝑓 (1)
𝑖𝐹 = −

𝑀𝑖
𝜈𝑖

𝐶2 −

(cid:19)

5
2

C · ∇x log 𝑇 +

𝑛
𝑛𝑖

C · d𝑖 +

(cid:18)

𝑚𝑖
𝑇

C ⊗ C −

(cid:19)

1
3

I

: ∇xv +

𝜈𝑖 𝑗
𝜈𝑖

𝑚𝑖
𝑇 C · v(1)

𝑖 𝑗

(cid:35)

,

∑︁

𝑗

(4.48)

where 𝑀𝑖 is again the Maxwellian for species 𝑖 with moments 𝑛𝑖, v, 𝑇, and 𝜈𝑖 = (cid:205) 𝑗 𝜈𝑖 𝑗 . The diffusive
velocity correction v(1)
𝑖 𝑗

is the solution to a symmetric linear system. As in the Euler case, we cannot

use the standard Chapman-Enskog result due to the presence of the kinetic distributions 𝑓𝑖𝐾 and

must determine its effect on the eventual hydrodynamic equations. To simplify the calculation, we

rewrite Eq. Eq(4.24) in terms of the diffusion velocity C. The time and space derivatives are then

expressed as 𝜕𝑡 → 𝜕𝑡 − 𝜕𝑡v · ∇C and ∇𝑥 → ∇𝑥 − (∇𝑥 ⊗ v) · ∇C respectively [22], and Eq. Eq(4.24)

becomes:

𝜕𝑡 𝑓𝑖𝐹 − D𝑡 (v · ∇C 𝑓𝑖𝐹) + (1 − ℎ) (C + v) · ∇𝑥 𝑓𝑖𝐹 − (1 − ℎ) (C ⊗ ∇C 𝑓𝑖𝐹) : (∇𝑥 ⊗ v)

+(1 − ℎ) c · ∇𝑥 𝑓𝑖𝐾 =

(1 − ℎ)
𝜖

𝑄𝑖 [ 𝑓 ].

(4.49)

Note that we did not modify the term including 𝑓𝑖𝐾, as there is no benefit to do so. This term is

passively carried along during the Navier-Stokes derivation for 𝑓𝑖𝐹 2. As in the Euler case, we add

and subtract a collision term that contains only 𝑓𝑖𝐹 and rewrite the collision operator as

𝑄𝑖 [ 𝑓 ] = Δ𝑖 +

𝑄 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹],

∑︁

𝑗

Δ𝑖 =

∑︁

𝑗

Δ𝑖 𝑗 =

∑︁

𝑗

(cid:0)𝑄 [ 𝑓𝑖, 𝑓 𝑗 ] − 𝑄 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹](cid:1) .

(4.50)

(4.51)

2One can simply write c = C + v to use the same set of variables as 𝑓𝑖𝐹

77

Here, Δ𝑖 is the difference between the collision operator for the full distributions 𝑓𝑖 and the operator

for the fluid distributions 𝑓𝑖𝐹. We write 𝑄𝑖 𝑗 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹] ≡ 𝑄 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹] for simplicity. Assuming that

the hydrodynamic approximation applies in the buffer region means that Δ𝑖 is of the order 𝜖 and

any following expansion produces terms of the order 𝜖 2. We further assume (and will later show)

that inserting the expansion of Eq. Eq(4.47) into the first term of the collision operator yields

𝑄 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹] ≈ 𝑄 (0) [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹] + 𝜖𝑄 (1) [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹]

(4.52)

and thus

𝑄 [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹] ≈ 𝜖𝑄 (1) [ 𝑓𝑖𝐹, 𝑓 𝑗 𝐹].

(4.53)

𝑄 (1) and related terms are defined in 4A (see e.g., Eq. Eq(4A.17)). As in the Euler case, at leading

order in 𝜖, Eq.Eq(4.24) results in

∑︁

(cid:104)

𝑄𝑖 𝑗

𝑖𝐹 , 𝑓 (0)
𝑓 (0)

𝑗 𝐹

(cid:105)

= 0.

𝑗

(4.54)

By the same argument, the leading order terms in the expansion must be Maxwellians. At the next

order, we have

𝜕𝑡 𝑓 (0)

𝑖𝐹 − D𝑡

(cid:16)

v · ∇C 𝑓 (0)
𝑖𝐹

(cid:17)

+ (1 − ℎ) (C + v) · ∇𝑥 𝑓 (0)
𝑖𝐹

− (1 − ℎ) (C ⊗ ∇C 𝑓 (0)

𝑖𝐹 ) : (∇𝑥 ⊗ v) + (1 − ℎ) c · ∇𝑥 𝑓𝑖𝐾 = (1 − ℎ) 𝑄 (1)

𝑖

[ 𝑓 (1)

𝑖𝐹 ] + (1 − ℎ) Δ𝑖.

(4.55)

From here, the usual Chapman-Enskog workflow is to compute an expression for 𝑓 (1)
Eq(4.37), Eq(4.43), and Eq(4.46). Once 𝑓 (1)

𝑖𝐹 is known, we can determine the various Navier-

𝑖𝐹 via equations

Stokes correction terms, i.e. V𝑖, P, q. The calculations to expand 𝑄 in terms of 𝜖 and subsequently

78

determine 𝑓 (1)

𝑖𝐹 can be found in 4A. Here, we present the final result 3

∑︁

𝑓 (1)
𝑖

=

𝜈𝑖 𝑗
𝜈𝑖

𝑓 (0)
𝑖𝐹

𝑚𝑖
𝑇 C · v(1)
𝑖 𝑗 +

(cid:32) 1
2

𝑚𝑖𝐶2
𝑇

−

(cid:33) 𝑇 (1)
𝑖 𝑗
𝑇

3
2














(cid:18)

𝑗

−

−

(cid:20) 𝑚𝑖
𝑇

𝑓 (0)
𝑖𝐹
𝜈𝑖
𝑚𝑖
𝜌𝑇 C · J𝐾 +

C ⊗ C −

: (∇𝑥 ⊗ v) + C ·

(cid:19)

𝐶2I

1
3
𝑚𝑖𝐶2
3𝑇

(cid:18)

1 −

(cid:19) (cid:18) 1
𝑛𝑇

𝐻𝐾 −

3
2𝑛

∇𝑥 · ˜K1

−

(cid:18)(cid:18) 𝑚𝑖
2𝑇
(cid:19)

𝐶2 −

(cid:19)

5
2

∇𝑥 log(𝑇)

(cid:19)

+

𝑛
𝑛𝑖

C · d𝑖

1
𝑛𝑖

∇𝑥 · ˜K1,𝑖

(cid:21)

−

1
𝜈𝑖

c · ∇𝑥 𝑓𝑖𝐾 +

1
𝜈𝑖

Δ𝑖.

(4.56)

We derive the conservation of total mass by inserting Eq. Eq(4.47) into Eq. Eq(4.49) and performing

the usual integration. The result is,

𝜕𝑡 𝜌 + (1 − ℎ) ∇𝑥 · (𝜌v) + (1 − ℎ)∇x · (𝜌V𝑖) + (1 − ℎ) ∇𝑥 · K1 = 0.

(4.57)

The species diffusion velocities V𝑖 are obtained by solving the system of equations

𝐴V𝑖 = w − ⟨𝑚C Δ𝑖⟩ ,

with

𝐴𝑖 𝑗 =

− (cid:205) 𝑗




w𝑖 ≡ 𝑛𝑇d𝑖 −

𝜌𝑖 𝜌 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖 𝜈𝑖 𝑗 +𝜌 𝑗 𝜈 𝑗𝑖

− 𝜌𝑖 𝜌 𝑗 ,

if 𝑖 = 𝑗

,

− 𝜌𝑖 𝜌 𝑗 ,

if 𝑖 ≠ 𝑗

𝜌𝑖 𝜌 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖 𝜈𝑖 𝑗 +𝜌 𝑗 𝜈 𝑗𝑖
𝜌𝑖
𝜌 J𝐾 + J𝑖𝐾,

J𝑖𝐾 ≡ ∇𝑥 · K2,𝑖 − v (∇𝑥 · K1,𝑖).

(4.58)

(4.59)

(4.60)

(4.61)

For the conservation of momentum, we insert the expansion for 𝑓𝑖𝐹 from Eq. Eq(4.47) into the

Eq. Eq(4.49), multiply by 𝑚𝑖c, integrate over c, and sum over all species. The resulting equation is

𝜕𝑡 (𝜌v) + (1 − ℎ) ∇𝑥 ·

(cid:33)

(cid:32)

𝑝I + 𝜖 ∑︁

P(1)
𝑖

𝑖

+ (1 − ℎ) ∇𝑥 · (v ⊗ v𝜌) + (1 − ℎ) ∇𝑥 · K2 = 0

(4.62)

3This expression is not closed, as v(1)

𝑖 𝑗 and 𝑇 (1)
𝑖 𝑗

implicitly depend on 𝑓 (1)

𝑖𝐹 . See 4A.3 for the linear systems that

must be satisfied to fully define 𝑓 (1)
𝑖𝐹

79

with:

P(1)
𝑖 = −

(cid:18)

𝑛𝑖𝑇
𝜈𝑖

(∇𝑥 ⊗ v) + (∇𝑥 ⊗ v)T −

2
3

(∇𝑥 · v) I

(cid:19)

∑︁

+

𝑗

𝜈𝑖 𝑗
𝜈𝑖

𝑇 (1)
𝑖 𝑗 𝑛𝑖 I

(4.63)

(cid:18)

(cid:28)

𝑚𝑖

𝑠𝑖 I −

C ⊗ C −

+

1
𝜈𝑖

(cid:19)

𝐶2 I

1
3

c · ∇𝑥 𝑓𝑖𝐾

(cid:29)

+

1
𝜈𝑖

⟨𝑚𝑖 (C ⊗ C) Δ𝑖⟩ .

(4.64)

The cross-species temperature

𝑇 (1)
𝑖 𝑗 =

𝑛𝑖𝜈𝑖 𝑗𝑇 (1)

𝑖

+ 𝑛 𝑗 𝜈 𝑗𝑖𝑇 (1)

𝑗

𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖

,

is obtained by solving the system of equations

𝐵 T(1) = s − ⟨𝑚𝑖𝐶2Δ𝑖⟩ ,




𝑛𝑖𝑛 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝑛𝑖 𝜈𝑖 𝑗 +𝑛 𝑗 𝜈 𝑗𝑖

− (cid:205) 𝑗

𝐵𝑖 𝑗 =

𝑛𝑖𝑛 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝑛𝑖 𝜈𝑖 𝑗 +𝑛 𝑗 𝜈 𝑗𝑖

− 𝑛𝑖𝑛 𝑗 ,

if 𝑖 = 𝑗

.

− 𝑛𝑖𝑛 𝑗 ,

if 𝑖 ≠ 𝑗

with

𝑠𝑖 ≡ −

𝐻𝑖𝐾 ≡

1
2

(cid:17)

(cid:16) 𝑛𝑖
𝑛

𝐻 − 𝐻𝑖

(cid:16) 𝑛𝑖
2
𝑛
3
v2 ∇𝑥 · K1,𝑖 − v · (cid:0)∇𝑥 · K2,𝑖(cid:1) + ∇𝑥 · K3,𝑖.

∇𝑥 · ˜K1 − ∇𝑥 · ˜K1,𝑖

+ 𝑇

(4.65)

(4.66)

(4.67)

(4.68)

(cid:17)

Finally, for the conservation of energy, we insert Eq.Eq(4.47) into the Eq.Eq(4.49), multiply by

𝑚𝑖
2

𝐶2, integrate over c, and sum over all species. This gives,

3
2

𝜕𝑡 (𝑛𝑇) + (1 − ℎ)

3
2

∇𝑥 · (𝑛𝑇v) + (1 − ℎ) ∇𝑥 · q + (1 − ℎ) P : (∇𝑥 ⊗ v) + (1 − ℎ) 𝐻𝐾 = 0,

(4.69)

where q = 𝜖 (cid:205)𝑖 q𝑖 and

q𝑖 =

𝑇
𝑚𝑖

5
2

(cid:18)

𝜌𝑖v𝑖 +

1
𝜈𝑖

J𝑖 −

𝑛𝑖
𝜈𝑖

(cid:19)

−

∇𝑥𝑇

1
𝜈𝑖

(cid:68) 𝑚𝑖
2

𝐶2 C (c · ∇𝑥 𝑓𝑖𝐾)

(cid:69)

+

1
𝜈𝑖

(cid:68) 𝑚𝑖
2

𝐶2C Δ𝑖

(cid:69)

.

(4.70)

Equivalent to the Euler closure, Eqn. Eq(4.23), Eq(4.57), Eq(4.62), and Eq(4.69) form the coupled

kinetic and Navier-Stokes equations.

80

4.4 Example for the BGK-Navier-Stokes Coupling for Two Species

As an illustrative example and for potential future applications, we present the coupled BGK-NS

equations for a system of two particle species. The conservation equations of mass densities 𝜌1

and 𝜌2 are:

with

𝜕𝑡 𝜌1 + (1 − ℎ) ∇𝑥 · (𝜌1v) + (1 − ℎ) ∇𝑥 · (𝜌1V1) + (1 − ℎ) ∇𝑥 · K1,1 = 0,

𝜕𝑡 𝜌2 + (1 − ℎ) ∇𝑥 · (𝜌2v) + (1 − ℎ) ∇𝑥 · (𝜌2V2) + (1 − ℎ) ∇𝑥 · K1,2 = 0,

(4.71)

V1 = −

V2 =

𝜌2
𝜌1
1
𝜌2

𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖
1
𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌2
𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖
𝜈𝑖 𝑗 𝜈 𝑗𝑖

(w1 − ⟨𝑚1C Δ1⟩) +

(w2 − ⟨𝑚2C Δ2⟩)

(w1 − ⟨𝑚1C Δ1⟩) −

(w2 − ⟨𝑚2C Δ2⟩).

1
𝜌2

𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖
𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖
𝜈𝑖 𝑗 𝜈 𝑗𝑖

𝜌1
𝜌2

1
𝜌2

Inserting in values for w1 and w2 that are obtained by Eq. Eq(4.60), produces:

V1
(cid:169)
(cid:173)
(cid:173)
V2
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

= −𝐷 (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑛𝑇 (cid:169)
(cid:173)
(cid:173)
(cid:171)

d1

d2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

+ (cid:169)
(cid:173)
(cid:173)
(cid:171)

J1𝐾

J2𝐾

(cid:170)
(cid:174)
(cid:174)
(cid:172)

− (cid:169)
(cid:173)
(cid:173)
(cid:171)

⟨𝑚1C Δ1⟩

⟨𝑚2C Δ2⟩

,

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

where the diffusion coefficients are

𝐷 =

1
𝜌2

𝜌1𝜈12 + 𝜌2𝜈21
𝜈12𝜈21









𝜌2/𝜌1

−1

−1

𝜌1/𝜌2

.









(4.72)

(4.73)

(4.74)

The equation for the conservation of total momentum is

𝜕𝑡 (𝜌v) + (1 − ℎ) ∇𝑥 ·

(cid:16)

𝑝I + 𝜖P(1)
1

+ 𝜖P(1)
2

(cid:17)

+ (1 − ℎ) ∇𝑥 · (v ⊗ v𝜌) + (1 − ℎ) ∇𝑥 · K2 = 0,

(4.75)

where

P(1)
1 = −

(cid:18)

𝑛1𝑇
𝜈1

(∇𝑥 ⊗ v) + (∇𝑥 ⊗ v)𝑇 −

(cid:19)

(∇𝑥 · v) I

2
3

(cid:18)

(cid:28)

𝑚1

C ⊗ C −

(cid:19)

𝐶2I

1
3

c · ∇𝑥 𝑓1𝐾

(cid:29)

+

P(1)
2 = −

(∇𝑥 ⊗ v) + (∇𝑥 ⊗ v)𝑇 −

(cid:19)

(∇𝑥 · v) I

2
3

+

𝑠1I −
(cid:18)

1
𝜈1
𝑛2𝑇
𝜈2

+

1
𝜈2

𝑠2I −

(cid:18)

(cid:28)

𝑚2

C ⊗ C −

(cid:19)

𝐶2I

1
3

c · ∇𝑥 𝑓2𝐾

(cid:29)

+

81

𝜈11
𝜈1

𝑇 (1)
11

𝑛1 I +

𝜈12
𝜈1

𝑇 (1)
12

𝑛1 I

⟨𝑚𝑖 (C ⊗ C) Δ1⟩,

(4.76)

𝜈21
𝜈2

𝑇 (1)
21

𝑛2 I +

𝜈22
𝜈2

𝑇 (1)
22

𝑛2 I

⟨𝑚𝑖 (C ⊗ C) Δ2⟩,

(4.77)

+

1
𝜈𝑖

+

1
𝜈𝑖

and the temperature corrections 𝑇 (1)

1

and 𝑇 (1)

2

are given by

𝑛2
𝑛1𝜈12 + 𝑛2𝜈21
1
𝑇 (1)
1 = −
𝜈12𝜈21
𝑛1
𝑛2
𝑛1𝜈12 + 𝑛2𝜈21
1
𝜈12𝜈21
𝑛2

𝑇 (1)
2 =

(𝑠1 − ⟨𝑚1𝐶2Δ1⟩) +
𝑛1
𝑛2

(𝑠1 − ⟨𝑚1𝐶2Δ1⟩) −

𝑛1𝜈12 + 𝑛2𝜈21
1
𝜈12𝜈21
𝑛2
𝑛1𝜈12 + 𝑛2𝜈21
1
𝜈12𝜈21
𝑛2

(𝑠2 − ⟨𝑚2𝐶2Δ2⟩),

(𝑠2 − ⟨𝑚2𝐶2Δ2⟩).

Plugging in the values for 𝑠1 and 𝑠2 according to Eq. Eq(4.67) produces:

𝑇 (1)
(cid:169)
1
(cid:173)
(cid:173)
𝑇 (1)
2
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

= 𝐴 (cid:169)
𝑇
(cid:173)
(cid:173)
(cid:171)









∇𝑥 · 𝐾1,1

∇𝑥 · 𝐾1,2









−

2
3

+

𝐻1𝐾

𝐻2𝐾

























⟨𝑚𝐶2 Δ1⟩

⟨𝑚𝐶2 Δ2⟩









(cid:170)
(cid:174)
(cid:174)
(cid:172)

, 𝐴 =

1
𝑛2

𝑛1𝜈12 + 𝑛2𝜈21
𝜈12𝜈21

𝑛2/𝑛1

−1









Next, we solve for the the cross-species temperatures𝑇 (1)

12 and see that,

(4.78)

−1

.








(4.79)

𝑛1/𝑛2

12 = 𝑇 (1)
𝑇 (1)

21 =

𝑛1𝜈12𝑇 (1)
1

+ 𝑛2𝜈21𝑇 (1)
2

𝑛1𝜈12 + 𝑛2𝜈21
(cid:18) 𝜈12 − 𝜈21
𝜈21𝜈12

(cid:19)

=

𝑛1
𝑛2

(𝑠2 − ⟨𝑚2𝐶2Δ2⟩) +

𝑛2
𝑛2

(cid:19)

(cid:18) 𝜈21 − 𝜈12
𝜈21𝜈12

(𝑠1 − ⟨𝑚1𝐶2Δ1⟩).

(4.80)

Finally, the conservation of energy is:

𝜕𝑡 (𝑛𝑇) + (1 − ℎ)

3
2

3
2

∇𝑥 · (𝑛𝑇v) + (1 − ℎ) P : (∇𝑥 ⊗ v)

+ (1 − ℎ) ∇𝑥 · q + (1 − ℎ) (𝐻1𝐾 + 𝐻2𝐾) = 0,

(4.81)

where q = 𝜖 (q1 + q2), and

q𝑖 =

(cid:18)

𝑇
𝑚𝑖

5
2

𝜌𝑖V𝑖 +

1
𝜈𝑖

J𝑖𝐾 −

(cid:19)

−

∇𝑥𝑇

𝑛𝑖
𝜈𝑖

1
𝜈𝑖

(cid:68) 𝑚𝑖
2

𝐶2C(c · ∇𝑥 𝑓𝑖𝐾)

(cid:69)

+

1
𝜈𝑖

⟨

𝑚𝑖
2

𝐶2C Δ𝑖⟩,

(4.82)

J𝑖𝐾 = ∇𝑥 · K2,𝑖 − v (∇𝑥 · K1,𝑖).

In addition to the above equations, we also have to solve:

𝜕𝑡 𝑓1𝐾 + ℎc · ∇𝑥 𝑓1𝐾 + ℎc · ∇𝑥 𝑓1𝐹 = 𝜖 −1ℎ 𝑄1 [ 𝑓 ],

𝜕𝑡 𝑓2𝐾 + ℎc · ∇𝑥 𝑓2𝐾 + ℎc · ∇𝑥 𝑓2𝐹 = 𝜖 −1ℎ 𝑄2 [ 𝑓 ].

(4.83)

(4.84)

(4.85)

Where 𝑓𝑖𝐹 = 𝑓 (0)

𝑖

or 𝑓𝑖𝐹 = 𝑓 (0)

𝑖

+ 𝜖 𝑓 (1)
𝑖

, depending on whether we are using the Euler or Navier-

Stokes closure, respectively.

82

4.5 Numerical Results

4.5.1 Sod Shock Simulation

We begin the verification of our hybrid model with a simple 1D-1V single-species Riemann

problem known as the Sod shock. This dimensionless test is defined by its left (L) and right

(R) initial conditions for mass density, material velocity, and pressure: 𝜌𝐿, 𝑢𝐿, 𝑝 𝐿 = [1, 0, 1] and

𝜌𝑅, 𝑢𝑅, 𝑝𝑅 = [0.125, 0.0, 0.1], respectively. Our hybrid approach combines a single species BGK

model with no external field (a = 0) and its Euler limit. The Euler model is implemented with a

local Lax-Friedrichs scheme and the single species BGK model is implemented with an operator

splitting approach, which disentangles the collisions from the phase space advection. We use a

first order upwind stencil for phase space advection and a Crank-Nicholson stencil for the BGK

collision terms. Because Crank-Nicholson is unconditionally stable, we set our time step based on

the advection CFL condition. The McBGK model is directly derived from and tested against the

MultiBGK code from [23].

We define the collision frequency for BGK using simple hard sphere scattering with

𝜈 = 𝜋𝑅2𝑛

√︂

2𝑘 𝑏𝑇
𝑚

.

(4.86)

The non-dimensional parameter 𝑅 is analogous to the radius of the hard spheres and allows us to

tune the collision frequency in order to capture both kinetic and hydrodynamic regimes. Figure 4.2

shows a snapshot of the density and velocity at time 𝑡 = 1 as obtained by the kinetic model for

different values of 𝑅 in comparison to the continuum solution. In general, the BGK calculation

converges to the continuum result with increasing 𝑅. For 𝑅 = 1, the collisionality is small enough

that the BGK model produces diffusion. For 𝑅 = 25 there is clear separation into rarefaction wave,

contact discontinuity, and shock that are found in the Euler calculation. The simulation with 𝑅 = 5

lies between the results with 𝑅 = 1 and 𝑅 = 25.

In the context of the Chapman-Enskog expansion, the non-dimensional Knudsen number can

be defined as the ratio of the mean free path between collisions and a macroscopic reference scale.

This definition quantifies the deviation of the distribution function 𝑓 from the Maxwellian and

83

Figure 4.2 Snapshots of the density (left) and velocity (right) at time t = 1 as obtained by the kinetic
model for different values of R. We increase collisionality by increasing the non-dimensional hard
sphere radius 𝑅. As 𝑅 increases, we recover the fluid limit. We emphasize that the Euler simulation
matches the Sod analytic solution and that the BGK simulation converges to the Sod analytic
solution in the large R (i.e. hydrodynamic) limit.

can be related to the expansion parameter 𝜖 in the Navier-Stokes closure. However, due to the

ambiguity of choosing a macroscopic scale, we instead define two effective Knudsen numbers to

quantify the deviation from a Maxwellian. We are able to do this since we have access to the

underlying distribution functions as part of our hybrid model. The effective Knudsen numbers are

defined as

𝐾𝑛1 =

∫

1
𝑛

𝐾𝑛2 = 1 −

𝑑𝑐 |M [ 𝑓 ] − 𝑓 |,

⟨𝑣4⟩
3⟨𝑣2⟩2

,

(4.87)

(4.88)

where M [ 𝑓 ] is the Maxwellian in Eq. Eq(4.2) that is associated with the moments of 𝑓 . The value

of 𝐾𝑛1 quantifies the absolute deviation of 𝑓 in phase space from its associated Maxwellian, while
𝐾𝑛2 characterizes the deviation of the distribution function’s 4𝑡ℎ moment from the one predicted by
𝑀 [24]. In both cases, a value of zero indicates that 𝑓 is a Maxwellian distribution. In Figure 4.3,

we plot the local values of 𝐾𝑛1 and 𝐾𝑛2 for the Sod problem and different BGK collisionalities.

As expected, both Knudsen numbers generally decrease for larger collision frequencies. Note that

at the shock front and the contact discontinuity, 𝐾𝑛1 and 𝐾𝑛2 show persistent peaks. This signals

that kinetic behavior is important around such sharp features in the Euler solution. Furthermore the

left side of the domain, which contains the fluid and buffer regions, has relatively small Knudsen

numbers which shows that this is a good choice for their placement.

84

(a) Knudsen number 𝐾𝑛1 according to Eq. Eq(4.87) (b) Knudsen number 𝐾𝑛2 according to Eq. Eq(4.88)

Figure 4.3 Estimates of the deviation from equilibrium for different collisionalities via the Knudsen
number definitions in Eqn. Eq(4.87) and Eq(4.88), varied via the nondimensional particle radius 𝑅,
for the Sod problem. For reference, we also plot the (scaled) density profile of the Euler solution.
Left: effective Knudsen number via integrated deviations from Maxwellian, see Eq. Eq(4.87).
Right: effective Knudsen number via moment ratio, see Eq. Eq(4.88). Both models for the
effective Knudsen number show that the deviation from a Maxwellian is greatest near the the shock.

In Figure 4.4, we plot the hybrid model which couples kinetic dynamics with 𝑅 = 1 to its

associated Euler model. In the left pane we show, if the buffer region is correctly placed then the

hybrid model produces an equivalent result as the BGK model without using kinetic dynamics over

the entire physical domain. Here ‘correctly placed’ means that the buffer region is placed where the

Knudsen number is near zero. In other words, where the hydrodynamic description is valid. In the

right pane we show, if the buffer region is incorrectly placed in a part of the domain where kinetic

effects are still important, then we suppress the kinetic phenomena. Here ‘incorrectly placed’

means that the kinetic dynamics are NOT used where the Knudsen number was shown to be large.

In effect, we have assumed that the continuum approximation applies where it actually does not.

As a consequence, even in the buffer region, the hybrid model largely resembles the hydrodynamic

density pattern. Only at around 𝑥 = 6.5 it finally begins to reproduce the density curve of the BGK

calculation. This demonstrates that the buffer region should be placed sufficiently far away from

regions with kinetic phenomena such as shock fronts and material interfaces. Of course, we want

to place the buffer region to define the smallest kinetic region possible. Additionally, the kinetic

and hydro models are run in the buffer region it more expensive than either. In our study of buffer

width, we recovered nothing substantively different than what is presented in Degond et al.’s study

[1] of the 1-D SOD shock solution.

85

Figure 4.4 Sod problem with 𝑅 = 1. Left: the kinetic-hydro buffer region is placed in the space
interval [1, 3], i.e. away from the shock. Right: the buffer region is placed in [5, 7], i.e., around
the shock. Since the coupled model is derived for a system near equilibrium, this assumption is
imprinted in the buffer region. While in the left plot, the coupled model correctly follows the
kinetic and continuum solutions where appropriate, in the right plot, it tracks the hydrodynamic
solution and only transitions to the reference kinetic solution near the edge of the buffer region.

However, it is important to remember that here we zoom into the shock area. In a real physics

setup that is targeted by the kinetic-hydro approach, such a shock would occupy a relatively small

region with the hydrodynamic regime dominating the simulation volume.

4.5.2 MARBLE pore preheat problem

To demonstrate the multi-species modeling capabilities of our hybrid approach we simulate an

experimental setup that was designed to measure the amount of mix in ICF-type implosions. The

MARBLE campaign sought to quantify mixing by using separated reactants in ICF capsules and

examining the thermonuclear burn output [17].

Instead of a typical capsule, MARBLE used a

deuterated plastic foam studded with pores. The pores were filled with a gas containing tritium (see

Figure 4.5). In this chapter, we are focusing the qualitative macro pore collapse; therefore we only

consider the HT gas in macro pores and do not consider HT gas in any CD micro-pores. While we

do not expect the additional tritium in the micropores to significantly affect the macropore dynamics

studied here, one would need to include this in a larger capsule simulation that estimates the total

yield. Since the deuterium and tritium start separated, then DD vs DT yield is a diagnostic tool for

the amount of mix. The initial experiments found that varying the pore sizes showed little of the

expected effect. It is proposed that the capsule’s preheat phase left the separated reactants more

mixed than predicted by simulations. These simulations were done using hydrodynamic codes

86

Table 4.1 Material location, number densities, densities, and ionization levels in the MARBLE
preheat problem.

Material
Deuterium
Carbon
Hydrogen
Tritium

Location Number density [1/cc]

Foam
Foam
Pore
Pore

3.61 × 1021
3.61 × 1021
8.80 × 1020
1.15 × 1020

density [1/cc]
1.20 × 10−2
7.22 × 10−2
1.47 × 10−3
5.77 × 10−4

Ionization Level
full
4 electrons ionized
full
full

which only considered mixing from fluid instabilities or turbulence. Since these results have been

released, others have shown that non-equilibrium effects can play a role in the mix morphology of

the D and T ions in the preheat stages of the experiment. In particular, preheat mixing of the pores

may have a strong effect on the DT fusion yield [18, 25, 26]. This emphasizes the need to include

kinetic dynamics in the simulations.

With that, we simulate a simplified model of a single MARBLE pore using the hybrid method

in 1D and planar geometry. We note that this choice of dimensionality will suppress certain kinds

of hydrodynamic instabilities and emphasize the atomic mixing. However, it is a reasonable initial

demonstration of our coupled method. Follow-up studies may explore simulation setups in 2D

or 3D along with more precise computational cost studies. Here, we assume that the material is

initially at rest with a 50 eV background temperature, which is a reasonable facsimile of the preheat

conditions before shock arrival. The initial number densities and densities for the carbon-deuterium

(CD) foam and hydrogen-tritium (HT) pore gas are given in Table 4.1. The background preheating

will expand the foam thereby compressing the gas in the pore. With that, atomic mixing is likely

to occur at the foam/gas interface. Based on the findings in the Sod test, we expect that kinetic

dynamics will be necessary to model the interfaces during pore compression. A transition function

ℎ(𝑥) and its associated hydrodynamic and kinetic regions are plotted in Figure 4.6. This transition

function places the pore and the interfaces within the kinetic region and places the buffer regions in

the foam which are expected to be hydrodynamic. Due to the high temperature of the experiment,

we include electric fields via the amipolar approximation in both the kinetic and fluid models. For

87

Figure 4.5 Left: Illustration of the Marble-type foam which is studied in this chapter. Orange
represents the CD foam while the purple disks represent the macro-pores that are filled with TH
gas. Note that the CD foam also contains many smaller micro-pores; for the purpose of this study
we consider the foam region to be a homogenous CD material. Right: Initial densities used in
the 1D planar MARBLE pore preheat problem (see Sec. (4.5.2)). We simulate a 200𝜇𝑚 slice of
carbon-deuterium foam with a 20𝜇𝑚 hydrogen-tritium pore located at the center.

Figure 4.6 Left: Transition function ℎ(𝑥) for the MARBLE pore preheat problem (Sec. 4.5.2). The
edges of the buffer regions are indicated by the dashed lines. Right: Material densities together
with the edges of the buffer regions. Due to the sharp interfaces which define the pore, we expect
kinetic effects to be important around the pore. To ensure our assumption that the buffer region is
placed in a hydrodynamic region is satisfied, we place the buffer edges away from where we expect
shock fronts (i.e in the foam).

the Euler model, this results in an additional term in the momentum equation:

𝜕𝑡 (𝜌𝑢) = ∇

(cid:16)

𝜌𝑢2 + (𝑛 + 𝑛𝑒) 𝑘 𝑏𝑇

(cid:17)

.

For the kinetic equations we include a Vlasov term:

a =

𝑒𝐸
𝑚

= −

1
𝑚 𝑛𝑒

∇ 𝑛𝑒𝑇 .

(4.89)

(4.90)

We focus on the simulation results for deuterium and tritium. Other species are still present,

but we plot the two species of greatest interest for clarity of presentation We start with Figure 4.7,

which shows the full time evolution of the deuterium and tritium densities as a heat map, simulated

88

Figure 4.7 Evolution of the material densities as compared across models for the MARBLE pore
preheat problem (Sec. 4.5.2). Each heat map has time evolution on vertical axis and spatial position
on the horizontal axis. In the hybrid simulation, we visualize the buffer region as dashed lines.
Note that initial conditions plotted in Figure 4.6 Right are a cross section from the heat map at time
0ps. The top row shows the deuterium in the foam while the bottom row gives the evolution of the
tritium densities. The left to right columns correspond to the BGK, hybrid, and Euler methods.
For every model, these heat maps show that the pore is compressed for approximately 150ps as
the foam/pore expands. For the Euler model the compression is not nearly as great as the others.
Eventually the pore is compressed enough to trigger a rarefaction wave. This rarefaction wave
looks sharpest for the BGK and hybrid model because the compression of the pore was greatest.

with the BGK, hybrid, and Euler methods. This figure, along with the equivalent figure plotting the

hydrodynamic velocities, Figure 4.9, are the major results for the MARBLE pore preheat example

problem. Similar to Figure 4.4 for the Sod shock, these plots demonstrate that the hybrid model can

replicate a full kinetic simulation without needing kinetic dynamics in all physical regions of the

simulation. The remainder of the section is dedicated to exploring and understanding the dynamics

which kinetic models include.

In the first few picoseconds of the simulation, there are very large electric fields at the pore

interface, which rapidly diminish greatly by 10 picoseconds. This is shown in Figure 4.8. Such

behavior matches results from atomistic interface simulations, as presented by Stanton et al. [10].

Depending on the model (i.e. BGK, hybrid, Euler) the electric field has different effects on the

89

Figure 4.8 Electric field profiles at early times in the MARBLE pore preheat problem. The electric
fields evolution for the kinetic model (BGK) and Euler model are indistinguishable so only BGK
is presented here. The strong electric fields accelerate positively charged HT ions The constituents
of the plasma quickly redistribute to diminish the electric fields. (i.e. less than 10 ps)

velocity profiles. As with the density evolution, the velocity profiles for kinetic and hybrid models

agree. Furthermore, they have significant differences with the continuum calculation. To see this,

examine the evolution of the velocity profiles for 𝐷 and 𝑇 across models in Figure 4.9. The figure

shows that the electric fields in the BGK and hybrid models accelerate some tritium (and hydrogen)

particles to large velocities and eject them into the foam. This separate, electric field induced

mixing is possible because the BGK and hybrid models allow each species to have individualized

velocity fields around the pore region, while the Euler model enforces a common bulk velocity

among all species over the entire physical domain. Two important conclusions arise. First, since

the tritium velocity profile differs from the mixture bulk velocity, which is used as the reference

equlibrium velocity assumed in the hydroynamic approximation; this these ejected particles are not

an equilibrium phenomenon. Second, because the Euler solution’s bulk velocity is not affected by

the electric field, we can conclude that the proportion of the mass density comprised by the TH

ejecta is negligible.

Next, we investigate how the ejecta, which violate the assumptions made when placing the

buffer regions, did not corrupt the simulation. In Figure 4.10, we plot 𝐾𝑛1 for the four species in

the simulation at 20 ps. This is 10 ps after the electric field has reduced and the pore ejecta are

traveling through the foam. We overlay the species density and see that a negligible amount of

hydrogen and tritium were ejected from the pore into the foam. In Figure 4.11, we plot the evolution

of tritium’s effective Knudsen number 𝐾𝑛1 throughout the simulation as a heat map. First, notice

90

Figure 4.9 Time evolution of the velocity profiles in the MARBLE pore preheat problem. The top
row shows the velocity in the deuterium foam material, while the bottom row gives the velocities
in the pore tritium. The left to right columns correspond to the BGK, hybrid, and Euler solutions.
Since the hydrodynamic model assumes a single velocity, the deuterium and tritium are both
In the kinetic and hybrid models, however, the tritium distribution shows an
propelled inward.
additional velocity jet at early times which corresponds to a very small amount of tritium ejected
from the pore by the electric field at a large velocity. Furthermore, tritium ions show a non-zero
velocity field beyond the edges of the pore (90𝜇m - 100𝜇m). This is due to the few ejected ions
which have been sprinkled throughout the CD foam. Note that for display purposes, the maximum
velocity in the color map is set to ±80 km/s to ensure a representative color map on the region of
interest; the velocities in the ejection ’plumes’ typically exceed ±200 km/s.

that the tritium Knudsen plot from Figure 4.10 (Bottom Right) is a horizontal cross section of the

BGK evolution in Figure 4.11 (Left) at time 20ps. Second, notice for the the BGK evolution in

Figure 4.11 (Left) that for all times the high Knudsen numbers exist in a region where number

density is less than 1 particle per cc. Therefore, even though the ejecta have a high Knudsen number

they are rarefied enough to not corrupt our assumption that the foam is hydrodynamic. Additionally,

notice in the hybrid evolution in Figure 4.11 (Center) that the choice of the buffer location can

impact the hybrid solution when compared to the reference kinetic model. This matches what

was seen in the Sod shock, in Figure 4.4. Indeed, the ejecta are forcibly equilibrated to the local

Maxwellian of the mixture, and any deviation from mixture velocity and temperature is lost (i.e.

ejecta are brought to a halt and never enter the hydrodynamic region). As with BGK, the curve

91

Figure 4.10 Comparison of effective Knudsen number Eq(4.87) across species for the MARBLE
pore preheat problem, at time 250ps. The (rescaled) density profile is shown in the background
for reference. As expected, the effective Knudsen number is larger in carbon and deuterium where
there rapid changes in density, i.e., where the gradient scale length is small. Additionally, the tritium
and hydrogen ejections produce a large Knudsen number which travels into the buffer region (i.e.
[50, 70] and [130, 150]). However the associated densities are negligible. Thus, the hydrodynamic
model does not cause the hybrid results to differ from the BGK results.

for 1 particle per cc indicates that the number density is again small enough to find this spurious

halting irrelevant.

Now consider the rarefaction wave which exists in the time range of 100 picoseconds and beyond

in this section’s main result, Figure 4.7. Right around peak compression, at approximately 150ps,

a rarefaction wave forms in all models. The propagation of this rarefaction wave constitutes the

remainder of the simulation. For the following discussion of this rarefaction wave it is important

to notice that the BGK and hybrid model lead to a stronger compression of the pore material.

This is visualized by the brighter, thinner width of the pore. In Figure 4.12 we plot the density

profiles (i.e. cross sections from Figure 4.7) at 250ps, 375ps, and 500ps. These cross sections

emphasize the form of the rarefaction wavefront as it propagates through the deuterium. As well

as, the expansion of the tritium as the rarefaction wave propagates. For 250ps and 375ps, kinetic

and hybrid models share the same form of the wavefront, but differ from the Euler model. It is

92

Figure 4.11 Plots of the effective Knudsen number 𝐾𝑛1 for BGK, hybrid, and Euler solutions of the
Marble preheat problem. For reference we have added a solid white contour which marks where the
material number density is less that 1 particle per cc. The minuscule amount of particles ejected by
the strong interface electric fields carry high Knudsen values as they propagate through the foam
until they collide with each other on the periodic boundary conditions. As can be seen by 𝐾𝑛1 in
the buffer region for the hybrid method, the transition to Euler in the buffer region suppresses the
high speed, uncollided ejected particles from penetrating further into the foam. In both cases, the
contour indicates that the high Knudsen values occur where a negligible amount of particles exist.
Thus, the Knudsen number of the ejecta does not corrupt our hydrodynamic assumptions.

the higher compression attained in the kinetic and hybrid models which leads them to have sharper

rarefaction wave fronts than the Euler model. In the deuterium cross section at 500ps, we see that

rarefaction has propagated into the hydrodynamic region. At this point, the BGK and hybrid model

differ from each other. This behavior indicates that the hybrid model differs from BGK because our

assumptions that 𝑓 can be approximated by a Maxwellian has broken down and we are artificially

suppressing kinetic phenomena. This behavior was studied in the Sod problem in Figure 4.4. As

before, the Eulerian contribution to the coupled solution is forcing the wave front to be less diffusive

(i.e. more sharp). This is an important sanity check.

Note that the Euler equations, as defined by the zeroth order Chapman-Enskog expansion,

require all species share same bulk velocity and bulk temperature. Allowing one or more species

to deviate from this would violate this definition of the Euler equation. If we violate the definition

of Euler then this flexibility would be possible, but the scheme would not be consistent. We believe

this species separation along with an adaptive buffer region are both needed for large scale projects,

but creating this separation is beyond the scope of this dissertation and will be the subject of future

work.

93

Figure 4.12 Each plot contains BGK, hybrid, and Euler density curves for a different species and
time. The rows separate deuterium (top) and tritium (bottom) density profiles. From left to right, the
plots are 250ps, 375ps, and 500ps. The top row illustrates the propagation of the rarefaction wave
through the foam. The bottom row illustrates the expansion of the pore after peak compression.
The rarefaction wave enters the buffer zone at around 500ps. We can see that a difference emerges
between the coupled model and BGK after that time. The difference indicates that our assumption
that tritium’s distribution function is at local equilibrium around the wavefront is incorrect and we
are therefore artificially suppressing kinetic dynamics.

4.6 Summary and outlook

We have presented a hybrid model for coupling the multispecies Bhatnagar-Gross-Krook (BGK)

kinetic model with their limiting Euler and Navier-Stokes hydrodynamic equations. The hybrid

model is not merely a weighted average of independent kinetic and hydrodynamic calculations

with the same initial conditions. Rather, the hybrid model is the simultaneous evolution of the

two models coupled together. Our technique uses a buffer region to impose meaningful boundary

conditions when modeling the transition from a kinetic into a fluid region, generalizing the single

species approach introduced by Degond et al. (see e.g. [1]). In the buffer region, both the kinetic

and hydrodynamic equations are solved simultaneously, and the solution to the hybrid model is a

weighted sum of the solutions to the coupled models. The smooth transition between models avoids

the need to find direct interface conditions, which can introduce unphysical effects in hydrodynamic

94

regions. Kinetic models, while expensive due to their dimensionality, are able to capture important

multispecies physics effects such as velocity and temperature separation. This hybrid method

allows one to localize the use of a high dimensional kinetic model only where it is needed, therefore

maximizing the computational efficiency. We validated our model with simple Sod shock problem

example and then applied the method to study the effect of kinetic multispecies mixing in the

preheat phase of a high energy-density physics experiment. We demonstrated that if the buffer

regions are placed correctly the hybrid model can produce kinetic simulations without needing

kinetic dynamics over the entire physical domain.

One advantage of this approach is that a direct mesh decomposition is not needed; especially in

higher spatial dimensions, the transition function can take care of any potentially geometric features

of the interface. Similarly the transition function could be modified to take dynamically evolution

of the kinetic region into account; see [2]. This will be the subject of future work.

Acknowledgement

Thomas Chuna would like to acknowledge F.D.C. Willard for insightful discussions and proof

reading. This work was supported by the U.S. Department of Energy through the Los Alamos

National Laboratory. Los Alamos National Laboratory is operated by Triad National Security,

LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract

No. 89233218CNA000001). Research presented in this article/presentation/report was supported

by the Laboratory Directed Research and Development program of Los Alamos National Laboratory

under project number 20190005DR.

95

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98

APPENDIX 4A

COMPUTING THE FLUID CORRECTION 𝑓 (1)

𝑖𝐹 IN BUFFER REGION

For the convenience of the reader we group the definitions which will be frequently used in this

section:

𝑛𝑖𝑇 ≡

3
2

(cid:68) 𝑚𝑖
2

(cid:69)

𝐶2 𝑓𝑖𝐹

K1,𝑖 ≡ ⟨𝑚𝑖 c 𝑓𝑖𝐾⟩ , K2,𝑖 ≡ ⟨𝑚𝑖 (c ⊗ c) 𝑓𝑖𝐾⟩ , K3 ≡

, ∇𝑥 · q𝑖 ≡ ∇𝑥 ·

(cid:68) 𝑚𝑖
2

𝐶2C 𝑓𝑖𝐹

(cid:69)

,

(cid:68) 𝑚𝑖
2
∇𝑥 · K2,𝑖 − v (∇𝑥 · K1,𝑖),

c 𝑐2 𝑓𝑖𝐾

(cid:69)

,

J𝐾 ≡

∑︁

J𝑖𝐾 ≡

∑︁

𝑖

𝐻𝐾 ≡

𝑖
∑︁

𝑖

𝐻𝑖𝐾 ≡

∑︁

𝑖

(cid:68) 𝑚𝑖
2

𝐶2 (c · ∇𝑥 𝑓𝑖𝐾)

(cid:69)

,

(4A.1)

(4A.2)

(4A.3)

(4A.4)

We start from Eq. 4.55, rewritten here for convenience.

𝜕𝑡 𝑓 (0)

𝑖𝐹 − D𝑡

(cid:16)

v · ∇C 𝑓 (0)
𝑖𝐹

(cid:17)

+ (1 − ℎ)(C + v) · ∇𝑥 𝑓 (0)
𝑖𝐹

− (1 − ℎ)(C ⊗ ∇C 𝑓 (0)

𝑖𝐹 ) : (∇𝑥 ⊗ v) + (1 − ℎ) c · ∇𝑥 𝑓𝑖𝐾 = (1 − ℎ)𝑄 (1)

𝑖

[ 𝑓𝑖𝐹] + (1 − ℎ)Δ𝑖.

(4A.5)

4A.1 Computing the 𝑓 (0)

𝑖𝐹 terms

We begin by focusing on reformulating the left hand side of Eq(4A.5) in terms of 𝑛𝑖, v, 𝑇, and

𝑓𝑖𝐾. To simplify many derivative calculations, we rewrite this expression using 𝜕𝑡 ln 𝑓 = 1/ 𝑓 𝜕𝑡 𝑓

(cid:34)

𝑓 (0)
𝑖𝐹

𝜕𝑡 log( 𝑓 (0)

𝑖𝐹 ) − D𝑡v · ∇c log( 𝑓 (0)
𝑖𝐹 )

+ (1 − ℎ)

(cid:16)

(C + v) · ∇𝑥 log( 𝑓 (0)

𝑖𝐹 ) − (C ⊗ ∇C log( 𝑓 (0)

𝑖𝐹 )) : (∇𝑥 ⊗ v)

(cid:35)

(cid:17)

+ (1 − ℎ) c · ∇𝑥 𝑓𝑖𝐾 = (1 − ℎ)𝑄 (1)

𝑖

[ 𝑓𝑖𝐹] + (1 − ℎ)Δ𝑖.

(4A.6)

99

(cid:21)

+ c · ∇𝑥 𝑓𝑖𝐾 − Δ𝑖,

(4A.7)

(4A.8)

Note that 𝑓 (0)

𝑖𝐹 is the Maxwellian, so logarithmic properties can used to avoid complicated chain

rules. With the aid of Eqs. (4.37), (4.43), and (4.46), the above equation becomes,

𝑄 (1)
𝑖

[ 𝑓𝑖𝐹] = 𝑓 (0)
𝑖𝐹

(cid:20) 𝑚𝑖
𝑇

(C ⊗ C −

C2I) : (∇𝑥 ⊗ v) + C ·

(cid:19)

5
2

∇𝑥 log(𝑇)

(cid:19)

+

𝑛
𝑛𝑖

C · d𝑖

𝑚𝑖
𝜌𝑇 C · J𝐾 +

−

(cid:18)

1 −

(cid:19) (cid:18) 1
𝑛𝑇

𝐻𝐾 −

3
2𝑛

∇𝑥 · ˜K1

∇ · ˜K1,𝑖

(cid:18)(cid:18) 𝑚𝑖

2𝑇 C2 −
(cid:19)
1
𝑛𝑖

−

1
3
𝑚𝑖C2
3𝑇

where

d𝑖 =

𝑛𝑖
𝑛

∇𝑥 log( 𝑝𝑖) −

𝜌𝑖
𝜌

∇𝑥 log( 𝑝)

is the diffusion driving force and 𝑝𝑖 = 𝑛𝑖𝑇. The first line contains the typical Chapman-Enskog

terms from [21], and the second line contains the new terms arising from the presence of 𝑓𝑖𝐾.

4A.2 Computation of 𝑓 (1)

Although the derivation of the correction to the BGK collision operator in the first part of this

section has been presented in earlier works [21], we repeat it here for completeness. We want to
find 𝑄 (1)

𝑖 𝑗 . For this we first write the bulk velocity as,

𝑖 = (cid:205) 𝑗 𝑄 (1)

𝑛𝑖v𝑖 = ⟨c 𝑓𝑖⟩ = ⟨c 𝑓 (0)

𝑖

⟩ + 𝜖 ⟨c 𝑓 (1)

𝑖

⟩ = ⟨(C + v) 𝑓 (0)

𝑖

⟩ + 𝜖 ⟨c 𝑓 (1)

𝑖

⟩ = 𝑛𝑖v + 𝜖 ⟨c 𝑓 (1)

𝑖

⟩ = 𝑛𝑖v + 𝜖 𝑛𝑖v(1)

𝑖

.

and note that v𝑖 = v + 𝜖 v(1)

𝑖

. Similarly, for the temperature:

+ 𝜖 𝑓 (1)
𝑖

(cid:69)

)

=

(C + v − v𝑖)2 𝑓 (0)

𝑖

𝑛𝑖𝑇𝑖 =

3
2

=

=

(cid:68) 𝑚𝑖
2
(cid:68) 𝑚𝑖
2
𝑛𝑖𝑇 +

3
2

𝑖

(c − v𝑖)2( 𝑓 (0)
𝐶2 + (v − v𝑖)2(cid:17)

(cid:16)

(v − v𝑖)2𝜌𝑖 +

1
2

3
2

(cid:68) 𝑚𝑖
2
(cid:68) 𝑚𝑖
3

𝜖

𝑓 (0)
𝑖

(cid:69)

+

3
2
𝜖𝑛𝑖𝑇 (1)
𝑖

(c − v𝑖)2 𝑓 (1)

𝑖

(cid:69)

(4A.9)

(cid:69)

+ 𝜖

(cid:68) 𝑚𝑖
2

(c − v𝑖)2 𝑓 (1)

𝑖

(cid:69)

(4A.10)

we use that v𝑖 − v = 𝜖v(1)

𝑖

so that 𝑇𝑖 = 𝑇 + 𝜖 𝑇 (1)

𝑖

. One may want to include 𝑓𝑖𝐾 since it is at order 𝜖 in

the buffer region. However, the collision operator solely depends on 𝑓𝑖𝐹 here. Thus the definitions

100

of v𝑖 𝑗 and 𝑇𝑖 𝑗 are defined as in [21]:

v𝑖 𝑗 =

𝑇𝑖 𝑗 =

𝜌𝑖𝜈𝑖 𝑗 v𝑖 + 𝜌 𝑗 𝜈 𝑗𝑖v 𝑗
𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖
𝑛𝑖𝜈𝑖 𝑗𝑇𝑖 + 𝑛 𝑗 𝜈 𝑗𝑖𝑇𝑗
𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖

= v + 𝜖

𝜌𝑖𝜈𝑖 𝑗 v(1)

𝑖 + 𝜌 𝑗 𝜈 𝑗𝑖v(1)
𝑗

𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖

= v + 𝜖 v(1)
𝑖 𝑗

𝜌𝑖𝜈𝑖 𝑗 (v2

𝑖 𝑗 ) + 𝜌 𝑗 𝜈 𝑗𝑖 (v2

𝑖 − v2
3(𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖)

𝑗 − v2
𝑗𝑖)

+

= 𝑇 + 𝜖 𝑇 (1)
𝑖 𝑗

.

Next, we will apply these terms in the expansion of the multi-species Maxwellian:

𝑀𝑖 𝑗 = 𝑀𝑖 𝑗

(cid:12)
(cid:12)𝜖=0

+ 𝜖

(cid:18) 𝜕
𝜕𝜖

(cid:19)

𝑀𝑖 𝑗

𝜖=0

= 𝑓 (0)
𝑖

+ 𝜖

(cid:18) 𝜕
𝜕𝜖

(cid:19)

𝑀𝑖 𝑗

𝜖=0

with [21]

so that

𝑀𝑖 𝑗 = 𝑛𝑖

(cid:19) 3/2

(cid:18) 𝑚𝑖
2𝜋𝑇𝑖 𝑗

(cid:32)

exp

−

(cid:33)

𝑚𝑖 (cid:0)c − v𝑖 𝑗 (cid:1) 2
2𝑇𝑖 𝑗

𝑚𝑖
(cid:16)
𝑇 + 𝜖𝑇 (1)
𝑖 𝑗

(cid:17)

2𝜋

= 𝑛𝑖

(cid:169)
(cid:173)
(cid:173)
(cid:171)

3/2

(cid:170)
(cid:174)
(cid:174)
(cid:172)

−

exp (cid:169)
(cid:173)
(cid:173)
(cid:171)

𝑚𝑖

(cid:17) 2

(cid:16)

2

c − v − 𝜖 v(1)
𝑖 𝑗
(cid:17)
(cid:16)
𝑇 + 𝜖 𝑇 (1)
𝑖 𝑗

(cid:170)
(cid:174)
(cid:174)
(cid:172)

(4A.11)

(4A.12)

(4A.13)

(4A.14)

(cid:18) 𝜕
𝜕𝜖

(cid:19)

𝑀𝑖 𝑗

𝜖=0

= 𝑓 (0)
𝑖








𝑚𝑖
𝑇 C · v(1)
𝑖 𝑗 +

(cid:32) 1
2

𝑚𝑖𝐶2
𝑇

−

(cid:33) 𝑇 (1)
𝑖 𝑗
𝑇

3
2








= 𝑀 (1)
𝑖 𝑗

(4A.15)

and 𝑀𝑖 𝑗 = 𝑓 (0)

𝑖

+ 𝜖 𝑀 (1)
𝑖 𝑗

. Inserting this expression into the multi-species BGK collision operator

results in:

𝑄BGK
𝑖 𝑗

= 𝜈𝑖 𝑗 (cid:0)𝑀𝑖 𝑗 − 𝑓𝑖(cid:1) = 𝜈𝑖 𝑗

(cid:16)

𝑓 (0)
𝑖

+ 𝜖 𝑀 (1)

𝑖 𝑗 − 𝑓 (0)

𝑖 − 𝜖 𝑓 (1)

𝑖

(cid:17)

= 𝜈𝑖 𝑗 𝜖

(cid:16)

𝑀 (1)

𝑖 𝑗 − 𝑓 (1)

𝑖

(cid:17)

.

(4A.16)

Considering that 𝑄BGK

𝑖 𝑗

= 𝜖𝑄 (1)

𝑖 𝑗 + 𝑂 (𝜖 2) we also can use the previously derived expression for 𝑄 (1)

𝑖

:

𝑄 (1)

𝑖 =

∑︁

𝑄 (1)

𝑖 𝑗 =

∑︁

𝜈𝑖 𝑗

𝑗

𝑗

which gives us an expression for 𝑓 (1)

𝑖

(cid:16)

:

𝑀 (1)

𝑖 𝑗 − 𝑓 (1)

𝑖

(cid:17)

=

∑︁

𝜈𝑖 𝑗 𝑀 (1)

𝑖 𝑗 −

𝑗

𝜈𝑖 𝑗 𝑓 (1)
𝑖

∑︁

𝑗

(4A.17)

𝑓 (1)
𝑖

=

1
(cid:205) 𝑗 𝜈𝑖 𝑗

(cid:32)

∑︁

𝑗

𝜈𝑖 𝑗 𝑀 (1)

𝑖 𝑗 − 𝑄 (1)

𝑖

(cid:33)

(4A.18)

101

Inserting expressions for 𝑀 (1)

𝑖 𝑗 and 𝑄 (1)

𝑖

∑︁

𝑓 (1)
𝑖

=

𝜈𝑖 𝑗
𝜈𝑖

𝑓 (0)
𝑖𝐹

𝑚𝑖
𝑇 C · v(1)
𝑖 𝑗 +

(cid:32) 1
2

𝑚𝑖𝐶2
𝑇

−

(cid:33) 𝑇 (1)
𝑖 𝑗
𝑇

3
2














(cid:18)

𝑗

−

−

(cid:20) 𝑚𝑖
𝑇

𝑓 (0)
𝑖𝐹
𝜈𝑖
𝑚𝑖
𝜌𝑇 C · J𝐾 +

C ⊗ C −

: (∇𝑥 ⊗ v) + C ·

(cid:19)

𝐶2I

1
3
𝑚𝑖𝐶2
3𝑇

(cid:18)

1 −

(cid:19) (cid:18) 1
𝑛𝑇

𝐻𝐾 −

3
2𝑛

∇𝑥 · ˜K1

−

(cid:18)(cid:18) 𝑚𝑖
2𝑇
(cid:19)

𝐶2 −

(cid:19)

5
2

∇𝑥 log(𝑇)

(cid:19)

+

𝑛
𝑛𝑖

C · d𝑖

1
𝑛𝑖

∇𝑥 · ˜K1,𝑖

(cid:21)

−

1
𝜈𝑖

c · ∇𝑥 𝑓𝑖𝐾 +

1
𝜈𝑖

Δ𝑖.

(4A.19)

Note that for ℎ → 0, the kinetic terms in the last line go to zero and we recover the standard
Navier-Stokes correction for 𝑓 (1)

from Eq(4.48).

𝑖

4A.3 Computation of v(1)

𝑖 𝑗 and 𝑇 (1)
𝑖 𝑗

According to Eq. 4A.11, to compute v(1)

𝑖 𝑗 we need to determine v(1)

𝑖

. To do this we multiply

both sides of the expression for 𝑓 (1)
𝜖 ⟨𝑚𝑖C 𝑓 (1)

𝑖 with 𝑚𝑖C and integrate over C (i.e. compute 𝜌𝑖V𝑖 = 𝜌𝑖𝜖v(1)
⟩). Most of the terms are zero because 𝑓𝑖𝐹 has no odd moments and we are left with:

𝑖 =

𝑖

𝜌𝑖v(1)

𝑖 =

𝜌𝑖
𝜈𝑖

∑︁

𝑗

𝜈𝑖 𝑗 v(1)

𝑖 𝑗 −

1
𝜈𝑖

w𝑖 +

1
𝜈𝑖

⟨𝑚𝑖C Δ𝑖⟩ , w𝑖 ≡ 𝑛𝑇d𝑖 +

𝜌𝑖
𝜌 J𝐾 − J𝑖𝐾 .

(4A.20)

Using the definition of v(1)

𝑖 𝑗 we obtain,

𝜌𝑖 𝜌 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖𝜈𝑖 𝑗 + 𝜌 𝑗 𝜈 𝑗𝑖

∑︁

𝑗

(V 𝑗 − V𝑖) = w𝑖 − ⟨𝑚𝑖C Δ𝑖⟩ .

(4A.21)

As deduced in previous works [21], if the system in eq.(4A.21) is subject to the constraint, (cid:205)𝑖 𝜌𝑖V𝑖 =

0, then the system of equations can be formulated as 𝐴v = w, where:

𝐴𝑖 𝑗 =





− (cid:205) 𝑗

𝜌𝑖 𝜌 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖 𝜈𝑖 𝑗 +𝜌 𝑗 𝜈 𝑗𝑖

− 𝜌𝑖 𝜌 𝑗 ,

if 𝑖 = 𝑗

𝜌𝑖 𝜌 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝜌𝑖 𝜈𝑖 𝑗 +𝜌 𝑗 𝜈 𝑗𝑖

− 𝜌𝑖 𝜌 𝑗 ,

if 𝑖 ≠ 𝑗

(4A.22)

Next, we focus on 𝑇 (1)
𝑖 𝑗

. According to Eq. 4A.12, to compute 𝑇 (1)

𝑖 𝑗 we need to determine 𝑇 (1)

𝑖

.

The mathematical steps here are the same as before (i.e. compute a moment and solve a system of
equations). We multiply both sides of the expression for 𝑓 (1)

𝑖 with 𝑚𝑖𝐶2 and integrate over c (i.e.

102

compute ⟨𝑚𝑖𝐶2 𝑓 (1)

𝑖

⟩). Most of the terms are zero because 𝑓𝑖𝐹 has no odd moments and we are left

with:

𝑛𝑖𝑇 (1)
𝑖

=

𝜈𝑖 𝑗
𝜈𝑖

∑︁

𝑗

𝑛𝑖𝑇 (1)

𝑖 𝑗 −

𝑠𝑖
𝜈𝑖

+

1
𝜈𝑖

⟨𝑚𝑖𝐶2Δ⟩ ,

𝑠𝑖 ≡ −

(cid:16) 𝑛𝑖
𝑛

2
3

𝐻𝐾 − 𝐻𝑖𝐾

(cid:17)

+ 𝑇

(cid:16) 𝑛𝑖
𝑛

∇𝑥 · ˜K1 − ∇𝑥 · ˜K1,𝑖

(cid:17)

Next, we subtract the left-hand side and reformulate,

𝑛𝑖𝑛 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝑛𝑖𝜈𝑖 𝑗 + 𝑛 𝑗 𝜈 𝑗𝑖

∑︁

𝑗

(cid:16)
𝑗 − 𝑇 (1)
𝑇 (1)

𝑖

(cid:17)

= 𝑠𝑖 − ⟨𝑚𝑖𝐶2Δ𝑖⟩ .

(4A.23)

(4A.24)

As deduced in previous works [21], if the system in eq.(4A.24) is subject to the constraint,
(cid:205)𝑖 𝑛𝑖𝑇 (1)

= 0, then the system of equations can be formulated as 𝐵T(1) = s, where:

𝑖

𝐵𝑖 𝑗 =





− (cid:205) 𝑗

𝑛𝑖𝑛 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝑛𝑖 𝜈𝑖 𝑗 +𝑛 𝑗 𝜈 𝑗𝑖

− 𝑛𝑖𝑛 𝑗 ,

if 𝑖 = 𝑗

𝑛𝑖𝑛 𝑗 𝜈𝑖 𝑗 𝜈 𝑗𝑖
𝑛𝑖 𝜈𝑖 𝑗 +𝑛 𝑗 𝜈 𝑗𝑖

− 𝑛𝑖𝑛 𝑗 ,

if 𝑖 ≠ 𝑗

(4A.25)

This system has a unique solution with a symmetric formulation.

103

APPENDIX 4B

CORRECTIONS TO THE STRESS TENSOR

To find the correction to the stress tensor, we have to calculate P(1)
terms in 𝑓 (1)

that contain an even power of C should be non-zero. The first term is [21]:

= ⟨𝑚𝑖 (C ⊗ C) 𝑓 (1)

𝑖

⟩. Only

𝑖

𝑖

(cid:20) 𝑚𝑖
𝑇
(cid:18)

(cid:28)

𝑚𝑖

−

1
𝜈𝑖

= −

𝑛𝑖𝑇
𝜈𝑖

(cid:18)

C ⊗ C −

(cid:19)

𝐶2I

1
3

: (∇𝑥 ⊗ v)

(cid:21)

(cid:29)

(C ⊗ C) 𝑓 (0)

𝑖

(∇𝑥 ⊗ v) + (∇𝑥 ⊗ v)𝑇 −

(cid:19)

(∇𝑥 · v)I

2
3

The second relevant term is:

𝜈𝑖 𝑗
𝜈𝑖

∑︁

𝑗

(cid:28)

𝑇 (1)
𝑖 𝑗

𝑚𝑖 (C ⊗ C)

(cid:18) 𝑚𝑖
2𝑇 2

𝐶2 −

(cid:19)

3
2

1
𝑇

(cid:29)

𝑓 (0)
𝑖

∑︁

=

𝑗

𝜈𝑖 𝑗
𝜈𝑖

𝑇 (1)
𝑖 𝑗 𝑛𝑖I

(4B.1)

(4B.2)

followed by:

−

1
𝜈𝑖

(cid:18) 1
𝑛𝑇

𝐻 −

3
2𝑛

(cid:19) (cid:28)

∇𝑥 · ˜K1

𝑚𝑖 (C ⊗ C)

(cid:18)

1 −

(cid:19)

𝑚𝑖𝐶2
3𝑇

𝑓 (0)
𝑖

(cid:29)

+

1
𝜈𝑖

(cid:18) 1
𝑛𝑖

(cid:19)

∇𝑥 · ˜K1,𝑖

⟨𝑚𝑖 (C ⊗ C) 𝑓 (0)

𝑖

⟩

𝐻𝐾 − 𝑇

𝑛𝑖
𝑛

∇𝑥 · ˜K1

(cid:19)

I +

𝑇
𝜈𝑖

∇𝑥 · ˜K1,𝑖I

(cid:16) 𝑛𝑖
𝑛

𝐻𝐾 − 𝐻𝑖𝐾

(cid:17)

− 𝑇

(cid:16) 𝑛𝑖
𝑛

∇𝑥 · ˜K1 − ∇𝑥 · ˜K1,𝑖

(cid:17) (cid:19)

I +

2
3𝜈𝑖

𝐻𝑖𝐾I +

1
𝜈𝑖

(cid:18) 2𝑛𝑖
3𝑛
(cid:18) 2
3

=

=

=

1
𝜈𝑖
1
𝜈𝑖
1
𝜈𝑖

𝑠𝑖I +

2
3𝜈𝑖

𝐻𝑖𝐾I

The final term can be used to eliminate the 𝐻𝑖𝐾 from the previous term:

−

1
𝜈𝑖

⟨𝑚𝑖 (C ⊗ C) c · ∇𝑥 𝑓𝑖𝐾⟩ = −

(cid:18)

(cid:28)

𝑚𝑖

1
𝜈𝑖

C ⊗ C −

(cid:19)

𝐶2I

1
3

c · ∇𝑥 𝑓𝑖𝐾

(cid:29)

−

2
3𝜈𝑖

𝐻𝑖𝐾I

(4B.3)

(4B.4)

(4B.5)

(4B.6)

(4B.7)

Combining these expressions with the Δ term which we didn’t alter, the correction to the stress

tensor is:

P(1)
𝑖 = −

(cid:18)

𝑛𝑖𝑇
𝜈𝑖

(∇𝑥 ⊗ v) + (∇𝑥 ⊗ v)T −

2
3

(∇𝑥 · v) I

(cid:19)

∑︁

+

𝑗

𝜈𝑖 𝑗
𝜈𝑖

𝑇 (1)
𝑖 𝑗 𝑛𝑖I

(cid:18)

(cid:28)

𝑚𝑖

𝑠𝑖I −

C ⊗ C −

+

1
𝜈𝑖

(cid:19)

𝐶2I

1
3

c · ∇𝑥 𝑓𝑖𝐾

(cid:29)

+

1
𝜈𝑖

⟨𝑚𝑖 (C ⊗ C) Δ𝑖⟩

(4B.8)

104

APPENDIX 4C

CORRECTIONS TO THE HEAT FLUX

To find corrections to the heat flux, we calculate q(1)
symmetry; only terms in 𝑓 (1)
𝑖 with odd powers of C contribute. The first non-zero term is
(cid:19)
(cid:28)

𝐶2C 𝑓 (1)

𝑖 = ⟨ 𝑚𝑖

(cid:21) (cid:19)

(cid:29)

2

𝑖

⟩. Many terms cancel due to

−

1
2𝜈𝑖

(cid:16)

𝑚𝑖

𝐶2C

(cid:17) (cid:18)

C ·

(cid:20) (cid:18) 𝑚𝑖
2𝑇

5
2

𝐶2 −

∇𝑥 log(𝑇)

𝑓 (0)
𝑖

For a vector A, not dependent on C, we know that [22],

⟨𝐶2𝑛C (C · A) 𝑓 (0)

𝑖

⟩ =

A ⟨𝐶2𝑛+2 𝑓 (0)

𝑖

⟩ .

1
3

Thus, Eq(4C.1) simplifies to

−

1
2𝜈𝑖

(cid:20) 35𝑛𝑖𝑇 2
2𝑚𝑖

−

(cid:21)

25𝑛𝑖𝑇 2
2𝑚𝑖

∇𝑥 log(𝑇) = −

1
𝜈𝑖

5𝑛𝑖𝑇
2𝑚𝑖

∇𝑥𝑇 .

Using the same identity, the remaining non-zero heat flux terms are given by

𝑛
1
𝑛𝑖
𝜈𝑖
𝑚𝑖
𝜌𝑇

−

1
𝜈𝑖

(cid:68) 𝑚𝑖
2
(cid:68) 𝑚𝑖
2

𝐶2C(C · d𝑖) 𝑓 (0)

𝑖

= −

𝐶2C(C · J𝐾) 𝑓 (0)

𝑖

=

(cid:69)

(cid:69)

d𝑖,

5𝑛𝑇 2
1
2𝑚𝑖
𝜈𝑖
5𝑛𝑖𝑇
2𝜌 J𝐾,

1
𝜈𝑖

(4C.1)

(4C.2)

(4C.3)

(4C.4)

(4C.5)

and

(cid:42) 𝑚𝑖
2

𝐶2C

(cid:32)

∑︁

𝑗

𝜈𝑖 𝑗
𝜈𝑖

𝑓 (0)
𝑖

(cid:104) 𝑚𝑖
𝑇 C · v𝑖 𝑗

(cid:105)

(cid:33) (cid:43)

∑︁

=

𝑗

𝜈𝑖 𝑗
𝜈𝑖

𝑚𝑖
𝑇

(cid:68) 𝑚𝑖
2

𝐶2C (cid:0)C · v𝑖 𝑗 (cid:1) 𝑓 (0)

𝑖

(cid:69)

=

∑︁

𝑗

𝜈𝑖 𝑗
𝜈𝑖

5
2

𝑛𝑖 𝑇 v𝑖 𝑗 .

(4C.6)

Combining these calculations, the heat flux correction is

𝜖q𝑖 = −𝜖 5
2

𝑛𝑖𝑇
𝑚𝑖𝜈𝑖

∇𝑥𝑇 + 𝜖 ∑︁

𝑗

𝜈𝑖 𝑗
𝜈𝑖

5
2

𝑛𝑖𝑇v𝑖 𝑗 −

𝜖
𝜈𝑖

𝑛𝑇 2
𝑚𝑖

5
2

d𝑖 +

𝜖
𝜈𝑖

5
2

𝑛𝑖𝑇
𝜌 J𝐾

𝜖
𝜈𝑖
𝑛𝑖𝑇
𝑚𝑖𝜈𝑖

(cid:68) 𝑚𝑖
2
∇𝑥𝑇 + 𝜖 5
2

𝐶2C(c · ∇𝑥 𝑓𝑖𝐾)
(cid:32) 𝜌𝑖
𝜈𝑖

𝑇
𝑚𝑖

(cid:69)

+

1
𝜈𝑖

⟨

𝑚𝑖
2

∑︁

𝜈𝑖 𝑗 v𝑖 𝑗 −

𝐶2C Δ𝑖⟩ ,

𝑛𝑇
𝜈𝑖

d𝑖 +

𝜌𝑖
𝜌 J𝐾 −

1
𝜈𝑖

1
𝜈𝑖

J𝑖𝐾 +

(cid:33)

1
𝜈𝑖

J𝑖𝐾

𝐶2C(c · ∇𝑥 𝑓𝑖𝐾)

𝐶2C Δ𝑖⟩ ,

∇𝑥𝑇 + 𝜌𝑖v𝑖 +

1
𝜈𝑖

J𝑖𝐾

(cid:68) 𝑚𝑖
2

𝐶2C(c · ∇𝑥 𝑓𝑖𝐾)

(cid:69)

+

1
𝜈𝑖

⟨

𝑚𝑖
2

𝐶2C Δ𝑖⟩ .

(4C.7)

= −

−

𝜖

5
2

−

= 𝜖 5
2

𝑇
𝑚𝑖

𝜖
𝜈𝑖
(cid:18)

(cid:68) 𝑚𝑖
2
𝑛𝑖
𝜈𝑖

−

𝑗

+

(cid:69)

⟨

1
𝜈𝑖
(cid:19)

−

𝑚𝑖
2
𝜖
𝜈𝑖

105

CHAPTER 5

DATA DRIVEN OBSERVATIONS OF SYSTEM EQUILIBRATION

5.1

Introduction

In near-vacuum-hohlraum inertial confinement fusion (ICF) experiments, the ions’ mean free

path is too long for the system to be sufficiently described by the hydrodynamic equations [1]. To

simulate such kinetic systems, extended moment hydrodynamic models can be employed [2]. By

including moments of the distribution function beyond the moments which define a Gaussian, i.e.,

density, velocity, temperature, these equations include more kinetic phenomena than typical fluid

dynamics equations.

An infinite number of moments are needed to describe the non-equilibrium distribution evolved

by a kinetic equation, but only a finite number can be simulated. Additionally, the 𝑁-th moment

depends on an unknown 𝑁 + 1-th moment. Thus, truncating the hierarchy of moment equations

requires an assumption which rewrites the 𝑁 + 1-th moment in terms of the previous 𝑁 moments,

often referred to as closure information. Often it is assumed that the system is near local equilibrium

and this limits the model’s predictive capacity. As such closures are a critical area of research for

modeling systems with kinetic phenomena.

Machine learning (ML) techniques are emerging approaches to the moment closure problem,

where a truncated system of moment equations is closed with a neural network [3]. This approach

is often used because neural networks can go beyond local equilibrium assumptions [4]. Neural

closures, a special type of Neural ODE (NODE) [5], are a leading approach to data-driven closures

[6]. Many investigators have found that enforcing some structure in the NODE results in lower

errors in long-term evolution, improving the predictive capacity [7, 8, 9]. For instance, Huang et

al. have enforced hyperbolicity in the neural closure of extended moment methods [10, 11]. These

constraints ensure that the system’s evolution adheres to an invariant manifold within the solution

space. However, Celledoni et al. indicate in their review of structure-preserving deep learning that

the " generalisation of most [NODEs] to the manifold setting is still missing" [12].

In extended moment models, the closures include dissipation that returns the system to equilib-

106

rium. This equilibration process has been formulated as the convergence to the invariant manifold

in the space of distributions [13]. Thus invariant manifold detection is a key criteria to assess the

quality of a closure. According to the Hartman-Grobman theorem, a dynamical system’s Jacobian

near an equilibrium characterizes an invariant manifold. Alternatively, dynamic mode decompo-

sition (DMD), rooted in the Koopman operator framework, is a computational approach that can

be characterize the invariant manifold of a system [14]. Furthermore, Lan and Mezic’s results

indicate that DMD’s linear operator may extend the Hartman–Grobman theorem, enlarging the

domain of linearity from near the equilibrium to the next equilibrium point [15]. A DMD-based

reconstruction error has been used to detect regime transition [16], but the authors do not make the

explicit connection to the invariant manifold.

In this work, we apply DMD and dimension reduction techniques, developed by Roweis and

Saul [17], to assess whether a closure guides a system towards an invariant manifold. We focus on

observing both the convergence to the slow manifold and the slow manifold. This is an alternative

to observing the constants of motion which characterize the invariant manifold. Our data-driven

investigation is done in the context of extended moment fluid dynamics equations implemented

with either Grad’s closure [18] or a neural closure. We find that our neural closure can equilibrate

towards a slow manifold.

The structure of this chapter is as follows. In Section 5.2 we introduce Grad’s moment equations.

In section 5.3 we introduce Neural ODEs and show how the technique is used to resolve the closure

problem. Our major results are contained in section 5.4, we introduce the methods for observing

system equilibration and apply these methods to investigate the slow manifold in Grad’s system

with and without a neural closure. We provide a summary and outlook in Section 5.5. Lastly, a

numerical comparison between neural ODEs and neural closures is provided in the appendix.

5.2 Grad’s Hydrodynamic Equations

Grad’s extended moment hydrodynamic equations are derived from the Boltzmann equation by

expanding the distribution function 𝑓 in a Hermite basis [18]. The near-equilibrium, 1-dimensional

107

version of Grad’s equations can be expressed [13]

𝜕𝑡 𝑓 + 𝐴( 𝑓 ) =

1
𝜖

𝑄( 𝑓 ).

(5.1a)

The vector of moments 𝑓 is defined as 𝑓 (𝑥, 𝑡) ≡ [𝜌(𝑥, 𝑡), 𝑢(𝑥, 𝑡), 𝑇 (𝑥, 𝑡), 𝜎(𝑥, 𝑡), 𝑞(𝑥, 𝑡)]𝑇 , where

𝜌 = 𝛿𝜌/𝜌0 is the relative density, 𝑢 = 𝛿𝑢/𝑢0 is the relative velocity, and 𝑇 = 𝛿𝑇/𝑇0 is the relative

temperature. Further, 𝜎 is the dimensionless pressure tensor, 𝑞 is the dimensionless heat flux, 𝑡 is

the dimensionless time, and 𝑥 is the dimensionless distance. The advection term 𝐴( 𝑓 ) is defined

𝐴( 𝑓 ) ≡



















𝜕𝑥𝑢

𝜕𝑥 ( 𝑝 + 𝑢 + 𝜎)

2
3
𝜕𝑥

𝜕𝑥

𝜕𝑥 (𝑢 + 𝑞)
(cid:17)
(cid:16)

𝑞

2𝑢 + 4
5
(cid:16) 5
𝑇 + 𝜎
2

(cid:17)

,



















(5.1b)

and the dissipative or “collisional” term 𝑄 is defined

0

















In this dissertation, we chose 𝑘 = 1 and 𝜖 = 𝜋/25, a decision that, according to dispersion relations,


















𝑄( 𝑓 ) ≡

−2𝑞/3

(5.1c)

−𝜎

0

0

.

decouples the hydrodynamic modes from the kinetic modes [13]. Eq(5.1) is one of many approaches

to the fluid dynamics moment closure problem, closing the typical hydrodynamics equations of 𝜌,

u, and 𝑇 by expressing the dynamics of the pressure tensor 𝜎 and the heat flux 𝑞 in terms of 𝑓 .

The conventions of Eq(5.1) are intentionally chosen to evoke connections between Grad’s extended

moment equations and kinetic equations.

Grad’s equations are known to enforce equilibration to a slow manifold [13]. By the Hartmann-

Grobmann theorem, we can expand about equilibrium to produce analytic estimates of the invariant

108

manifold. Let 𝜖 be small, expanding Eq(5.1) to linear order about

𝑓𝜖=0 ≡ 𝑀 𝑓 = [𝜌, 𝑢, 𝑇, 0, 0],

where 𝑀 is the matrix which projects the equilibrium moments, yields

𝜕𝑡 𝑓𝜖 = 𝐹 ( 𝑓0) + ∇ 𝑓 𝐹 ( 𝑓 )

(cid:12)
(cid:12)
(cid:12) 𝑓 = 𝑓0

𝛿 𝑓 .

(5.2)

(5.3)

In this expression, 𝛿 𝑓 = 𝑄−1(𝐼 − 𝜕 𝑓0

𝑓 · 𝑀) 𝐴( 𝑓0) and ∇ 𝑓 𝐹 ( 𝑓 )

(cid:12)
(cid:12)
(cid:12) 𝑓 = 𝑓0

is the Jacobian. Evaluating the

RHS produces the Navier-Stokes equations,

𝜕𝑡

𝜌

𝑢

𝑇

𝜎

𝑞



































= −


















𝜕𝑥𝑢

𝜕2
𝑥 𝑢

𝜕𝑥 ( 𝑝 + 𝑢) − 𝜖 4
3
𝜕𝑥𝑇

𝜕𝑥𝑢 − 𝜖 5
2

2
3

0

0


















(5.4)

This equilibrium expansion is structured to emphasize the connection between the dynamic system’s

Jacobian and the Chapman-Enskog expansion used to produce the Navier Stokes equations. From

these derivations, we see that, for rapidly dissipating systems 𝜖 ≪ 1, Grad’s moment equations are

well described by its invariant manifold.

In this work, we consider the Fourier transform 𝜌(𝑟, 𝑡) = (cid:205)+∞

−∞ 𝜌𝑘 (𝑡)𝑒𝑖k·r of the these PDEs.

For these near-equilibrium equations, the Fourier transform reduces Grad’s equations to a system

of ODEs. The eigenvalues of this ODE update matrix define the dispersion relations. A table

organizing the equations and their Fourier transforms is presented in Table 5.1

5.3 Neural Closures

Neural ODEs (NODE) use neural networks (NN) to solve differential equations [5]. NODEs

emerged as the continuous time version of ResNets [19]. NODEs were originally formulated to

predict the difference between an input 𝑥0 and the desired output 𝑥1 as 𝑥1 = 𝑥0 + 𝛿𝑡𝑁 𝑁 (𝑥0), where

𝑁 𝑁 (𝑥0) is the output of the neural network. The structure resembles an Euler update, where

the neural network is learning to predict the derivative. NODEs are particularly suited to inform

109




(cid:121)

𝜕𝑡

𝜕𝑡

𝜌
𝑢
𝑇
𝜎
𝑞























= −














𝜕𝑥𝑢
𝜕𝑥 ( 𝑝 + 𝑢 + 𝜎)
𝜕𝑥 (𝑢 + 𝑞)
(cid:17)
(cid:16)
𝑞
2𝑢 + 4
5
(cid:17)
(cid:16) 5
𝑇 + 𝜎
2

2
3
𝜕𝑥

𝜕𝑥














− 1
𝜖

0


0


0


𝜎


2𝑞/3














Fourier
→ 𝜕𝑡

𝜎 ≈ −𝜖 4
3

𝜕𝑥𝑢, 𝑞 ≈ −𝜖 15
4

𝜕𝑥𝑇

−𝑖𝑘𝑢𝑘
−𝑖𝑘 ( 𝑝𝑘 + 𝑢𝑘 + 𝜎𝑘 )

3 (𝑢𝑘 + 𝑞𝑘 )
(cid:17)
𝑞𝑘
(cid:17)

−𝑖𝑘 2
(cid:16)
2𝑢𝑘 + 4
5
(cid:16) 5
𝑇𝑘 + 𝜎𝑘
2

− 𝜎𝑘 /𝜖

− 2𝑞/3𝜖

−𝑖𝑘

−𝑖𝑘

= −



























𝜎 ≈ −𝑖𝑘𝜖 4
3

𝑢, 𝑞 ≈ −𝑖𝑘𝜖 15
4

𝑇

𝜌𝑘
𝑢𝑘
𝑇𝑘
𝜎𝑘
𝑞𝑘


























(cid:121)

𝜌


𝑢


𝑇𝑘










= −








𝜕𝑥𝑢

𝜕𝑥 ( 𝑝 + 𝑢) − 𝜖 4
𝜕2
𝑥 𝑢
3
𝜕𝑥𝑇
𝜕𝑥𝑢 − 𝜖 5
2

2
3








Fourier
→ 𝜕𝑡

𝜌𝑘
𝑢𝑘
𝑇𝑘















= −








−𝑖𝑘𝑢𝑘
−𝑖𝑘 ( 𝑝𝑘 + 𝑢𝑘 ) − 𝜖 𝑘 2 4
3

−𝑖𝑘 2
3

𝑢𝑘 − 𝜖 𝑘 2 5
2

𝑇

𝑢








Figure 5.1 A grid visualizing the relation between Grad’s equations (upper) and Navier-Stokes’
equations (lower) in both spatial (left) and Fourier space (right).

closures since closure information is often expressed as an ODE. A neural closure is a type of

NODE, in which the neural net estimates a subset of the derivatives needed to update the system’s

current state. Such approaches often lead to better stability [6].

In the context of Grad’s equations, we use our neural closure to estimate (cid:164)𝜎𝑘 and (cid:164)𝑞𝑘 while (cid:164)𝑛𝑘 ,

(cid:164)𝑢𝑘 , (cid:164)𝑇𝑘 are determined using Grad’s equations Eq(5.1). Therefore, our derivative estimator is

(cid:164)𝑓 𝑁 𝑁 (𝑡) =

(cid:164)𝜌𝑘 (𝑡)

(cid:164)𝑢𝑘 (𝑡)

(cid:164)𝑇𝑘 (𝑡)

(cid:164)𝜎𝑘 (𝑡)

(cid:164)𝑞𝑘 (𝑡)



































= −

𝑖𝑘 2

𝑖𝑘𝑢𝑘 (𝑡)





𝑖𝑘 ( 𝑝𝑘 (𝑡) + 𝑢𝑘 (𝑡) + 𝜎𝑘 (𝑡))













3 (𝑢𝑘 (𝑡) + 𝑞𝑘 (𝑡))
𝑁 𝑁 (inputs)

𝑁 𝑁 (inputs)

.


















(5.5)

In this derivative estimate, mass, momentum, and energy are conserved and the neural network is

relegated to where traditional methods may be insufficient. We compare the neural closure to its

NODE counterpart in Appendix 5 and demonstrate the neural closure has lower error and better

stability.

We use a neural network comprised of a single layer of complex valued 500 Rectified Linear

Units (ReLU), this is known as a complex valued neural network (CVNN) [20]. The CVNN predicts

110

two outputs (cid:164)𝜎𝑘 (𝑡) and (cid:164)𝑞𝑘 (𝑡) but takes as input the current state 𝑓𝑘 (𝑡) ≡ [𝜌𝑘 (𝑡), 𝑢𝑘 (𝑡), 𝑇𝑘 (𝑡), 𝜎𝑘 (𝑡), 𝑞𝑘 (𝑡)]𝑇 ,

the 5 preceding positions, and the 5 preceding derivatives. The training/testing data is gathered

from 15 trajectories of Grad’s moment equations evolved 𝑡 ∈ [0, 4] with 𝛿𝑡 = 0.01. Each trajectory

is initialized with coordinates that are randomly sampled from inside a complex 5-dimensional unit

ball, i.e., 10 real numbers. For both training and testing data, (cid:164)𝑓𝑘 is estimated from the trajectories

(with no added noise) using fourth order symmetric finite difference. The sequential inputs and

outputs are presented in Figure 5.1.

Table 5.1 Visualization of the complex valued neural network’s sequential input data (x) and output
data (Y).

time
f
𝜕𝑡 𝑓

𝑡 − 5𝛿𝑡
×
×

𝑡 − 4𝛿𝑡
×
×

𝑡 − 3𝛿𝑡
×
×

𝑡 − 2𝛿𝑡
×
×

𝑡 − 𝛿𝑡
×
×

𝑡
×
Y

This neural closure is a derivative approximation, which can be used inside the multi-step update

scheme [21]; this approach is known as a multi-step neural closure (MsNC). Multi-step updates

recycle previous derivative estimates to improve accuracy. The second order Adams-Bashforth

(AB2) multi-step scheme is

𝑓 (𝑡 + 𝛿𝑡) = 𝑓 (𝑡) + 𝛿𝑡

(cid:18) 3
2

(cid:164)𝑓 𝑁 𝑁 (𝑡) −

1
2

(cid:164)𝑓 𝑁 𝑁 (𝑡 − 𝛿𝑡)

(cid:19)

.

(5.6)

We expect increasing the order will decrease the update error.However, we also expect that in-

creasing the multi-step order has diminishing returns as the update error becomes on par with the

CVNN prediction error. Numerical tests are conducted in Appendix 5 and these expectations are

verified. Sample trajectories, using either Grad’s closure Eq(5.1) updated with RK7 or the neural

closure Eq(5.5) updated with AB2, are presented in Figure 5.2 Left. For 20 randomly sampled

initial conditions, the mean squared error incurred at each time step is plotted in Figure 5.2 Right.

The plots demonstrate that error does not grow exponentially at long times.

5.4

Invariant Manifolds

Previous data-driven investigations used manifold learning to observe dimension reduction

during the equilibration process. Ellison et al. detected the onset of hydrodynamic evolution in

111

Figure 5.2 Left: neural closure Eq(5.5) evolved with second order multi-step neural closure, labeled
as MsNC, plotted alongside Grad’s closure Eq(5.1) evolved with seventh order Runge-Kutta, labeled
as “correct”. The difference between the trajectories is too small to see. Right: The mean square
error incurred at each update for 20 trajectories, plotted alongside the smoothed median error, and
cumulative smoothed median error. The plot demonstrates that the neural closure does not have
exponentially increasing error.

kinetic equations by assessing the reducibility of an ensemble of kinetic evolutions (i.e., ability

to reduce the ensemble’s dimensionality while preserving its structure) [22].

In this section,

we extend this work, complementing it with dynamic mode decomposition to characterize the

invariant manifold. We apply both tools to Grad’s extended moment hydrodynamics equations

with and without a neural closure.

To investigate reducibility, we create two ensembles of trajectories (𝑁 = 10, 000) evolved ac-

cording to Eq(5.1) with RK7 and Eq(5.5) with AB2. Each trajectory in an ensemble is initialized

with coordinates randomly sampled from a complex 5-dimensional unit ball, (i.e., 10 real dimen-

sions) and evolved to 𝑇max = 1. At early times the ensemble of points is irreducible, but at late times

the ensemble can be reduced from 10 dimensions to 6 dimensions. The reducibility of the data was

assessed using Saul and Rowes’ inverse reconstruction error 𝐸 with a principal component analysis

(PCA) dimension reduction technique [17]

𝐸 (𝑡, 𝑛) = 𝐸2(𝑡, 𝑛)/𝐸1(𝑡),

𝐸1(𝑡) = ∥ 𝑋 (𝑡) − 𝑊𝐵𝐶 (𝑡) 𝑋 (𝑡) ∥2,

𝐸2(𝑡, 𝑛) = ∥ 𝑋 (𝑡) − 𝑊𝑅𝐵𝐶 (𝑛, 𝑡) 𝑋 (𝑡) ∥2.

(5.7a)

(5.7b)

(5.7c)

Here 𝑋 (𝑡) is a matrix containing the coordinates of all 𝑁 trajectories at time 𝑡, 𝑊𝐵𝐶 (𝑡) is the barycen-

112

ter weight matrix computed from 𝑋 (𝑡). 𝑊𝑅𝐵𝐶 (𝑛, 𝑡) is the barycenter weight matrix computed from

the data after it has been reduced onto 𝑛 principal vectors [23, 24]. We plot the reconstruction error

𝐸 (𝑡, 𝑛) in Figure 5.3.

Figure 5.3 Plot of the reducibility of an ensemble of trajectories, as assessed by the inverse
reconstruction error [23]. The ensemble is more reducible at later times. The ensemble is reduced
using PCA into a various number of dimensions (varied along the y-axis) the evolution time is
plotted along the x-axis. The yellow and orange regions indicate that reducing and un-reducing
the data destroys the ensemble’s local structure and the purple and blue regions indicate that local
structure is preserved. Both Grad’s closure and the neural closure’s evolve to a reduced subspace,
though the neural closure is not as reducible as Grad’s closure at late time.

To characterize the invariant manifold, we again create two ensembles, initialized the same as

in the reducibility investigation. However, each ensemble is composed of 20 simulations evolved

to 𝑇max = 10, which is 10× longer than the reducibility trajectories. The invariant manifold can be

characterized using dynamic mode decomposition (DMD) estimates of the Koopman operator 𝐾

[14]. The Koopman operator is estimated from a single trajectory, not on the entire ensemble of

points. Originally proposed by Schmid, the estimator is defined [25],

(cid:13)
(cid:13)
(cid:13)

𝐾𝑌 − 𝑌 ′(cid:13)
(cid:13)
(cid:13)𝐹

.

min
𝐾

(5.8)

𝑌 and 𝑌 ′ are constructed from the data in a sliding window (i.e., a sub-sequence of a trajectory).

𝑌 is a matrix comprised of the initial values and 𝑌 ′ is a matrix comprised of the updated values.

The sliding window, 𝑌 , and 𝑌 ′ are all visualized in Figure 5.4. The solution to this minimization

problem can be computed explicitly as

𝐾 = 𝑌 ′𝑌 + = ΞΛΞ𝑇 .

(5.9)

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Figure 5.4 Visualization of DMD on a sliding window. DMD is conducted on only the subsequence
of data contained in the blue band. Left: View of the sliding window from the global time series
level. Right: View of sliding window from the localized data level. The blue window contains 5
time steps, the first 4 comprise 𝑌 and the last 4 comprise 𝑌 ′.

The eigenvalues of 𝐾’s are represented by Λ = diag(𝜆1, . . . , 𝜆𝑛) and the eigenvectors are represented

Ξ = [𝜉1, . . . , 𝜉𝑛]. From each window of a trajectory we gather eigenvalues and vectors, yielding a

time series of eigensystems Λ(𝑡) and Ξ(𝑡). For both Grad’s closure and the neural closure. the two

smallest eigenvalues decrease orders of magnitude by 𝑡 = 1. This indicates dissipation on a similar

time scale as seen in the dimension reduction plots. A plot of the eigenvalue time series can be

seen in Figure 5.5. Additionally, only one of the neural closure’s eigenvalues decays to numerical

zero indicating the neural closure carries an additional basis vector in its invariant manifold,

demonstrating that the neural closure has an inferior convergence to the invariant manifold. This

parallels the neural closure’s inferior reducibility. Now consider the eigenvectors which characterize

the invariant manifold. To assess whether the sub-sequence of data is discovering the same invariant

manifold at each step, we observe the similarity 𝑆 between the DMD eigenvectors at time 𝑡𝑖 and 𝑡 𝑗 .

𝑆𝑖 𝑗 =

1
𝑁

∑︁

⟨𝜉𝑘 (𝑡𝑖)|𝜉𝑘 (𝑡 𝑗 )⟩

trajectories

(5.10)

where 𝑁 is the number of trajectories. since the system is transient (non-stationary), we do not

observe the autocorrelation. Rather, we are looking for the emergence of stationarity, hence our

choice of 𝑆 Eq(5.10). We observe that after 𝑡 = 1, DMD rediscovers the same eigenvector; A plot

of the eigenvector similarity across the time series (averaged across 20 random initial conditions)

can be seen in Figure 5.6.

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Figure 5.5 Time series of DMD eigenvalues. Left: Sliding Window DMD conducted on full Grad’s
equations Eq(5.1). Right: Sliding Window DMD conducted on Grad’s equations. For both Grad’s
closure and the neural closure, the eigenvalues separate by orders of magnitude by 𝑡 = 1. This
indicates both Grad’s closure and the neural closure evolve towards a slow manifold. However, the
neural closure converges worse than Grad’s closure because the smallest eigenvalues do not reduce
to numerical zero.

Figure 5.6 Plots of the similarity between a DMD eigenvector at time 𝑡𝑖 and at time 𝑡 𝑗 averaged
across 20 random initial conditions (similarity measured by complex dot product). The eigenvector
corresponding to the largest eigenvalue is labeled “1”, while the eigenvector corresponding to the
second largest eigenvalue is labeled “2”. A clear transition occurs in both Grad’s closure and
the neural closure, where the DMD eigenvectors discovered after 𝑡 = 1 are similar to each other.
However, with the eigenvectors associated to the neural closure display weaker similarity than the
eigenvectors associated to Grad’s closure.

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In conclusion, the ability to assess convergence to a characterized invariant manifold allows for

quantitative comparisons between dissipative closures. Taken together Figures 5.2, 5.3, 5.5, 5.6

indicate that while neural closure incurs a cumulative relative error of less than a tenth of a percent,

its overall convergence to an invariant manifold is inferior to Grad’s closure. These figures also

confirm a visualization of the equilibration process provided in Figure 5.7.

Figure 5.7 Representation of an ensemble of simulations equilibrating according to Grad’s eqns
towards a slow manifold (image altered from [13]).

5.5 Summary and outlook

In this work, we have presented data-driven approaches to observing a dynamical system’s

convergence to its slow manifold. These approaches complement previous analytic work done

by Gorban and Karlin [13], providing visualizations of the equilibration to the invariant manifold.

When the DMD eigenvectors are similar across long time intervals this indicates that the trajectory is

evolving on an invariant manifold. When the eigenvectors are dissimilar, we can assess whether the

trajectory is exponentially decaying towards or away from an invariant manifold, by the magnitude

of the DMD eigenvalues. Together these approaches assess whether the trajectory is transitioning

to the invariant manifold characterized by the DMD eigenvectors.

Further, we applied these data-driven techniques to assess the quality of a multi-step neural

closure. Our results indicated the neural closures can evolve a system toward an invariant manifold.

However, the invariant manifold for the neural closure is a lower quality. This suggests that

116

the naive neural closure introduces some error which pushes the trajectory off of its invariant

manifold, even after the trajectory has converged. In this work, the neural net was not explicitly

constrained to converge to the invariant manifold. thus, exploring how to incorporate these data-

driven observations as a regularization on the prediction fidelity will be the subject of future

work.

117

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[13] Alexander Gorban and Ilya Karlin. Hilbert’s 6th problem: exact and approximate hydro-
dynamic manifolds for kinetic equations. Bulletin of the American Mathematical Society,
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[14] Steven L Brunton, Marko Budisic, Eurika Kaiser, and J Nathan Kutz. Modern koopman

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APPENDIX 5

UPDATE ERRORS IN BOTH MULTI-STEP NODES AND MULTI-STEP NEURAL
CLOSURES

In this appendix, we compare the multi-step neural closure (MsNC) to its unguided multi-step

neural ODE (MsNODE) counterpart. For a NODE, every entry on the RHS of Eq(5.5) is estimated

by the neural network. Thus, the NODE has 5 outputs, whereas the neural closure has only 2

outputs. To compare these two approaches, we assess the error incurred at each update (mean

squared distance from the full Grad’s equations updated with RK7) for 20 randomly sampled initial

conditions. Comparisons are made for first, second, and third order multi-step update schemes.

The first order multi-step update is the common forward Euler update, the second order update is

given in Eq(5.6), and the third order update is

𝑓 (𝑡 + 𝛿𝑡) = 𝑓 (𝑡) + 𝛿𝑡

(cid:18) 23
12

(cid:164)𝑓 𝑁 𝑁 (𝑡) −

16
12

(cid:164)𝑓 𝑁 𝑁 (𝑡 − 𝛿𝑡) +

(cid:164)𝑓 𝑁 𝑁 (𝑡 − 2𝛿𝑡)

(cid:19)

.

5
12

(1)

We expect that when conservation of mass, momentum, and energy is enforced the simulation

will have lower error and better stability. This is a result that has been demonstrated in countless

works on ODEs and constitutes the primary justification for structure preserving methods. We

also expect that increasing the order of the multi-step update will decrease the relative difference

from the RK7 update because of the order conditions enforced. However, as the update error

becomes comparable to the error from the neural network’s predictions, increasing the multi-step

order yields diminishing returns. Typically, the derivative (cid:164)𝑓 is calculated at the level of numerical

precision. However, this is not the case for a NODE or neural closure because the neural net is

trained on derivatives computed via finite difference and the neural net has error in its predictions.

These errors, unique to MsNC and MsNODE, are distinct from those incurred by finite step size.

Therefore, once the error from finite step size is negligible compared to the error from the neural

network’s derivative estimate, increasing the multi-step order does not decrease the error.

In

numerical tests, the MsNC update has lower error and better stability than the MsNODE update.

Additionally, both updates improve with increasing order, but diminishing returns are observed.

Sample trajectories and error estimates are given in Figure 5.1.

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Figure 5.1 Left: A sample trajectory of Grad’s full equations Eq(5.1) updated with RK7, labeled
as “correct”, alongside a multi-step neural closure trajectory, labeled as MsNC, and a multi-step
neural closure trajectory, labeled as MsNODE. The difference between the MsNODE trajectories
and the correct trajectory is large enough to be visible. Right: The relative mean square error
(MSE) incurred at each update for 20 different trajectories, plotted alongside the smoothed median
error and cumulative smoothed median error. The plot demonstrates that the error decreases as
the order of the multi-step update increases. This plots demonstrates that using a multi-step neural
ODE to update the system leads to a growing relative error across all orders. Alternatively, the
relative error decreases, across all orders, when conservation of mass, momentum, and energy is
enforced.

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CHAPTER 6

SUMMARY AND OUTLOOK

This dissertation begins by pointing out phenomena that are unique to plasmas and defining

classification parameters that categorize different plasmas. Then we map the 𝜌𝑇 space of hydrogen

plasmas and spotlight the high energy density plasmas generated through inertial confinement

fusion. We conclude the introduction by motivating the dissertation’s major results in the context

of high energy density plasmas. In particular, we highlight the demand for models of a plasma’s

dynamical response and the problems with existing models. We also emphasize that many HED

plasmas manifest non-equilibrium conditions which violate the assumptions made in typical HED

codes.

In chapter 2, we detail Hilbert’s sixth problem and demonstrate, within the many body formalism,

how to aggregate microscopic degrees of freedom into macroscopic degrees of freedom. In effect,

we show how this aggregation gives rise to closures in molecular dynamics equations, kinetic

equations, and fluid dynamics equations. The major results of this dissertation, which are all

closures, are then presented in Chapter 3, Chapter 4, and Chapter 5.

In chapter 3, we start from the multi-component Bhatnagar-Gross-Krook kinetic equation and

produce a multi-species susceptibility which conserves number and momentum, referred to as the

“completed Mermin” susceptibility and explore its properties and uses. We show that the com-

pleted Mermin susceptibility satisfies the frequency sum rule. We apply the model to a carbon

contaminated deuterium and tritium plasma at NIF direct drive hot spots 𝑛𝑇 conditions and find

that momentum conservation qualitatively impacts the DSF’s shape. In our appendices, we provide

numerical implementations of the completed Mermin susceptibility for the reader’s convenience.

Further, we produce a new non-Drude conductivity model, by introducing free parameters on

the number and momentum conservation terms of our completed Mermin susceptibility’s single

species limit. To illustrate how number and momentum conservation impact the dynamical conduc-

tivity shape, we apply our conductivity model to dynamical gold conductivity measurements [1].

Finally, comparing our model to the Drude-Smith conductivity model, we conclude that Smith’s

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phenomenological parameter violates local number conservation. To use this model, the collision

rate and the inter-species potential must be supplied. Thus, the next step is to ascertain collision

rates and potentials for a particular experiment and apply this model.

In chapter 4, we present a hybrid model which uses a buffer region to transition from a

kinetic into a fluid description. Following the original work of Degond et al. [2], we extend the

original method of Degond to flows with multiple particle species in 3D. We derive the coupled

equations for the multispecies Bhatnagar-Gross-Krook model and its limiting Euler or Navier-

Stokes hydrodynamic equations. In the buffer region, both the kinetic and hydrodynamic equations

are solved simultaneously while being coupled via a so-called transition function. The transition

function ensures a smooth conversion from the coupled model to either the kinetic or continuum

approach at the interfaces of the buffer region. With that, the method avoids the need to find direct

interface boundary conditions and allows one to localize the use of a high dimensional kinetic model

only where it is needed. To validate our model numerically, we simulate a Sod shock problem.

Then we apply the hybrid model to investigate kinetic multi-species mixing in the preheat phase of

a high energy-density plasma physics experiment. We identify persistent velocity and temperature

separation between the species and electro-diffusion at the interfaces. The next step for this work is

to include this adaptive capability into our hybrid models. Degond et al. [3] have developed time

dependent buffer regions allowing for adapative degrees of freedom.

In chapter 5, we employ Dynamic Mode Decomposition (DMD) [4] and dimension reduction

techniques [5], to evaluate whether a closure steers a trajectory towards an invariant manifold and

then remains on the slow manifold. These approaches complement previous analytic work done

by Gorban and Karlin [6], providing visualizations of the equilibration to the invariant manifold

and providing an alternative to evaluating constants of motion to assess whether the simulation is

remaining on the invariant manifold. The findings from our data-driven analysis reveal that our

neural closure can equilibrate towards a slow manifold, but the quality is inferior to Grad’s closure.

The next step for this work is to use these data-driven observations to regularize the cost function

used to train the neural network.

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of fluid mechanics, 656:5–28, 2010.

[5] Sam T Roweis and Lawrence K Saul. Nonlinear dimensionality reduction by locally linear

embedding. science, 290(5500):2323–2326, 2000.

[6] Alexander Gorban and Ilya Karlin. Hilbert’s 6th problem: exact and approximate hydrodynamic
manifolds for kinetic equations. Bulletin of the American Mathematical Society, 51(2):187–246,
2014.

124