. . .

fiaFWr& . .nu“ _ E. .

Ania-«um. r. . . m. .3...- fl:

.. . 3:3...2 . .. A.

. . .r. , a...“ «El 1!. . .wcsaui ‘2
m .i

‘ u 5‘!
.uxr. us :5 as. r

391.-

.2. u.“

a... afifirkigzl
. ‘ 3.....131.
«p. ..

v
.r...
.1...

1...?

\ .PI
“15.5.... a
I: r Eur

Y“! 1:! . J:
.11... vi. 3.
z. .51.

..
N

t. . y :
$5.7.

.x
1.0“" V
L3... x. z .. . .9
01.2.3. 18.3.: a.
a: 31.3?
¢\ £33.. .

. :6...

. . “#35:...

1 7...»).(1 .

. 1.... 12...
OF 315.11%

4 .

Iv]? 7% it“

.3 ~11$3$éar 3&2 v. in.

.321- .2 1.... 5......

a: . \i... ..£.ix.ryex...x . ~
0.... d .

2'3

q
VI

.0
‘ I

, 3..
‘1

. it... 11

.. y. 1 h .
1..“ #94...
1 . {2.8155
.5: an. unrivrn
. , . . .‘n
‘Xil
turn)...
.: L. .

.\ t} .
31?...
‘oA

53....

iii-.0111}?

.0?! 1‘. s33?! ‘51
.5?! ..

a i 3d...)

9
l u I... .
v.7?!
\l 1

.. . : 1. ..
infilist

. A! I t.

1.91...

I. 1. . .. v. 1.
:1... . v 3.1.91

Vii. .37 v. var).
€2.90 .5 i9

901‘ 1...:

§ 7?
1...: 1
.. ‘1. .

 

{o
aau$~
v.21...

 

 

:51... , . .
,. 3.1a.

 

 

~ «masts

This is to certify that the
dissertation entitled

ACTIVATED ESCAPE
IN PERIODICALLY MODULATED SYSTEMS

presented by

DMITRI RYVKINE

has been accepted towards fulfillment
of the requirements for the

 

 

 

 

PhD. degree in Physics
/(l1 ' Dy Gmo‘m
Major Professor’s Signature
0 7/0 5/1504
Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

LIBRARY
I.‘iichigan State
U niVersity

 

 

 

—-—- .c-o-o-a-n--o-o--o-o-o-o--c-.-o--o-o-o-u--.-—

.-.—.--o--o-u-o-u-u—u-oco-c-o-o -

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 p:/ClRC/DateDue.indd-p.1

ACTIVATED ESCAPE
IN PERIODICALLY MODULATED SYSTEMS

By

Dmitri Ryvkz'ne

DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2006

ABSTRACT

ACTIVATED ESCAPE
IN PERIODICALLY MODULATED SYSTEMS
By
Dmitri Ryvkine

Noise-induced escape from a metastable state is studied for an overdamped pe—
riodically modulated system. We develop an asymptotic technique that gives both
the instantaneous and period-average escape rates for an arbitrary modulation fre-
quency w F and amplitude A. Using this technique, we reveal new system independent
features of escape in the periodically modulated systems.

The period-average escape rate has the form W = vexp(—R/ D), where 1/ is the
prefactor, R is the activation energy, and D is the noise intensity. Near a bifurcation
point the system experiences critical slowing down. This leads to scaling behavior
of fluctuations. We find that, in contrast to previously studied stationary systems, a
periodically driven system may display three scaling regimes and scaling crossovers
near a saddle—node bifurcation point where a metastable state disappears.

The activation energy scales with the driving field amplitude A as R o< (Ac — A)E ,
where Ac is the bifurcational value of A. With increasing field frequency the critical
exponent 5 changes from 5 = 3/2 for stationary systems to a dynamical value 6 = 2

and then again to g = 3/2.

The prefactor z/ depends on A nonmonotonically. Near the bifurcation amplitude
AC it scales as 11 cc (AC — A)C, with crossovers C = 1/4 to —1, and to 1/2.

We find the parameter range where escape is strongly synchronized by the mod-
ulation and the instantaneous escape rate W(t) displays sharp periodic peaks. The
peaks’ shape varies from Gaussian to strongly asymmetric with increasing modulation
frequency or amplitude.

The trajectories followed in escape form periodically repeated in time diffusion
broadened tubes. We Show that these tubes can be directly observed and find their
shape. Quantitatively, the tubes are characterized by the distribution of trajectories
that, after escape, pass through a given point in phase space for a given modulation
phase. This distribution may display several peaks shifted by the modulation period.

All analytical results are in good agreement with numerical Simulations for a model

system.

h.

“1'

ACKNOWLEDGMENTS

Most of all, I would like to thank my advisor and teacher Mark Dykman. The high
standards he set for himself as a person and researcher will always be a source of
admiration to me.

Brage Golding exposed me, as much as he could, to the experimental view of
physics.

My other Dissertation Committee members, Bhanu Mahanti, Scott Pratt, and
C.-P. Yuan, have been enthusiastic about my work and helpful with their comments.

The staff of the Department of Physics and Astronomy have been very responsive
to routine needs of an international graduate student.

I should also mention Christian Hicke, with whom I was lucky to share office 4252
for two years. We discussed everything: business, politics, history, religion, and even
physics.

Special thanks go to my wife, Anastasia Semykina, my parents and friends in my
home town, Ekaterinburg, and my “second home” town, Prague, for all the beautiful

moments in between.

iv

TABLE OF CONTENTS

LIST OF FIGURES vii
1 Introduction 1

2 Scaling and crossovers in activated escape near a bifurcation point 7

2.1 Introduction .................................. 7
2.2 Activated escape: general formulation .................... 12
2.3 Dynamics near a bifurcation point ..................... 16
2.3.1 The adiabatic approximation ....................... 17
2.3.2 Locally nonadiabatic regime ........................ 19
2.3.3 Fast-oscillating field ............................ 23
2.3.4 Connection to the locally nonadiabatic regime .............. 25
2.4 Activation energy of escape ......................... 28
2.4.1 Activation energy in the adiabatic approximation ............ 29
2.4.2 Nonadiabatic correction to the activation energy ............ 30
2.4.3 Activation energy in the locally nonadiabatic region ........... 32
2.4.4 Activation energy for wptr >> 1 ...................... 34
2.4.5 Scaling crossovers near a critical point .................. 35
2.5 Scaling crossovers for a model system .................... 35
2.5.1 Activation energy .............................. 38
2.5.2 Simulations ................................. 41
2.6 Conclusions .................................. 43
3 Noise-induced escape of periodically modulated systems: From weak

to strong modulation 46
3.1 Introduction .................................. 46
3.2 The model and the boundary layer distribution .............. 50
3.2.1 Motion near periodic states ........................ 51
3.2.2 Distribution near the unstable state ................... 52
3.3 Instantaneous and period-average escape rate ............... 53
3.3.1 General expression for the escape rate .................. 53
3.3.2 Synchronization of escape ......................... 55
3.4 Matching the intrawell and boundary—layer distributions ......... 56
3.4.1 Intrawell distribution near the basin boundary .............. 56
3.4.2 Matching the exponents and prefactors .................. 59
3.5 Time dependence of the escape rate ..................... 63
3.5.1 Adiabatic limit ............................... 63

3.5.2 N onadiabatic regime ............................ 67

3.5.3 Nonlinear current propagation ....................... 69
3.6 Period-average escape rate .......................... 71
3.7 Scaling near the bifurcation point ...................... 72
3.7.1 Adiabatic scaling .............................. 73
3.7.2 Locally nonadiabatic scaling ........................ 74
3.7.3 High-frequency scaling ........................... 75
3.8 Results for a model system .......................... 76
3.8.1 The adiabatic regime ............................ 77
3.8.2 Locally nonadiabatic regime near the bifurcation point ......... 77
3.9 Conclusions .................................. 79
4 Pathways of activated escape in periodically modulated systems 83
4.1 Introduction .................................. 83
4.2 Escape of a periodically modulated system ................. 90
4.2.1 Dynamics near the periodic states ..................... 92
4.2.2 Most probable escape paths ........................ 93
4.3 Prehistory Probability Distribution near the basin boundary ....... 94
4.3.1 Transition probability density ....................... 95
4.3.2 General expression for the PPD near qb(t) ................ 96
4.4 Adiabatic regime near the basin boundary ................. 98
4.4.1 Weak distortion, 9 << 1 ........................... 99
4.4.2 Strong distortion, 0 >> 1 .......................... 101
4.5 Nonadiabatic regime near the basin boundary ............... 104
4.6 The PPD inside the attraction basin .................... 107
4.7 Results for a model system .......................... 111
4.8 Conclusions .................................. 115
5 Conclusions 117
APPENDICES 121
A Variational equations for the escape problem 122
A.1 Escape in systems with a slow variable ................... 124
B Reduced equation of motion for slow driving 127
C Reduced equation of motion for fast driving 131
D Distribution matching for dynamically weak modulation 136
E Nonadiabatic corrections for slow modulation 140
BIBLIOGRAPHY 146

vi

1.1

1.2

1.3

1.4

2.1

LIST OF FIGURES

(Reproduced from Ref. [7]) The phase of the reflected signal from a driven
Josephson junction exhibits hysteresis as a function of the drive cur-
rent. The two coexisting metastable states correspond to the super-
conducting and dissipative states of the junction. ...........

(Reproduced from Ref. [24]) (a) A three-dimensional double-well optical
trap created by two laser beams. (b) A time series of the zit—coordinate
of a small silica particle in the trap clearly Shows switching between
the two metastable states. See also Ref. [134] for details. .......

(Reproduced from Ref. [40]) For T = 0, magnetization of a Single-domain
magnet switches when the applied field reaches certain value HEW; for
T > 0 the switching field st is random, with a distribution around
a value, which is generally different from ng- The figure shows ex-
perimentally measured distributions of switching fields as a function of
e = 1 - st/ng for several temperatures ................

(Reproduced from Ref. [65]) A cross-sectional schematic of the microme-
chanical torsional oscillator with electrical connections and measure-
ment circuitry (not to scale). When driving is weak, the system
response is linear; for stronger driving there emerge two coexisting
metastable states [106] (circles) ......................

(a) An oscillating potential barrier. In the limit of slow modulation, the
stable and unstable periodic states qa and qb are the instantaneous
positions of the potential minimum and barrier top, respectively. (b)
For slow modulation, when the modulation amplitude A is close to
its adiabatic bifurcational value A‘c‘d, the states qa,b(t) come close to

each other once per period. (c) As A further increases beyond Agd,
the barrier of U disappears for a portion of the modulation period, but
the system may still have coexisting periodic states qa,b(t). AS seen in
((1), they become skewed compared to the adiabatic picture, to avoid

crossing. In the critical range, the form of qa,b(t) is model-independent.

2.2 Nonadiabatic stable and unstable states Qa('r) and Qb(7') = —Qa(—T)

for slow modulation as given by the equation dQ/dT = C(Q, n, 7') for
77 = 0.2. The functions Qa’bfr) are strongly asymmetric, in contrast to
the adiabatic states (2.19) which are even functions of T. .......

vii

9

22

2.3

2.4

2.5

2.6

2.7

The stable and unstable states, qa(t) and qb(t), close to the bifurcation

point. For wptr >> 1 the states are close to each other throughout the
modulation period. The figure refers to a one-dimensional overdamped
particle in a potential U(q,t) = ;]-q — 313q3 — Aq cos(wpt) for (Ac —

A) /Ac z 0.01. The modulation is comparatively slow, wptso) = 1, but
for chosen A the relaxation time becomes long, wptr z 9.8 .......

The activation energy R vs. 17 cc Ac — A for slow driving, wptr << 1. The

thick solid line Shows the numerical solution of Eq. (2.28). The dashed
line is the adiabatic activation energy (2.32), Rad cc (7] — 1)3/2. The
thin solid line shows the corrected adiabatic activation energy Rad+6R
It is close to the numerical result for 17 2 3. The correction 6R diverges
at the adiabatic bifurcation point 77 = 1. ................

The activation energy if = ——Dan on a logarithmic and linear scale

(inset) vs. 7) oc AC — A for slow modulation, wptr << 1. Thick solid
lines Show the numerical solution of the variational problem (2.28). It
describes the crossover between different scaling regions. The thin solid
line shows the adiabatic scaling for large n, R oc 776 with 6 = 3 / 2. The
full result of the adiabatic approximation is shown by the dashed line.
The dash-dot line Shows the nonadiabatic result (2.37) that applies for
7] << 1; here R oc 775 with 5 = 2. .....................

The critical amplitude Ac as a function of the modulation frequency tap

for the system (2.39). Numerical results are shown by thick solid lines.
The dashed line shows the linear in w F nonadiabatic correction to AC
described by Eq. (2.16). The thin solid line in the inset describes a
correction obtained from the self-consistent local analysis, Eq. (2.41).
The dash-dot line describes the high-frequency asymptotic that follows
from Eq. (2.42) ...............................

The activation energy of escape B vs. modulation amplitude A on the

logarithmic and linear (inset) scales for a Brownian particle in a mod-
ulated potential (2.39). The values of w F are indicated on each panel.
The thick solid lines show the results of the numerical solution of the
variational problem for R. The dashed lines for (up = 01,025 Show
the adiabatic approximation, whereas for w F = 0.5, 1.0 they Show the
approximation of effectively fast oscillations: in both cases the scaling
exponent is 5 = 3 / 2 (for top = 0.1 this asymptotic scaling is Shown by
the thin solid line). The dash-dot lines show the 6 = 2 scaling (2.37).
The dots Show the results of numerical simulations of Eq. (2.39).

2.8 The scaled probability density of the dwell time pdw(t) = pdw(t)7'p ob-

tained by numerical simulations of a Brownian particle in a modulated
potential, Eq. (2.39). The parameters are A = 0.1,D = 0.05,wp =
0.25 (solid line). The dashed line Shows the exponential fit of the en-
velope with decrement WTF = 0.008. ..................

viii

24

32

33

37

40

42

3.1

3.2

3.3

3.4

3.5

(a) An oscillating potential barrier. In the limit of slow modulation, the
stable and unstable periodic states qa and 9b are the instantaneous
positions of the potential minimum and barrier top, respectively. The
instantaneous escape rate is characterized by the current at an “ob-
servation point” located at a sufficiently large distance Q from qb. (b)
The dependence of the prefactor V in the period-average escape rate
W- : Vexp(—R/ D) on the modulation amplitude A (schematically).
For A —-> 0, V is given by the Kramers theory. In regions II and III
escape is synchronized and V 0: Dl/z, where D is the noise intensity.
In region III, close to the critical point Ac where the metastable state
disappears, the prefactor scales as V oc (Ac — A)"1. In region IV
V oc (Ac — A)”2 is independent of D ...................

(a) Pulses of escape current in the adiabatic approximation as functions
of time scaled by the relaxation time. With increasing parameter 0 the
pulses change from Gaussian to strongly asymmetric. (b) The same
pulses as functions of time scaled by the modulation period, (b = wpt.

The prefactor V in the average escape rate W (3.45). The results refer to
the model (3.52) with top = 0.1 and describe escape in the regime of
strong synchronization, where V oc Dl/Z. The solid line for small A
shows the scaling V o< A’1/2. The solid lines for small 5A = A —— Ac in
the main figure and in the inset Show the scaling (3.51). The dashed
line shows the result of the numerical solution of Eq. (3.22). The
squares and crosses Show the results of Monte Carlo simulations for

R/ D = 5 and R/ D = 6, respectively. ..................

The prefactor V in the average escape rate W (3.45) close to the bifurcation
point A = Ac. The results refer to the model (3.52) with tap = 1.
The squares and crosses Show the results of Monte Carlo simulations
for R/ D = 4 and R/ D = 5, respectively. The solid line Shows the

asymptotics V = [flaflbll/2/27r o< (Ac — A)1/2. .............

Different regions of escape behavior in modulated overdamped systems

depending on the modulation frequency cap and amplitude A; A”) is
the relaxation time in the absence of modulation. The smeared bound-
aries between the regions are shown by dashed lines. The bold solid
line indicates the bifurcational amplitude where the metastable state
disappears. The shaded region below it indicates the range where the
activation energy of escape R g D. The transition between the regions
of exponentially strong and nonexponential synchronization occurs for

oooooooooooooooooooooooooooooooo

48

68

76

78

4.1

4.2

4.3

4.4

4.5

Activated motion leading to detection of an escaped particle at point q f
at time t f (schematically). Panels (a) and (b) illustrate escape from a
static and a periodically modulated potential well, respectively. In the
latter case qa(t) and qb(t) are the periodic stable state and the basin

boundary. The trajectories qégzu) (n = 1,. . . , —2) are the periodically
repeated most probable escape paths. The four major stages of motion
A, B, C, and D on the way to qf are discussed in the text. ......

The prehistory probability density (PPD) ph(q,t|qf,tf) and its contour
plot for a noise-driven overdamped system with equation of motion
(j = q2 — 0.25 + Acos wpt + f(t), where f(t) is white noise of intensity
D. The parameters are A = 0.7, wp = 2, D = 0.01, qf = 0.8,
t f = (TF/2)(mod TF), where T}? = 27r/wF is the modulation period.
The shadowing (color code on line) corresponds to the 4 regions of the
height of the distribution separated by the values ph = 0.5, 2, 7.

The reduced position of the maximum (left) and the reduced width (right)
of the PPD (4.23) in the adiabatic regime as a function of the reduced
time llbm(tf — t). .............................

The positions of the maxima of the PPD ph(q, thf, t f) in Fig. 4.2, which
show the most probable paths followed by the system in escape. The
data of simulations are shown by full squares where the PPD has one
peak and by crosses where two peaks are well resolved. Solid lines

Show periodically repeated most probable escape paths qégflt) for the
model (4.1), (4.37). Dashed lines Show the basin boundary qb(t) and
the attractor qa(t). ............................
Cross-sections of the PPD ph(q, th f, t f) shown in Fig. 4.2 as functions of
the coordinate q for the time t f —t = 0.37'F, 0.5TF, 7p, 27F. The results
refer to the model system (4.1), (4.37) with A = 0.7, (12}: = 2, D = 0.01,
qf = 0.8, tf = 0.57p. The point (qf, tf) is close to the expected
maximum of the distribution behind the basin boundary. Solid lines
Show the expression (4.36). Squares Show the results of simulations.

86

89

103

113

114

Chapter 1

Introduction

Noise—activated processes are at root of many physical, chemical, biological and, ar-
guably, social phenomena: nucleation, diffusion in solids, chemical reactions, protein
folding, and epidemics are examples. One of the most important activated processes is
escape, in which a noise-driven system leaves the basin of attraction to its metastable
state. It is important to understand how activated escape occurs, particularly in sys-
tems away from thermal equilibrium. Full understanding would include a description
of the underlying system dynamics and escape probability. Owing to its exponential
sensitivity to the system parameters, the escape probability provides an important
means of characterizing a system. In particular, it determines the lifetime of the
metastable state.

The notion of escape is meaningful when it refers to a rare event: the lifetime of
a metastable state Should be large compared to all other characteristic times in the
system, such as its relaxation time or a period of external modulation. In this case,
the system spends most of the time fluctuating around its metastable state, and only
occasionally there occurs an unusually large fluctuation leading to escape.

The goal of the present thesis is to reveal and study universal features of

 

(I -

S‘x’h

dial.

Stat

Illt'

 

 

fluctuation-induced escape in nonequilibrium systems. In contrast to equilibrium
systems, where fluctuation probabilities are known at least in principle [1], no general
principles have been established for nonequilibrium systems. Therefore it is especially
important to find system-independent properties of fluctuation phenomena far from
equilibrium.

Fluctuation phenomena in nonequilibrium systems have been attracting much
attention in recent years, particularly because fluctuations become more important
with reduction of the system size. Of special interest in this respect are systems with
coexisting stable states. Examples include modulated Josephson junctions [2, 3, 4, 5,
6, 7] (see Fig. 1.1), current-driven nanomagnets [8, 9, 10], semiconductor lasers [11, 12,
13], tunneling systems and nanowires [14, 15, 16, 17, 18, 19], nanoelectromechanical
systems [20, 21, 22, 23] and optically trapped colloidal particles [24, 25, 26, 27] (see
also Fig. 1.2). Fluctuations in such systems lead to switching between coexisting
states, and therefore even where they are weak on average, they largely determine

the system behavior.

 

 

 

200 I I I 3 lat 7 i I if f i Tl I I I l
‘ -
a 150 4». . V I0—1.172 “A
.8 I RX" 1 0n=0.122
V “- 1 5“ A}-_ _
e. 100 ' g I Kq- '0”
8 a 5 ' Film;
(3 50‘_ I: ' | 5’ l '—
t
'C _ if ' O , 5 n
L 1 4‘ LI 1 '44 I L 1 1 a n .I .

 

0.00 0.05.B 0.10 015 la 0.20

drive current id/ l0

Figure 1.1: (Reproduced from Ref. [7]) The phase of the reflected signal from a driven
Josephson junction exhibits hysteresis as a function of the drive current. The two
coexisting metastable states correspond to the superconducting and dissipative states
of the junction.

(‘3‘.
ha

Ila,

 

(b’zzxMfl “MATH
x, [ml Half

 

Time(s)

Figure 1.2: (Reproduced from Ref. [24]) (a) A three—dimensional double—well optical
trap created by two laser beams. (b) A time series of the x~coordinate of a small
silica particle in the trap clearly shows switching between the two metastable states.
See also Ref. [134] for details.

The rate of activated escape W 0( exp(—AU/kBT) has been studied in depth for
systems in thermal equilibrium, starting with Kramers’ work [29]. The activation
energy AU displays a universal behavior near bifurcation, or critical, points where
the number of stable states of the system changes. An example is a saddle—node
bifurcation where, for a critical value nc of the control parameter 17, a stable state
and an unstable stationary state of the system merge together in phase space. For a
Brownian particle this occurs where a minimum and a saddle point of the confining
potential U (r) coalesce. Such bifurcation is an analog of a spinodal point in the mean
field theory. Here, for small |n—nc| the activation energy scales as ln—nc|£ with critical
exponent a5 = 3/ 2 [30, 31, 32, 33]. This characteristic behavior of thermal activation
has been carefully studied in superconducting tunnel systems [34, 35, 36, 37, 38] and
nanomagnets [39, 40, 41, 42, 43] (see also Fig. 1.3), among other systems.

For nonequilibrium systems escape rates can be obtained from the analysis of the

3

 

NH) __

 

 

 

 

 

 

 

0 0.001 0.002 0.003 0.004 0.005 8

Figure 1.3: (Reproduced from Ref. [40]) For T = 0, magnetization of a single-domain
magnet switches when the applied field‘reaches certain value ngi for T > O the
switching field st is random, with a distribution around a value, which is gener-
ally different from ng- The figure Shows experimentally measured distributions of
switching fields as a function of c = 1 — st/HQW for several temperatures.

system kinetics. They are also often described by an activation-type law, where the
role of temperature is played by the intensity of the external Gaussian noise D or the
total number of reacting particles [14, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56].
The form of the activation barrier R depends on the details of the kinetics. However, it
was predicted that where the system is stationary (or stationary in the rotating frame,
for high-frequency modulation), close to the bifurcational value 176 of the control
parameter 77, the activation barrier should still display a power-law dependence [56,
57, 58, 59, 61, 62, 63]. The nonequilibrium 5 = 3/2 scaling near a saddle-node
bifurcation has been observed recently in resonantly driven Josephson junctions [64]
and micromechanical resonators [65] (see also Fig. 1.4). The supercritical scaling
6 = 2 has been observed for atoms in a parametrically driven magneto-optical trap

[66].
The scaling behavior is far richer for nonequilibrium systems than for equilibrium

4

 

 

 

Vnoise dc2

Figure 1.4: (Reproduced from Ref. [65]) A cross-sectional schematic of the microme-
chanical torsional oscillator with electrical connections and measurement circuitry
(not to scale). When driving is weak, the system response is linear; for stronger
driving there emerge two coexisting metastable states [106] (circles).

ones. For periodically modulated systems, we identify new scaling regimes close
to a saddle-node bifurcation point, with crossovers between different values of the
scaling exponent. We also find a complete solution of the time-dependent problem

and calculate the instantaneous escape rate W'(t).

A basic physical feature of activated escape is that the trajectories followed in
escape form a narrow tube centered at a certain path qopt(t) in the phase space of
the system (different components of the vector q are coordinates and momenta of the
system), which is the most probable escape path (MPEP) [45, 47, 49, 52, 55, 67]. In a
general case where fluctuations are induced by Gaussian noise this path goes from the
initially occupied metastable state qa to an unstable stationary (or periodic) state qb
on the boundary of the basin of attraction to the metastable state [51]. In particular,
for a Brownian particle in a one-dimensional potential well the Optimal path goes
from the minimum of the well to the barrier top. There is a qualitative difference
between MPEPS in equilibrium and nonequilibrium systems. In equilibrium systems
the MPEP is simply related to the dynamical trajectory of the system that goes from

qb to qa in the absence of fluctuations [68]: the equations for these paths differ just

5

by the Sign of the friction force. This property holds not only for the MPEP, but
for most probable trajectories qopt(t]q) for fluctuations from qa to any state q in its
attraction basin [69].

A generic feature of nonequilibrium systems is lack of detailed balance. As a
consequence, there is no Simple relation between qopt(t]q) and the dynamical tra-
jectory from q to qa. This limits [70] the applicability of fluctuation theorems
[71, 72, 73, 74, 75]. Moreover, the pattern of optimal paths has observable and
hidden singularities [76] related to nonintegrability of the equations of motion for the
optimal paths [47, 52, 77, 78, 79].

Optimal fluctuational paths can be observed in experiment. This can be done by
accumulating paths along which the system arrives to a vicinity of a given state q and
studying the distribution of these paths backward in time from the instant of arrival,
the prehistory probability distribution (PPD) [80]. It should peak at qut (th). For
a one-dimensional system in equilibrium — a semiconductor laser — such a peak has
indeed been observed [81].

A different situation happens if the system is modulated in time with period of
order of its relaxation time tr. As we Show, in this case the PPD peak is narrow not
only in phase space, but also in time. Moreover, as a consequence of nonequilibrium
modulation, the PPD may display several peaks inside the attraction basin.

The thesis is organized as follows. In Chapter 2 we discuss scaling of the activation
energy of a periodically modulated system close to a saddle-node bifurcation point.
In Chapter 3 we present a complete solution of the Kramers’ problem for periodically
modulated systems. In Chapter 4 we explore the dynamics of escape trajectories and

find the prehistory probability density for such systems. Chapter 5 concludes.

Chapter 2

Scaling and crossovers in activated

escape near a bifurcation point

2. 1 Introduction

The barrier for escape from a metastable state is reduced when the system is close
to a bifurcation (critical, or spinodal) point where the state disappears. For systems
that display hysteresis such a bifurcation point corresponds to the switching point on
the hysteresis loop. The idea of bringing the system close to the bifurcation point
[30] has been used in studying activated switching in Josephson junctions [34, 35, 36,
37], where it has become a standard technique for determining the critical current.
This idea is also used in studies of activated magnetization reversals in nanomagnets
[39, 41, 82].

Experiments on nanomagnets and Josephson junctions are often performed by
ramping the control parameter (magnetic field or current) and measuring time dis-
tribution of escape events [30]. In interpreting the data it is usually assumed that,

for sufficiently slow ramp rates, the system remains quasistationary. In this approxi-

7

a

(Ci

mation the barrier height, i.e., the activation energy of a transition R, usually scales
with the control parameter 17, measured from its critical (bifurcational) value 176 = 0,
as 173/2. The 173/2 behavior near a termination point 17,; = O of a metastable state de-
scribes the vanishing of the meanfield free-energy barrier with the control parameter
17 [83]-

Scaling of R near a bifurcation point is related to slowing down of one of the
motions [84], i.e., the onset of a “soft mode”. The relaxation time of the system tr
diverges as the control parameter 1; —+ 0. Therefore if 17 depends on time, even where
this dependence is slow the assumption of quasistationary may become inapplicable

for small 17.

In this Chapter (see also Refs. [62, 85]) a theory of activated transitions is devel-
oped for periodically modulated systems. In such systems the notion of a stable state
is well-defined irrespective of the modulation rate, and the applicability of the qua-
sistationary approximation can be carefully studied. It turns out that, unexpectedly,
near a critical point this approximation breaks down even where the relaxation time

tr is still much smaller than the driving period T}: = 21r/wp.

We show that an interplay between the critical slowing down and the slowness
of time-dependent modulation leads to a rich scaling behavior of the transition rate
and to crossovers between different scaling regions. This behavior near a bifurcation
point is system-independent and has no counterparts in stationary systems. We find
three regions in which the activation energy scales as R o< 175. As the parameters
change, for example with the increase of the modulation frequency cap, the critical
exponent E varies from 3/ 2 to 2 and then again to 3/ 2. Numerical calculations and

Monte Carlo simulations for a model system agree with our predictions.
Activated transitions in periodically driven systems were investigated earlier in

8

 

A1

 

 

various contexts [2, 6, 54, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96], stochastic resonance
and diffusion in modulated ratchets being recent examples [97, 98, 99, 100]. In this
Chapter we study the previously unexplored region of driving amplitudes close to

critical and reveal the universality that emerges.

 

.\ (a) (b) ‘. /"- /‘

U(q.t)
/

 

 

U(qit)

 

 

 

 

 

 

Figure 2.1: (a) An oscillating potential barrier. In the limit of slow modulation, the
stable and unstable periodic states (1a and qb are the instantaneous positions of the
potential minimum and barrier top, respectively. (b) For slow modulation, when the
modulation amplitude A is close to its adiabatic bifurcational value Agd, the states
qa,b(t) come close to each other once per period. (c) AS A further increases beyond

A‘c‘d, the barrier of U disappears for a portion of the modulation period, but the
system may still have coexisting periodic states qa’b(t). As seen in ((1), they become
skewed compared to the adiabatic picture, to avoid crossing. In the critical range,
the form of qa,b(t) is model-independent.

A qualitative picture of motion near a bifurcation point can be obtained if one
thinks of the system as a particle in a potential U (q, t) that oscillates in time with

period TF1 see Fig. 2.1 (a). Such particle has periodic stable and unstable states,

9

qa(t) and qb(t). In the adiabatic limit ”F -—> 0 they lie at the minimum and local
maximum of the potential in Fig. 2.1 (a). As the modulation amplitude A increases,
the states become close to each other for a portion of the period TF, see Fig. 2.1 (b).
The barrier height reaches its minimum during this time, and this is when the system
is most likely to escape from the potential well. The driving amplitude Agd for which

the barrier disappears in the limit 0) F ——+ 0 determines the adiabatic bifurcation point.

However, for nonzero cap, as A approaches Agd the periodic states qa,b(t) become
distorted to avoid crossing and may coexist even where the barrier has completely
disappeared for a portion of a period, see Fig. 2.1 (c,d). The adiabatic approximation

becomes inapplicable for such modulation.

The parameter range where adiabaticity is broken can be estimated by noticing
that the adiabatic relaxation time t, (i) is a function of the instantaneous modu-
lation phase 40 = wpt, and (ii) sharply increases near the bifurcation point. As a
consequence, tr sharply increases when 03 approaches the value where (10,1) are at their
closest, because this corresponds to approaching the bifurcation point. The quasis-
tationary (adiabatic) approximation requires that latr/éltl << 1. It is this condition
that limits the range of adiabaticity, rather than a much less restrictive condition
ter << 1.

In the nonadiabatic region, a sufficiently large fluctuation is still required to move
the system away from the stable periodic state. For A 2 Agd, there emerges the new
scaling of the activation energy R. The control parameter is now 17 o< Ac — A, where
Ac is the “true” bifurcation value of the modulation amplitude where the states qa(t)

and qb (t) coalesce.

In the limit wptr >> 1 the behavior near a bifurcation point is in some sense

simpler. In this case qa(t) and qb(t) come close to each other everywhere on the

10

cycle, not just for a part of the period. The motion of the system in the vicinity of
(Ia,b(t) is oscillations with a slowly varying amplitude. The amplitude change can be
described by averaging the complete dynamics over the period. It is then mapped
onto motion in an effectively stationary potential. Not surprisingly, the scaling of the
escape activation energy R with the distance to the bifurcation point is the same as

for stationary systems.

In Sec. 2.2 and Appendix A we provide a general formulation of the problem
of activated escape in periodically modulated systems driven by Gaussian noise. In
Sec. 2.3.1 and Appendix B we discuss the dynamics near a bifurcation point in the
adiabatic limit wptr -+ 0. In Sec. 2.3.2 we consider the strongly nonadiabatic dynam—
ics that emerges where still wptr << 1. In Sec. 2.3.3 and Appendix C the dynamics
near a bifurcation point is described in the parameter range where the field becomes
effectively fast-oscillating, i.e. wptr >> 1, even though the relaxation time in the ab-
sence of modulation 15,”) may be g l/wp. The connection between the nonadiabatic
local theory of See. 2.3.2 and the theory of Sec. 2.3.3 is discussed in Sec. 2.3.4. In
Sec. 2.4 the activation energy is explicitly evaluated in the three regions discussed
in Sec. 2.3, and the scaling laws for the activation energy R o< (Ac — A)E in these
regions are obtained. The scaling crossovers are discussed. We also find nonadiabatic
corrections to the escape rate in the adiabatic region. In Sec. 2.5 we consider a pe-
riodically modulated Brownian particle. Numerical results for the activation energy

are compared to the results of Monte Carlo simulations and to the predictions of

Sec. 2.4. Sec. 2.6 contains concluding remarks.

11

2.2 Activated escape: general formulation

We will adopt a phenomenological approach in which a multidimensional system with

dynamical variables q(t) is described by the Langevin equation
('1 = K(q; At) + f0), K(q; A,t + TF) = K(q; At). (2-1)

The function K is periodic in time, with the modulation period TF 2 21r/wp; A
is a control parameter that characterizes the modulation strength. For example, in
the case of an overdamped particle in a potential U0(q) modulated by an additive

periodic force F(t), the vector K becomes
Kfq; Aat) = -VU0(Q) + F(t) (2-2)

(here and below, V E B/Oq). In this case A = max IF] is the modulation amplitude
(note that the force F(t) = F(t + 7F) does not have to be Sinusoidal).

The function f (t) in Eq. (2.1) is zero—mean Gaussian noise with correlation matrix

901.10 — t’) = <fi(t)fj(t,)>- (2-3)

The characteristic noise intensity D can be defined as the maximal value of the power

Spectrum,

D = max @nn(W), (Du-7,7,(LU) : /dteiwtf.pnyn(t). (2.4)

For noise from thermal source D is o< kBT. The noise intensity D is the smallest
parameter of the theory. Smallness of D leads to the rate of noise-induced escape W
being much smaller than tfl and (Up.

In the absence of noise, Eq. (2.1) may have different periodic solutions qper,
which can be stable (attractors), unstable (repellers), or hyperbolic (saddles). We

are interested in the parameter range where one of the stable periodic solutions

12

qa(t) = qa(t + TF) comes close to a saddle-type periodic solution qb(t) with the same
period (period 1, for concreteness). For slow modulation, these states are sketched in
Fig. 2.1. They merge together at the saddle-node bifurcation point A = Ac. In what
follows we will assume that A is close to the critical value AC.

Escape from a metastable state qa(t) occurs as a result of a large fluctuation. The
fluctuational force f (t) has to overcome the restoring force K and drive the system
away from the basin of attraction to qa(t) [e.g., away from the potential well in
Fig. 2.1 (a)]. We will assume that the required force f (t) is much larger than the
typical noise amplitude o< Dl/Z.

The motion of the system during escape is random. However, different trajectories
have exponentially different probabilities. The system is most likely to move along
a particular trajectory called the optimal path qopt(t) [68]. It is determined by the
most probable noise realization fopt(t). In the case of a periodically modulated 1D
system driven by stationary Gaussian noise, a way to find the optimal paths was
discussed earlier [94]. We now briefly outline a generalization of the formulation
to multidimensional systems, following the arguments in Ref. [51] (more details are
provided in Appendix A).

For a stationary Gaussian noise, the probability density of realizations of f (t) is

given by the functional (cf. Ref. [101])

7’lf(t)] = eXP(-R0[f(t)l/D)a (2-5)

where R0 is quadratic in f,

Rolf] = g [#1074th1 — t’m-(t’). (2.6)

The matrix F is the inverse of rpm-(t — t’ ) / D,
fdt’Fij-(t — t')n,9jk(t' — t”) = D6,k6(t — t”). (2.7)

13

"'l
I

I 'I”

 

\L

a,“

 

f1

(‘1

Ill

0i

l0

lld

of

()])1

5N1

 

 

We are interested in noise realizations that lead to escape, and therefore f (t)
largely exceeds its root—mean-square value. From Eq. (2.5), the probabilities of such
noise realizations are exponentially small and exponentially strongly depend on the
form of f (t) AS a consequence, escape trajectories should form a narrow ”tube”
centered at an optimal path fopt(t) that maximizes ’P[f(t)], i.e., minimizes R0[f(t)].
The minimum of R0 should be found with the constraints that (i) the system and
noise trajectories, qopt (t) and fopt(t), are interrelated by the equation of motion (2.1),
(ii) the path qopt(t) starts in the vicinity of the stable state qa(t) and ends behind or
on the boundary of the basin of attraction to qa(t), and (iii) the force fopt(t) is equal
to zero before the escape event happens and becomes equal to zero once the system
has escaped, so that, as fopt(t) decays, it does not drag the system back to the basin
of attraction to qa.

As explained in Appendix A, these conditions lead to boundary conditions for

optimal paths of the form

qa(t) for t —> —oo,
goptit) ’7
qb(t) fort —-> oo,

fopt(t) —> 0 fort ——> :too. (2.8)

[Note that qopt(t) ends on the basin boundary, not on another attractor]

The variational problem for optimal paths is reduced to minimizing the functional
12a f] = Rolf} + f dt’Mt’) - [qua — Km; A. 1') - £00] (2.9)

with boundary conditions (2.8). The function Mt) is a Lagrange multiplier. The
boundary condition for it is Mt) -—> 0 for t ——> 21:00.
It follows from Eqs. (2.8), (2.9) and from the results of Appendix A that the

Optimal trajectories qopt(t), fopt(t) are instanton-like [102, 103]. The typical duration

14

of motion is given by the relaxation time of the system tr and the noise correlation
time tcorr. In stationary systems instantons are translationally invariant with respect
to time, i.e., if qopt(t),fopt(t) is a solution, then qopt(t + T),fopt(t + ’r) is also a
solution, for an arbitrary 7'. In contrast, in periodically modulated systems this is
true only for 1' = TF. The instantons are synchronized by the modulation: generally

there is one instanton per period that would provide a global minimum to R.

From Eq. (2.5), we obtain for the escape probability

W oc exp(—R/D), R = min’R[q, f]. (2.10)

The activation energy R is equal to the value of the functional R0[fopt] calculated for

the optimal noise trajectory for escape.

For small noise intensity, the escape rate II'V << wp. It periodically depends on
time. However, in the small-D limit this dependence is seen only in the prefactor
[92, 93, 54]. Here we are interested in the exponent, which gives the period-averaged

escape rate W. It is equal to the probability of escape over the time T}? divided by
TF

In the general case, the variational problem for the activation energy can be solved
only numerically. Therefore it is particularly important to find model-independent
properties of R. So far they have been found for comparatively weak modulation,
where it was shown that R has a term linear in the modulation amplitude [94]. In
this Chapter we analyze the activation energy R in a previously unexplored region
near a bifurcation point and Show that R displays a nontrivial scaling behavior in

this region.

15

2.3 Dynamics near a bifurcation point

The dynamics near a saddle-node bifurcation point has universal features related to
the occurrence of a slow variable, or a “soft mode” [84]. For periodically modulated
systems, closeness to the bifurcation point in the parameter space usually implies that
the merging states are close to each other in phase space throughout the modulation
period.

If the modulation frequency (up is small compared to the reciprocal relaxation
time in the absence of modulation 1/150), there emerges a situation where the stable
and unstable states come close to each other only for a portion of a period. During
this time, the system behaves as if it were close to a bifurcation point. Then it is
possible to single out a slow variable that controls the system dynamics. Escape from
a metastable state is most probable when q(,,(t) and q7,(t) are closest to each other.

On the other hand, if the modulation frequency (.11sz / ti"), near the bifurcation
point the states qa(t) and qb(t) are close to each other throughout the modulation
period. Then escape can happen with nearly same probability at any modulation
phase, i.e., synchronization of escape by the modulation is essentially lost.

Because the dynamics near a bifurcation point is slow, the system filters out
high-frequency components of the noise. As a result, the noise becomes effectively
6-correlated (we will not consider here the situation where the noise power spectrum
has singular features at high frequencies). The reduction to one slow variable driven
by white noise can be done directly in the equations of motion. For slow modulation
(wptgo) << 1), this reduction is local in time (see Appendix B), otherwise it has to
be done globally over the cycle (see Appendix C). Alternatively, the dimensionality
reduction can be done directly in the variational problem for the optimal escape path

(see Appendix A).

16

2.3.1 The adiabatic approximation

In the limit of Slow modulation, where the period of the field T1: is large compared

(0)

to the system relaxation time tr , a convenient starting point of the analysis is the
adiabatic approximation. The adiabatic periodic states of the system qg‘g, are given

by the equation
Kean/1.1) = o, (2.11)

which is obtained by disregarding q and the noise f in the equation of motion (2.1).

(1: ad

The adiabatic stable state (attractor) qf] __ q2d(t) is the solution qper for which

the real parts of the eigenvalues of the matrix [1,

My = (010/391) ,

are all negative. These eigenvalues give the “instantaneous” relaxation rates, for a
given phase of the modulation (,0 = wpt. For the periodic adiabatic saddle-type state
q2d(t) one of the eigenvalues of [1 has a positive real part.

In the adiabatic approximation, the saddle-node bifurcation occurs in the following
way. At the critical value of the control parameter A = Afid, the periodic trajectories
q3d(t) and qg‘d(t) given by (2.11) merge, but it happens only once per period. One
can picture it by looking at Fig. 2.1 (b) and imagining that the states qa(t) and
qb(t) touch each other. We set the corresponding instant of time equal to t = 0 (or
t = 11.770), i.e., we assume that q3d(0) = q2d(0) for A = Afld. Additionally, we set
qzdw) = qzdm) = 0.

At the adiabatic bifurcation point A = AfldJ = 0 one of the eigenvalues ,ul of the
matrix [1 is equal to zero, whereas all other eigenvalues 717)] have negative real parts.

The adiabatic approximation means that -Re 71,-)1 >> (up, or equivalently, that the

17

1111

relaxation time max [lRe 71,->1|‘1] is small compared to TF. This relaxation time is
typically of the order of tin).

We now write the dynamical variables q in the basis of the right eigenvectors of
the matrix 11 at the bifurcation point and expand K in the equation of motion (2.1)
in a series in q, t, and A — Afid. As shown in Appendix B, the motion described by
the variable ql is much slower than the motion described by the variables q,:>1. Over
the time ~ ti”) they “adjust” to the value of ql, i.e., they follow (11 adiabatically. The

variable (11 is the soft mode. It satisfies the equation of motion

(11 = K((11;A,t) + f1“), (2-12)

K = or]? + [36Aad —- (172(wpt)2.

Here (5Aad = A — Aild; the parameters (1,3,7 are expressed in terms of the derivatives
of K at the adiabatic bifurcation point and are given by Eqs. (B3), (B4).

The stable and unstable adiabatic periodic states in the absence of noise exist for
(113 (5Aad < 0. For concreteness and without loss of generality we set a > 0. For small

prtl the adiabatic states can be found by setting K((ql)2flb; A, t) = 0. This gives
ad (1
((11)a b _ _q:(20ta )-1
where 1;?“ is the instantaneous adiabatic relaxation time. It is given by l8K/8q1|_1

evaluated for (11 = (qflzd,

tad:

]_1/2 (2.13)

:- [— afidAad-i- (a'ywpt)2
ad
,1)

[the explicit expression for ((11): (t) is given by Eq. (B5)].

Validity of the adiabatic approximation

The applicability of the adiabatic approximation requires not only that 021M?d << 1,

but also [fit‘i‘d/Bt] << 1. If this latter condition is not met, the system cannot follow

18

the modulation without delay, its state depends on how the parameters were varying
in time. From Eq. (2.13), near the bifurcation point the time dependence of ti‘d is

pronounced, so that
max [mild/8t] = 373/2'ywp/I736Aad| >> wthr‘d.
Therefore the inequality Iatffd/Bt] << 1 is much stronger than wptixl << 1. It holds if
titd << t7, t7 = (aywprl/z, (2.14)

i.e., WF <<I136Aad]/'y.

The inequality (2.14) restricts the range of validity of the adiabatic approximation.
It is the time t, rather than New that imposes an upper bound on the adiabatic
relaxation time of a periodically driven system where this approximation still applies.
This time sets a new dynamical time scale. It fully characterizes the dynamics beyond

the adiabatic limit in the range wptr << 1, as discussed in Sec. 2.3.2.

2.3.2 Locally nonadiabatic regime

As A approaches A‘C‘d, the criterion (2.14) is violated. The periodic stable and unstable
states are pressed against each other. Since they cannot cross, they become distorted,
as shown in Fig. 2.1 (d). Ultimately they merge, but along a line rather than at a
point. From Eq. (2.12) one can see that this is just a straight line, which is described

by the equation

quit) = vat- (2-15)
Eqs. (2.12), (2.15) define the new nonadiabatic bifurcational value of the modu-

lation amplitude for slow driving,

A21 = Agd + a-lw. (2.16)

19

The corrections to A21 of higher order in 1.0 F are discussed in Sec. 2.3.4.

The change of the bifurcation amplitude described by Eq. (2.16) is somewhat
similar to the effect of bifurcation delay [104, 105], which occurs when the bifurcation
parameter is ramped through the bifurcation point. Here, however, we consider a
periodically driven system; the bifurcation is not delayed in time, rather it is the
bifurcational modulation amplitude that is changed compared to the adiabatic limit.

It is convenient to change in Eq. (2.12) to dimensionless variables Q : athl and

7' = t / t7, and to introduce the control parameter
n = awn-1M2} — A). (2.17)

We note that 17 is the only parameter of a slowly driven system in the critical
region. It describes both adiabatic and nonadiabatic behavior and gives the reduced
distance to the bifurcation point. Interestingly, the distance along the modulation-
amplitude axis A21 -A is scaled by the modulation frequency cap; both these quantities
are supposed to be small, but their ratio 17 can be arbitrary.

The equation of motion for the reduced variable takes the form

1Q_

dT —G(C.?,11,T)+f(r)a G=Q2—12+1-n. (2.18)

The function f(T) = (7wF)—lf1(t7'r) in Eq. (2.18) describes reduced noise. It is
effectively 6—correlated on the slow-time scale, ( f (1') f (1" )) = 2D6(T-— T'), as explained
in Appendix B. The noise intensity D is given by Eq. (B7).

The stable and unstable states Qa,b(7') are given by the equation dQ/dT = G. In
the absence of noise this equation has symmetry Q —-» —Q, T —> —T. As a consequence,

the stable and unstable states are antisymmetric, Q(,(T) = —Qa(—T). Therefore it

suffices to find only Qa(7').

20

We start with the adiabatic approximation. It applies for 17 >> 1. The adiabatic
stable and unstable states in the reduced variables are given by the equation G = 0

and have the form
1/2
03?, = a [T2 + (11— n2] . (2.19)

Each of these states is symmetric with respect to 'r = 0, where they are closest to
each other. The adiabatic bifurcation point is 17 = 1, which corresponds to A = A2“.
The region 17 g 1 is nonadiabatic, and 17 = 0 (or A = A?) is the nonadiabatic
bifurcation point for slow driving. At this point Qa_,b('r) merge into the straight line
QC(T) = 7"
Close to the nonadiabatic bifurcation point, where 17 << 1, one can find Qa,b("')
by perturbation theory in the whole range —00 < 1' < I111 17]” 2 /2. The linearized
(150

equation for the difference 6Q(1‘) = (20(1) — 7' has a form 7,;- = 21' 6Q+17. By solving

it we obtain

0.0) = —Qb(-T) z r —- n11... 6111672412- (220)

In the region of large negative 1' the function Qa(1') = —Qb(—1') has a simple
form Qa(1') z —7’ — 17(21')"1. The states Qa and Q), are closest to each other, with
separation ~ 17, in the range |1| < |ln 17]” 2 / 2. The interstate separation decreases as
17 approaches the bifurcational value 17 = 0. At the same time, the range of 1' where
62,, (1') and Qb(1') stay close to each other increases with decreasing 17.

As 1' increases beyond a: ] In 17] 1/ 2 / 2, there occurs a sharp crossover from the nearly
linear in 1' solution for Qa(1') (2.20) to the adiabatic solution (2.19) Qa oc —'r. The
functions Qa,b(r) for a specific value of 17 are shown in Fig. 2.2.

The interval of the real time |t| S tr 2 t7] ln 7]]1/2, where the states QaJ) are nearly

linear in t, should be much smaller than l/wp in order for Eq. (2.12) to apply. This

21

 

 

 

 

 

Figure 2.2: Nonadiabatic stable and unstable states Qa(1') and Qb('r) = —Qa(—1)
for slow modulation as given by the equation dQ/ d1 = G(Q,17,1') for 17 = 0.2. The
functions Qa,b(7') are strongly asymmetric, in contrast to the adiabatic states (2.19)
which are even functions of 1'.

imposes a restriction on 17,
17 >> exp(—C]a]"7/wF), C N 1. (2.21)

For smaller 17 oc [A2] — A] the local approximation, where the coefficients are expanded
about the adiabatic bifurcation point, no longer applies. The relaxation time tr
becomes comparable to the modulation period, and the behavior of the system during
the whole cycle becomes important. It follows from Eq. (2.21), however, that for low
frequencies the local approximation is extremely good.

On the whole, the locally nonadiabatic regime is limited in 17 by the condition
175.1 and by Eq. (2.21). The width of the amplitude range A21 — A imposed by
the first condition linearly increases with the field frequency, in the approximation
(2.16). Therefore locally-nonadiabatic critical behavior is more pronounced for higher
frequencies. However, the appropriate frequency range is limited from above by the
condition (2.21). For higher (up there should occur a crossover to a fully nonlocal

picture, which is discussed in the next section.

22

2.3.3 Fast—oscillating field

Sufficiently close to the “true” critical value of the modulation amplitude Ac, the
relaxation time of the system becomes large compared to the modulation period even

:0) g 1 far from the bifurcation point. The inequality wptr >> 1 defines the

if wpt
third region, in addition to the adiabatic and locally nonadiabatic, where we could
analyze the dynamics near a bifurcation point. The analysis of this region is simplified
by the fact that here the modulating field is effectively fast-oscillating.

For wptr >> 1, near the bifurcation point the periodic stable and unstable states
qa(t) and qb(t) stay close to each other throughout the cycle, see Fig. 2.3. For A = Ac,
they coalesce into a periodic critical cycle qc(t) = qc(t + 1F). When A is close to AC
and q is close to qc, we can simplify the equations of motion (2.1) by expanding the

function K in (Sq = q — qc and 6A = A — Ac (cf. Ref. [84]),

aq = [1(5q + éwq. V)2K + Mai/4K + f(t). (2.22)

Here, as before, 71,:j E 71,: j(t) 2: BKi/Oqj, but all derivatives of K are 110w evaluated for
A = AC and q = qC-(t). Therefore all coefficients in Eq. (2.22) are periodic functions
of time.

If initially the system is close to qc(t), its distance from qc(t) will oscillate with
frequency WF and with an amplitude that slowly varies over the period TF- This
amplitude is a slow variable, QS‘"(t). The equation for Qsm(t) can be obtained by an
appropriate averaging method explained in Appendix C. After rescaling to dimen-
sionless coordinate Q oc Qsm and time 1' oc t, see Eq. (C.11), this equation takes a

form which is similar to Eq. (2.18),

i9 = 002.11) + f(r). - (2.23)

(11'

G=Q2-n. 11=fi’(Ac—A)

23

 

 

 

 

 

Figure 2.3: The stable and unstable states, qa(t) and qb(t), close to the bifurcation
point. For wptr >> 1 the states are close to each other throughout the modulation
period. The figure refers to a one-dimensional overdamped particle in a potential
U(q,t) = zliq ~— §q3 —— Aq cos(wpt) for (Ac —— A)/Ac z 0.01. The modulation is com-

paratively slow, wptgo) = 1, but for chosen A the relaxation time becomes long,
wFtr z 9.8.

[in contrast to Eq. (2.18), the function G here is independent of time]. The coefficient
6’ is given by Eq. (C7).

The parameter 17 in Eq. (2.23) is the scaled distance to the bifurcation point. The
stationary states QaJ, = 27:171/2 exist for 17 > 0. They merge for 17 = 0. The noise
f (1') is effectively white on the time scale that largely exceeds 01;} and the noise

correlation time too". Its intensity D is given by Eq. (0.12).

The results of this Section and Appendix C refer to the case wptr >> 1, but
arbitrary wptfpo). Therefore the problem is different from the standard problem of
slow motion in a fast-oscillating field [106], where of interest is the smooth term in
the oscillating coordinate. In contrast, here we are interested in the slowly varying
oscillation amplitude. If wptgo) >> 1, a transition to slow and fast variables can be

made already in the original equation of motion (2.1), by separating q into slow and

fast oscillating parts. The equation for the slow part near the bifurcation point will

24

again have the form (2.23), but the expressions (C.7) for 01', 6’ will be simplified; in

particular, the factor [$11 in Eq. (C.7) will be equal to one.

2.3.4 Connection to the locally nonadiabatic regime

Eq. (2.22) allows us to look from a different perspective at the locally nonadiabatic
regime that emerges for wptr << 1. In contrast to the approach of Sec. 2.3.2, where
the starting point was the adiabatic approximation, here we will assume that A is
close to the true bifurcational value of the amplitude Ac and that q(t) is close to the

critical cycle qc(t), at least for a part of the period TF-

For wptr << 1, one can think of a local in time description of the dynamics near the
cycle qc(t). From Eq. (2.22), this dynamics is determined by the eigenvalues 71,,(t)
of the matrix 71(15). In contrast to the analysis of Sec. 2.3.1, we consider here the
matrix [1 calculated for the critical cycle qc(t) rather than the two similar matrices

calculated separately for the adiabatic stable and unstable states.

For much of the driving period the real parts of 71,,(t) are all large, [Re 71”] ~
l/tEO) >> cup. Then, when the system is in the stable state, it follows the field
adiabatically. The adiabaticity is broken where one of the eigenvalues, say 711 (t), goes
through zero. As we will see, at this time the stable and unstable states are most

close to each other. We set the time when it happens equal to zero, i.e., 711(0) = 0.

For small [t] < TF the analysis of the system dynamics is in many respects
similar to that in Sec. 2.3.2 and Appendix C. First, 6q(t) in Eq. (2.22) is written
as 2,, (Mum/(0), where eV(0) are the right eigenvectors of the matrix 71(0). The
component 6q1 of (Sq along the eigenvector e1(0) of [1(0) will be the slow variable, or

the soft mode.

25

The matrix [1(t) can be expanded about t = 0 for small It],

1(1) 5 12(0) +1401. (224)

where the time derivative is taken for t = 0. This derivative is small, its matrix
elements on the eigenvectors éu(0),eur(0) are lam/1] ~ wp/tgo) << (t]()))_2 [here,
62,/(0) are the left eigenvectors of the matrix 71(0)].

With (2.24), Eq. (2.22) can be solved for the “fast” components dqu>1. Over a
short time ~ 40) they approach their quasistationary values for given 6q1. Those are
small, of order 6q%,6A,6q1wFt, and follow 6q1 adiabatically. Noise-induced fluctua-
tions of 6gu>1 about the quasistationary values are also small for small noise intensity.
Therefore the effect of 6q,,>1 on the dynamics of 6q1 can be disregarded.

The equation of motion for 6q1 has a form of the Riccati equation with a random

force,

641 7.115.“ + aéqf +1114 + 11(1). (225)

f1(t) = é1(0) ' f(t): 111 = 51(0) ° 13(0)91(0)-
Here, a = (1/2)(e1(0) - V)2K1, fl = BKl/aA, with K1 2 é1(0) - K being now the
component of K in the direction e1(0). All derivatives of K are calculated on the
critical cycle qc(t) for t = 0. Because I 7'11] is small, relaxation of 6q1 is slow compared
to relaxation of 6q,,>1, for typical It] << TF-
Eq. (2.25) describes the stable and unstable states of the original equation of
motion (2.1) in the region It] << TF- It is seen that these states, (rim); and (5q1)b,

exist provided
711 > 0, 0166A < 0. (2.26)

In this range Eq. (2.25) is equivalent to Eq. (2.18). This can be seen if, on the one

hand, Eq. (2.18) is written for the deviation 6Q = Q — 1' of Q from its value on the

26

critical cycle QC = 1', and on the other hand, in Eq. (2.25) one changes to scaled
variables (IQ = o'(2/7'11)1/26q1 and 1 = (711/2)1/2t. The control parameter 17 in (2.18)

becomes

1, = —2a,371;16A. (2.27)

The analysis of Eq. (2.18) then applies to Eq. (2.25). In particular, the statement
in the beginning of this subsection that the stable and unstable states are at their
closest for t = 0 is an immediate consequence of the explicit expression for these

states (2.20).

There is an important difference between this approach and the approach of
Sec. 2.3.2. Because here we do not start from the adiabatic approximation, we for-
mally have not specified how small is the difference between the critical amplitude
Ac and its adiabatic value Agd. In Eq. (2.16) we only obtained the linear in wptgo)
term in Ac -— A3“. In general, AC — Agd has also higher-order terms. They can be
obtained by taking into account the dependence of the coeflicients a, B, 7 in Eq. (8.3)
on A, which was previously disregarded. This is illustrated for a particular model in
Sec. 2.5. It is for the renormalized critical amplitude, i.e., for the control parameter
given by Eq. (2.27), that there holds the exponential limit (2.21) on the range where
the local nonadiabatic approximation applies. The inequality (2.21) indicates that,
for small frequency, the critical amplitude found from the local theory is exponen-

tially close to the exact Ac. This is confirmed by numerical calculations for a model

discussed in Sec. 2.5.

27

2.4 Activation energy of escape

It follows from the results of Sec. 2.3 that, near a bifurcation point, a periodically
driven system has a soft mode Q, and the noise that drives this mode is effectively
white. The equation of motion is of the form dQ/ (11' = G + f (T) (2.18), where the
function G is given by G = Q2 + 1 — 17 — 1'2 for wptr << 1 [cf. Eq. (2.18)] and
C = Q2 — 17 for wptr >> 1 [cf. Eq. (2.23)]. The intensity D of the noise f(T) has the
form (B.7) and (C12) in these two cases.

For a white-noise driven system, the variational problem (2.9), (2.10) of calculating

the period—averaged rate of activated escape W can be written in the form

— I.

IV = const x exp(wR/D), R = min R[Q], (2.28)

2
R=/dTL(Q,Ell—Clg-,T), L=%(%g——G>.

(cf. Appendix A). In contrast. to the standard formulation [55], the function G
here may depend 011 time 1 and is not time-periodic, in the actual range of T. The
minimization is carried out over the paths Q(1') that start at the stable state Qa(1)
for 1' —> —00 and end at the unstable state Q7,(1') for 1' ——> +00. The non-stationarity
emerges for slow modulation, where wFtr << 1, and is related to the assumptions that
(i) escape is most likely to occur during a portion of the period where the states me
are close, and (ii) the duration of motion along the optimal escape path Qopt(7') is
much less than the modulation period.

We have solved the variational problem using the Hamilton equations of motion

for Q and P = 3L/0(dQ/d7'),

a—Q=2P+G,
01'

aP_ as

5; .. 4350' (229)

We then verified the assumptions made in obtaining Eqs. (2.28), (2.29).

28

Equations (2.29) were solved both analytically and numerically. 111 numerical
calculations, we chose the initial conditions on the optimal path close to Qa(1) with
large but finite negative 1'. In this range Eqs. (2.29) can be linearized in Q —— Qa(1').

On the solution that goes away from Qa the momentum P is linear in Q — Qa,

P C“ [Q — Q(1(7)]/0’g(7') (230)
03(7) : 2/_T (17" exp [11,/:- (IT’IQa(T”)]

We used the shooting method: we sought such initial Q — Qa,('r) for given 1' that the
trajectory approaches Q7,(1') for large 1, cf. Ref. [94].

Numerical results for the activation energy in the whole range of slow driving,
where wptr << 1, are Shown below in Figs. 2.4, 2.5 on linear and logarithmic scales,
respectively. Note that the activation energy is a function of a single control parameter
17 oc Ac — A, and in this sense the results are universal, i.e. system-independent. In the
rest of this Section we discuss analytical results and compare them with the numerical

results.

2.4.1 Activation energy in the adiabatic approximation

The adiabatic regime applies when the driving is slow, wptr << 1, and the system
is sufficiently far from the bifurcation point, so that tr << t7 [cf Eq. (2.14)] or
equivalently |6Aad| E IAifd — A] > wpl'y/fil. Respectively, the dimensionless control
parameter 17 cc Ac - A is large, 17 —— 1 >> 1, [of Eq. (2.17); we note that the actual
parameter in the adiabatic range is not 17 but 17 — 1]. In this case we expect that
escape occurs when the adiabatic states (2.19) are closest to each other, i.e., for

1' = 0. Then, in the first approximation, the term 1'2 in the function G in Eq. (2.18)

29

can be disregarded, and G becomes
Gad = Q2 + 1 — 17. (2.31)

The solution of Eq. (2.29) with G of the form (2.31) and with boundary conditions

Q(T) ——> 212(1) —- I)”2 for 1' —-> 27:00 is well known. It is an instanton (kink)
033.0. 10) = (11 -1)1/2tanh[(n — 111/20 — 10)]

centered at an arbitrary 1'0.

The characteristic duration of motion along the path Qggth') in dimensionless time
is A1 ~ (17—1)_1/2, which corresponds to At ~ tr in real time. Since A1 << (17—1)1/2,
the term 12 in the function G [Eq. (2.18)] can be disregarded compared to 17— 1, which

justifies replacing G with Gad as long as |1'0| << (17 — 1)1/2.

ad

The activation energy (2.28) calculated along the path QOpt is

Rad = 3(1) — 1)3/2 oc (A2“ — A)3/2. (2.32)

This equation shows that the activation energy of escape scales with the distance to

the bifurcation point as (AC — A)é with g = 3/ 2 in the adiabatic region.

2.4.2 Nonadiabatic correction to the activation energy

We now consider the lowest—order correction to the adiabatic activation energy. Two
factors have to be taken into account. First is that, because of the nonzero duration
of motion along the escape path AT, the equilibrium states QZ‘fbh) change, which was
disregarded in the analysis of Sec. 2.4.1. However, the corresponding correction to R
is exponentially small. Indeed, if we consider R as a function of the end point Q on

the optimal path, we have [mi/am 2 IF], where P is the momentum on the optimal

30

path. For an instantonic solution, the momentum goes to zero as Qopt. —> me, see

Eq. (2.30). Therefore a small change of Q01, affects the activation energy very weakly.

The major nonadiabatic correction to R comes from the time-dependent term in
G 1' Gad -— 1'2 [cf. Eq. (2.18)]. This term lifts the time invariance of the instanton
Qopt('r,1'0) with respect to 10,

2

To first order in 1' , i.e., to lowest order in (17 — 1)_1, the correction 6R can be

found from Eq. (2.28) by integrating the term 1'2 along the zeroth order trajectory

Q3313 (T1 T0)1

 

- dead (1'. T0)
6R 2 min/d1" Opt 1 1’2. (2.33)
7'0 617'
(we used that nggt / (11' = —Ga‘d). Minimization here is done over 1'0, the center of

the instanton. It is necessary because R is the absolute minimum of the functional
7%.
A direct calculation shows that the minimum of (SR is reached for 1'0 = 0, and

2
(SR: 7%(17— 1)‘1/2. (2.34)

The correction 6R rapidly falls off with increasing 17 — 1. On the other hand, as
17 decreases and becomes z 1, the term 6R increases very fast, which indicates a

breakdown of the adiabatic theory in this region.

The analytical results in the adiabatic region are compared with the numerical
solution for the activation energy R in Fig. 2.4. The corrected adiabatic theory works
well in the whole range where the control parameter 172,3, but for smaller 17 nonadi-
abatic effects are significant and have to be taken into account in a nonperturbative

way.

31

 

 

 

 

 

Figure 2.4: The activation energy R vs. 17 o< Ac — A for slow driving, wptr << 1. The
thick solid line shows the numerical solution of Eq. (2.28). The dashed line is the

adiabatic activation energy (2.32), R“? o< (17 ~— 1)3/2. The thin solid line shows the
corrected adiabatic activation energy Rad + (SR. It is close to the numerical result for
17 Z 3. The correction 6R diverges at the adiabatic bifurcation point 17 = 1.

2.4.3 Activation energy in the locally nonadiabatic region

Standard techniques do not allow to solve equations (2.29) for the optimal path an-
alytically in the general case 17 ~ 1. This is because the function G in Eq. (2.29)
explicitly depends on time 1'. However, a solution can be obtained close to the bifur-
cation point, where 17 cc AC — A is small, but not exponentially small, cf. Eq. (2.21).

Unusually for an instanton-type problem, and because of the strong time depen-
dence of G, the optimal path can be found by 1111601121119 the equations of motion

(2.29) about the critical state QC = 1'. This gives
6Q = 21'6Q —— 17+ 2P, R = —2PT, (2.35)

where 6Q E Q — T. The solution of these equations with boundary conditions Q(T) —>

Qa,b(7') for 1' —> 27:00 is

2

Q0pt(T) ___ T _ 77/0 dT,[1 _ fie—TIQ]8T2~TI ,
Popt(T) = ne_T2/\/§1 (2'36)

32

where we took into account the explicit form of Qa,b(t) (2.20).

It is seen from Eq. (2.36) that the momentum on the ‘optimal path Popt has a shape
of a Gaussian pulse centered at 1' = 0, with width ~ 1. The coordinate Qopt(1') over
the dimensionless time 1 ~ 1 switches between the equilibrium states Qa,b- The
typical duration of motion in real time is t7.

From Eqs. (2.36), the nonadiabatic activation energy of escape for wptr << 1 is
Rnonad : (”/8)1/2T]2 CX (AC _ A)2 (237)

Here, the critical amplitude Ac is given by Eq. (2.16), to first order in (up.

 

 

 

 

 

 

 

 

 

Figure 2.5: The activation energy R = —Dan on a logarithmic and linear scale
(inset) vs. 17 oc Ac — A for slow modulation, wptr << 1. Thick solid lines Show
the numerical solution of the variational problem (2.28). It describes the crossover
between different scaling regions. The thin solid line shows the adiabatic scaling for
large 17, R oc 175 with 5 = 3/2. The full result of the adiabatic approximation is
shown by the dashed line. The dash-dot line shows the nonadiabatic result (2.37)
that applies for 17 << 1; here R oc 176 with 5 = 2.

It is seen from Eq. (2.37) that, in the locally nonadiabatic region, the activation
energy again displays scaling behavior, R oc (Ac — A)€. But the scaling exponent is
5 = 2, it differs from the adiabatic exponent 5 = 3/ 2 (2.32) that has been known
for stationary systems. This is a result of the complicated nonadiabatic dynamics

associated with avoided crossing of the stable and unstable states, cf. Fig. 2.1. The

33

onset of this scaling behavior is the central result of this Chapter.

The predicted 5 = 2 scaling is compared with the result of the numerical calcula-
tion in Fig. 2.5. The analytical and numerical results are in quantitative agreement

in the whole range 1752.

2.4.4 Activation energy for wptr >> 1

It was shown in Sec. 2.3.3 that, sufliciently close to a bifurcation point, there holds a
condition wptr >> 1, even where the modulation frequency is less than the relaxation
rate far away from the bifurcation point, wptgo) g 1. Finding the activation energy
of escape R ‘—_= Rfa‘St for wptr >> 1 is formally similar to that in the adiabatic approx-
imation. The only difference is that 17 — 1 in Eq. (2.31) should be now replaced by 17.

This gives

Rfast = (473)-173/2 oc (A, — 103/2. (2.38)

Both the coeflicient [3’ that relates 17 to Ac — A [see Eqs. (2.23) and (C.7)] and
the noise intensity D (C.12) depend on the arbitrary initial time ti. It enters the
weighting factor 1111(t, ti) which was used in obtaining the equation of motion for the

slow variable (2.23). A straightforward analysis shows that ti drops out from the ratio

6’ 3/ 2 / D, which gives the escape rate W oc exp(—Rfas'c / D).

Eq. (2.38) shows that the activation energy displays scaling behavior with the
distance to the bifurcation point in the range wptr >> 1. The scaling exponent is

5 = 3/ 2, as in the adiabatic case.

34

 

 

 

 

fm
fast
rapj
regi
Uri

film

I
H

 

T0

 

811].

I
min

 

2.4.5 Scaling crossovers near a critical point

Eqs. (2.32), (2.37), and (2.38) show the onset of three regions where the activation
energy of escape displays scaling dependence on the modulation amplitude, R oc
R oc (AC —— A)€. The adiabatic and locally nonadiabatic regions emerge only if the
modulation frequency is slow compared to the relaxation rate far from the bifurcation
point, 101715.”) << 1. In this case, as seen from Fig. 2.5, as the bifurcation point is
approached, the system displays first the adiabatic scaling 5 = 3 / 2, which for smaller
Ac — A goes over into the scaling 5 = 2. As the bifurcation point A; is approached
even closer, there emerges the fast-oscillating regime where 5 = 3 / 2 again.

The widths of the regions of different scaling strongly depend 011 the modulation
frequency. For small 027st?) << 1 the range of amplitudes where motion is effectively
fast oscillating, wptr >> 1, is exponentially narrow. However this range increases very
rapidly with increasing cap. The particular way in which the widths of different scaling
regions vary with (up depends on the system dynamics, as illustrated in Sec. 2.5.

(0)

Ultimately, for wptr 21, the regime of effectively fast oscillations becomes the only

observable regime near a bifurcation point.

2.5 Scaling crossovers for a model system

To test the occurrence of three scaling regions and the scaling crossovers, we have
studied activated escape for a model system, an overdamped Brownian particle in a

modulated potential well. It is described by the Langevin equation

_0U(q, t)

(1: 7(1— + f(t). (f(t)f(t')> = 20505 — tI),
U(q,t) = —%q3 + iq — choswpt. (2.39)

35

The Shape of the potential U (q, t) is shown schematically in Fig. 2.1 (a). In the
absence of modulation, A = 0, the system has a metastable state at the bottom of the
potential well, qa = ——1 / 2, and an unstable equilibrium at the barrier top, qb = 1 / 2.
The relaxation time is 19” = 1/U”(qa) = 1. In the presence of modulation, the states
gay, oscillate in time. AS the modulation amplitude A increases to the critical value
Ac (the saddle-node bifurcation), the states merge, and then, for A > Ac, disappear.

The frequency dependence of the critical amplitude Ac is shown in Fig. 2.6. In
the limit wp = 0 we have AC E Agd = 1 / 4. The linear in cup correction to Ac can
be obtained from Eq. (2.16) by noticing that the adiabatic bifurcational value of the

coordinate is qffd = 0, and the adiabatic bifurcation occurs for t = 0 (or equivalently,

t = 1117:). Near the adiabatic bifurcation point we have
1
q: q2 + and —— 5.41.2212 + f(t), (2.40)

with 6Aad = A — Agd. This equation will have the same form as Eq. (2.12) if we
replace the factor A in AW%t2 with Agd = 1/4 [as it was done in Eq. (2.12)].

From Eq. (2.40) it follows that, for the model under consideration, the parameters
in Eq. (2.12) are a = fl = 1,7 = (Aé‘d/2)1/2 = 273/2. Therefore, from Eq. (2.16), to
first order in tap the critical amplitude is A21 = 1/4 + 2-3/ 2112p. It is Shown in the
main part of Fig. 2.6 with the dashed line.

As discussed in Sec. 2.3.4, the local nonadiabatic theory allows us to find higher-
order terms in the critical amplitude. This is done by noticing that the critical state
qc(t) into which the stable and unstable states merge at the bifurcation point is linear
in t for small t, i.e., qc(t) — qffd oc t. By substituting this solution into Eq. (2.40)

(without noise) we obtain
2 2 ”2
A612: 1+wF+wF (cup—+2) /4, wptr << 1. (2.41)

36

 

 

 

Fig
for
lim
Th
loo,-
tlla

fro

do

()1

8117

in

 

 

 

 

 

Figure 2.6: The critical amplitude Ac as a function of the modulation frequency 02 F
for the system (2.39). Numerical results are shown by thick solid lines. The dashed
line shows the linear in tap nonadiabatic correction to Ac described by Eq. (2.16).
The thin solid line in the inset describes a correction obtained from the self-consistent
local analysis, Eq. (2.41). The dash-dot line describes the high—frequency asymptotic
that follows from Eq. (2.42).

(0)

This equation is in good agreement with the numerical data for (aptr 5.0.4, as seen
from the inset in Fig. 2.6. The difference between the numerical and analytical Ac

decreases exponentially fast with decreasing wp.

We also evaluated for slow driving the time derivative of the eigenvalue 711 = 2qc(t)
on the critical cycle qc(t). For the model (2.40) the stable and unstable states are at
their closest for t = 0. Eqs. (2.17), (2.27) show that, at this time, 7'11 = (2Ac)1/2wp.

This value agrees with the numerical values of 7'11 to better than 10% for 1.12 F < 0.5.

In the high frequency limit, wptgo) >> 1, the motion of the system (2.39) is a

superposition of slow motion and fast oscillations at frequency (.077. To lowest order

1

in w; we have (1 a: Q + (A/wp)sinwpt. The equation for the slow variable Q

37

becomes

0 = 02 — — + ——,— + f(t). (2.42)

It shows that, for large 017:, we have Ac 3 wF/\/2. This is in good agreement with
numerical data in Fig. 2.6 for 027232.

In the intermediate range, wpt]0)Sl, the motion may not be separated into slow
and fast-oscillating for weak modulation, but the separation becomes possible near
a critical point, wptr >> 1. Here, the coefficients in the equation of motion for the

slow variable and the effective noise intensity (C.7), (C.12) are nonlocal and had to

be evaluated numerically as functions of (up.

2.5. 1 Activation energy

For a periodically modulated overdamped Brownian particle described by Eq. (2.39),
the activation energy Of escape R can be found from the variational problem (2.9),
(2.10) or, equivalently, (2.28) [94]. The variables Q, P, 1', and the function G in the
Lagrangian L (2.28) and the Hamilton equations (2.29) should be changed to q, p, t,
and —8U (q, t) / 6q, respectively. As explained in Sec. 2.2, there is one optimal path
per modulation period. The initial condition for the momentum p on the Optimal
path is given by Eq. (2.30), with Qa(1') replaced by qa(t) [the expression for 0?, can
be further simplified taking into account the periodicity of qa(t)]. Then Eqs. (2.29)
can be solved numerically.

The Obtained activation energy R as function of the modulation amplitude for
four characteristic values of cap is shown in Fig. 2.7. The solid lines on this plot
correspond to the results of the numerical solution of the variational problem. The

(0) _

dashed lines in the panels for 01F = 01,025 (we remind that tr — 1) Show the

38

adiabatic approximation,

3/2
Rad : min [U(qb(t)) — U(qa(t))] = g (i — A) .

The dash-dot lines on all panels show the locally nonadiabatic approximation near

the bifurcation point, which gives

nonad _ 71. 1/2 2 1/2 2
R _(g) (’71—) (AC—A).

111 plotting this expression we used the values of Ac and 7'11 found numerically (they
were very close to the analytical expressions given above).
Finally, the dashed lines in Fig. 2.7 in the panels for 02p = 0.5, 1 Show scaling for

the effectively fast-oscillating regime near the bifurcation point, with

Rfast ___ gifll3/2g (AC _ A)3/2 .

The coefficients 13’ and D as given by Eqs. (C.7), (C.12) were obtained numerically.

It is seen from Fig. 2.7 that, for small (121:, the adiabatic approximation applies
over a broad region Of driving amplitudes. Near the bifurcation point it gives scaling
R oc (Ac — A)6 with 5 = 3/2 (cf. the panel for (up = 0.1). However, close to the
bifurcation point this scaling does not work. Instead there emerges the nonadiabatic
dynamic scaling with 5 = 2. For top = 0.1 the range of the nonadiabatic scaling is
comparatively narrow.

As the frequency increases, the amplitude range characterized by the 5 = 2-scaling
dramatically increases. For 1.0;: = 0.25 this is practically the only scaling seen near
the bifurcation point.

With further increase of 1127:, close to the bifurcation point there emerges a region
of the fast-oscillation scaling R 0< (Ac — A)é where again 5 = 3/ 2. The panel for

wp = 0.5 shows a crossover from the scaling 5 = 3/ 2 very close to the bifurcation

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

-6 ' 33 ' 0 ' .31 32 '
|n[(Ac-A)/Ac] |n[(Ac-A)/AJ

Figure 2.7: The activation energy of escape R vs. modulation amplitude A on the
logarithmic and linear (inset) scales for a Brownian particle in a modulated potential
(2.39). The values of wp are indicated on each panel. The thick solid lines Show
the results of the numerical solution of the variational problem for R. The dashed
lines for “F = 0.1, 0.25 show the adiabatic approximation, whereas for (up = 0.5, 1.0
they show the approximation of effectively fast oscillations: in both cases the scaling
exponent is 5 = 3/ 2 (for WF = 0.1 this asymptotic scaling is shown by the thin solid
line). The dash-dot lines Show the 5 = 2 scaling (2.37). The dots show the results of
numerical simulations of Eq. (2.39).

40

point to the scaling 5 = 2 further away from Ac. Note that the frequency 0) F = 0.5 is
neither small nor large, and therefore there is a noticeable difference in the coeflicients
at (Ac — A)2 Obtained from the full variational problem for R and from the locally
nonadiabatic theory near the bifurcation point.

When wFt9)Zl we do not expect to see scaling for either adiabatic or locally
nonadiabatic regime. The only scaling to be expected near the bifurcation point is
the fast-oscillation one, with 5 = 3/ 2. This is indeed seen in the panel for wp = 1 in

Fig. 2.7. We note that the global fast-oscillation approximation (2.42) does not apply

for wpto) = 1.

2.5.2 Simulations

An additional test of the results can be obtained by directly simulating the Brownian
dynamics described by Eq. (2.39). We conducted such simulations using the second
order integration scheme of stochastic differential equations developed in Ref. [107].
As a result of the simulations we obtained the probability distribution of the dwell
time in the metastable state Pdw (t). It gives the probability density (over time) for
a system prepared at t = 0 close to the attractor to stay in the basin of attraction
until time t and leave at that time.

In practice we calculated pdw(t) by detecting the system at time t at a point (7 that
lied well beyond, but not too far from, the oscillating boundary qb(t). It is Simply
related to the time-dependent escape probability W(t), which gives the probability
current from the attraction basin at time t if the system was in the stable state at

tzopa

t
pdw(t) = we) exp [— /0 dt’ W(t’)] . (2.43)

41

 

5d,“)

"

‘ A = 0.1
0'4‘ 1 mp: 0.25

‘ 0:005
02 \

 

 

 

 

C:

 

0 U ] fill/Ullflflllflfimm {an m:- -
10

0 20 30
[117:

Figure 2.8: The scaled probability density of the dwell time 75d“,(t) = pdw(t)1'p ob—
tained by numerical simulations of a Brownian particle in a modulated potential,
Eq. (2.39). The parameters are A = 0.1,D = 0.05,wp = 0.25 (solid line). The
dashed line shows the exponential fit of the envelope with decrement WTF = 0.008.

The average escape rate is given by the mean dwell time,

_ oo --1
IV = ]/ dttpdw(t)] . (2.44)
0

We studied small noise intensities so that W << 027:; then W was independent of the
position of the “observation” point (7.

In most simulations the system was prepared initially at the stable state qa(t); we
found that W was independent of the initial state provided it was close to qa(t).

The dwell-time distribution for a particular set of parameter values in Eq. (2.39)
is Shown in Fig. 2.8. The data refer to modulation at a comparatively low frequency
and with comparatively small amplitude. The function pdw(t) is strongly modulated
in time, with period TF- This means that escape events are strongly synchronized by
the modulation, in agreement with the analytical results for W (t) obtained for the
same model in Ref. [92].

The average escape rate W was found from the data of the type shown in Fig. 2.8

42

by calculating the mean dwell time (2.44) and also from an exponential fit of the
envelope of pdw(t). These two approaches gave the same result. For each set of
A,wp, D we observed ~ 105 escape events. Then D was changed. The activation
energy of escape was found from an for 2—4 values of D. We tested that it was

independent of D in the range R/D26.

The data of simulations are shown in Fig. 2.7 by dots. For all parameter values
they are in excellent agreement with the results of the numerical solution of the

variational problem (2.9).

2.6 Conclusions

I11 conclusion, we have identified three regions near a bifurcation point where the
activation energy of escape displays scaling behavior as a function of the amplitude
of periodic modulation. The main results refer to slow modulation, where wptr << 1.
We show the emergence of nonadiabatic behavior in this region. The nonadiabaticity
leads to a crossover from the scaling with exponent 5 = 3/ 2, previously found for
stationary systems, to a new dynamical scaling with 5 = 2. The 5 = 2 region first
emerges near the bifurcation point and then expands with increasing modulation
frequency. With further increase of WF the crossover 5 = 2 to 5 = 3/ 2 can be
observed. Again, the effectively fast-oscillating region with 5 = 3/2-scaling emerges
first near the bifurcation point. Even though the widths of the regions of different
scaling depend on the parameters of a system, the phenomenon of scaling crossovers

should be universal.

The onset of the 5 = 2—scaling is a consequence of the slowing down of motion near

a bifurcation point. The adiabatic relaxation time of the system t1- strongly depends

43

on the distance to the bifurcation point, tr oc (Ac — A)_1/2. The nonadiabatic scaling
0
[ )11/2

emerges where tr becomes ~ [TFt . For smaller [AC — A] the dependence of tr on
Ac — A becomes weak, while i,- still largely exceeds the modulation period TF- This is
associated with avoided crossing of the stable and unstable states, which occurs with

decreasing Ac — A as these states are pressed against each other when the system

approaches the adiabatic bifurcation point, see Fig. 2.2.

Both in the adiabatic regime and the nonadiabatic regime for wptr << 1 escape is
most likely to occur while the equilibrium states are close to each other. The escape
rate is therefore determined by the behavior of the system for a small portion of the

modulation period, i.e., locally in time.

The regime of effectively fast oscillations near the bifurcation point emerges for
wptr >> 1. It can arise even where the modulation period 1p exceeds the relaxation
time far from the bifurcation point. In this regime the motion is controlled by a slow
variable, but the dynamics of this variable is no longer determined by local (in time)
characteristics. The relevant characteristics are obtained by averaging the appropriate
parameters along the critical periodic trajectory of the system into which the stable

and unstable periodic cycles merge at the bifurcation point.

We have developed a general formulation of escape of periodically modulated
systems driven by colored Gaussian noise. Near a bifurcation point slow motion of
the system effectively filters out high-frequency components of the noise spectrum and
makes the noise effectively white. From the theoretical point of view it is interesting
that, in the locally-nonadiabatic regime of 5 = 2 scaling, the instanton-like optimal

escape path can be found from linear equations of motion.

We expect that the new 5 = 2 scaling and the scaling crossovers can be seen in

various systems. Examples are modulated Josephson junctions, nanomagnets, and

44

optically trapped Brownian particles, where escape in the presence of modulation has

been already studied experimentally, albeit in different regimes.

Chapter 3

Noise-induced escape of
periodically modulated systems:

From weak to strong modulation

3. 1 Introduction

The most frequently used types of nonequilibrium modulation are ramping of a con-
trol parameter and periodic modulation. Ramping is usually done slowly, and it is
assumed that the system remains quasistationary [30]. Periodic modulation is con-
ceptually simpler as periodic metastable states are well-defined irrespective of the
modulation frequency. However, a theory of the escape rate is more complicated, be-
cause the system is away from thermal equilibrium [95]. Recently significant attention

was attracted also to escape over a randomly fluctuating barrier [108, 109].

In the present Chapter (see also Refs. [110, 111]) we study periodically modulated

systems and extend to them the analysis of the escape rate done by Kramers for

46

systems in thermal equilibrium [29]. Our approach gives the full time—dependent
escape rate W(t) as well as the period-average rate W = Vexp(—R/ D), where R is
the activation energy of escape and D is the noise intensity, D = kBT for thermal
noise. We find W (t) for an arbitra1y modulation amplitude A and an arbitrary
interrelation between the modulation frequency (up and the relaxation time of the
system tr. We show that the prefactor V depends on A strongly and nonmonotonically.
It displays scaling behavior near the bifurcational modulation amplitude Ac where

the metastable state of the system disappears.

In the absence of modulation escape can happen at any time with the same prob-
ability density. For systems in thermal equilibrium, the activation energy equals
the free energy barrier height. The prefactor V is given by the generalized attempt

frequency and does not depend on D for not too small damping [29].

Even a comparatively weak driving can exponentially strongly modulate the es-
cape rate leading to strong escape synchronization for small D [94]. This is easy to
see for a Brownian particle in a slowly modulated potential well, see Fig. 3.1(a): it
is most likely to escape once per modulation period when the barrier is at its lowest.
The “time window” for escape is diffusion-broadened. Therefore the period-average

escape rate is W oc D1/2 [92].

For an overdamped Brownian particle a transition, with increasing modulation
amplitude A, from the D-independent prefactor in the absence of modulation [29] to
V 0c D”2 was discussed in Ref. [92]. However, the results were limited to compara-
tively weak modulation, region I in Fig. 3.1(b). The range of intermediate modulation,
region II, was discussed in Refs. [54, 93, 112]. However, the obtained prefactor in the
period-average rate diverges for A —-> 0. The approach of Ref. [93, 112] is inapplicable

for slow modulation compared to the relaxation time of the system. In contrast, the

47

 

 

    

U(q.t)

 

 

 

 

50".
f"; (b) #1, ,L
.. q. 0 A A

C

 

 

Figure 3.1: (a) An oscillating potential barrier. In the limit of slow modulation, the
stable and unstable periodic states qa and qb are the instantaneous positions of the
potential minimum and barrier top, respectively. The instantaneous escape rate is
characterized by the current at an “observation point” located at a sufficiently large
distance Q from qb. (b) The dependence of the prefactor V in the period-average
escape rate W = Vexp(—R/ D) on the modulation amplitude A (schematically). For
A ——> 0, V is given by the Kramers theory. In regions II and III escape is synchronized
and V oc Dl 2, where D is the noise intensity. In region 111, close to the critical point
Ac where the metastable state disappears, the prefactor scales as V at (AC — A)“1. In

region IV V oc (Ac — A)”2 is independent of D.

technique developed in this Chapter is free from these limitations. The scaling regions
III and IV and the strongly nonmonotonic behavior of the prefactor have not been

previously identified.

To find the instantaneous escape rate W(t) we relate it, in the spirit of Kramers’
approach, to the current well behind the boundary qb(t) of the basin of attraction
to the initially occupied metastable state (q is the system coordinate). We call the
current away from the attraction basin the escape current. In stationary systems and
for the time t < W“1 the escape current is independent of the coordinate and is the
same on the basin boundary and behind it. In periodically modulated systems this

is no longer the case. A particle that crossed the boundary at one time may cross

48

it back at a later time, because the boundary itself is moving. In experiments the
position of the instantaneous basin boundary is not necessarily known. The current is
usually detected well behind the boundary, for example, close to another metastable
state. The functional form of this current is qualitatively different from that at qb(t)

calculated in Refs. [54, 93, 112].

The escape current can be obtained by relating the probability distributions of
the system p(q, t) behind the boundary (77,(t) and close to the attractor qa(t) from
which the system escapes. We do this in two steps. First, we find the general form
of the current-carrying distribution p(q, t) in the boundary layer about qb(t), where
the equation of motion of the system can be linearized. Then we match it to the
distribution inside the attraction basin but well outside the diffusion layer around
(77,(t). This distribution can be obtained in the eikonal approximation for small D. It
has singular features [47, 77, 113]. The matching is performed using these singular

features.

In Sec. 3.2 we describe the model and give the general form of the boundary-
layer distribution. In Sec. 3.3 this distribution is used to Obtain general expressions
for the instantaneous and period-average escape rate. Matching of the intrawell and
boundary-layer distributions is discussed in Sec. 3.4. In Sec. 3.5 we study the pulse
shape of synchronized escape current in different regimes. Section 3.6 provides a brief
discussion of the period-average escape rate in the regions I and II in Fig. 3.1(b). In
Sec. 3.7 we identify three different types of the scaling behavior of the prefactor V
close to the bifurcation point and find the critical exponents. In Sec. 3.8 the general
results are compared with Monte Carlo simulations for a specific system. In Sec. 3.9
we summarize the results and sketch a surprisingly rich map of different types Of

escape behavior in the plane of modulation parameters.

49

3.2 The model and the boundary layer distribu-
tion

Escape from a metastable state of a periodically modulated system is well character-
ized if the noise is weak, so that the escape rate W << $1,015». In this case, over the
relaxation time tr the periodically modulated system will most likely approach its peri-
odic metastable state (attractor) with the coordinate qa (t) = qa(t + 170) (1'1: = 21r/wp
is the modulation period). Then most likely, it will be performing small fluctuations
about qa and will “forget” the initial state (7(0). Eventually there will occur a large
fluctuation in which the system will go over the boundary qb(t) and leave the basin
of attraction, i.e. escape.

The instantaneous escape rate W(t) is characterized by the current ((7(t)) away
from the metastable state. The current has to be measured well behind the boundary
qb(t), so that the system practically does not return to the metastable state.

The probability distribution p(q, t) of a periodically modulated overdamped Brow-

nian particle is given by the Fokker-Planck equation (FPE)
ap = —a [Kemp] + 00312. (3.1)

Here, K (q, t) is the periodic force driving the particle, K (q,t) = K (q,t + 17:) E
—(9qU (q, t), where TF = 21r/wp is the modulation period and U (q, t) is the instanta-
neous potential.

The equation of motion of the particle in the absence of noise is (j = K (q, t). The
metastable state qa(t), from the vicinity of which the system escapes due to noise,
and the basin boundary qb(t) are the stable and unstable periodic solutions Of this
equation, respectively.

We will assume that the noise intensity D is small. Then in a broad time range

50

tr << t << l/W the distribution p(q,t) is nearly periodic in the basin of attraction
to qa(t). The current away from this basin, and thus the escape rate W(t), are also

periodic.

3.2.1 Motion near periodic states

The distribution p is maximal at the metastable state qa(t) and falls Off exponentially
away from it. In the presence of periodic driving it acquires singular features as
D ——> 0 [47, 77, 113], some of which have counterparts in wave fields [114], with D
playing the role of the wavelength. The singularities accumulate near qb(t). In order
to find W(t) one has to understand how they are smeared by diffusion.

In the absence of noise the motion of the system close to the periodic states

q,:(t) (i = (1,1)) is described by the equation (7 = K with K linearized in (Sq = q — q.,-(t):

61: 1.10611. 1.0) = 11.0 + 1F) 2 [0mm (32)

The evolution of (517(t) is given by (5(7(t) = 1:7(t, t’)6q(t’), where

111-(t, 1’) = exp [ft (11 111(1)] (1' = a, b). (3.3)

t,
Over the period 7F the distance (Sq decreases (for i = a) or increases (for i = b) by

the Floquet multiplier
A1,; = 1167(15 + TF, t) = exp ([177]?) , i: a, b, (3.4)

where 71,- is the period-average value Of 717(t), with 77.0 < 0, 717, > 0.
For weak noise the linearized force K can be used to find p(q, t) near qa,7,(t). Near

the metastable state qa, the distribution 7) is Gaussian [115],

[q — (110112} . (3,5,

( ) 211Daa(t) 2003.“)

51

The time-periodic variance is
2 —2 ‘1 TF —2
a, (t) = 2'11, — 1] (11115,, (t + 11, t) (3.6)
0

with i = a; in the absence of modulation 0?, = 1/]71.a|.

3.2.2 Distribution near the unstable state

The periodic distribution near the unstable state qb(t), i.e., the boundary-layer dis-
tribution, is substantially non-Gaussian. It corresponds to a periodic current away
from the attraction basin. The distribution can be found from Eq. (3.1) using the

Laplace transform, similar to the weak-driving limit [92]:

00
MM) =/0 dpe“”Q/D 1501.0, Q = q - (Ii(t)- (37)

We assume that Q is small, IQ] << mint |q7,(t) — (7a(t)]. Using the expansion K =
(7,,(1) + ,ub(t)Q, we Obtain from the FPE (3.1) a first-order equation for the Laplace
transform 75(7), 1‘.)

01/3 = item/3 + 02/0125. (3.8)

This equation can be solved by the method of characteristics, giving

130. t) = 50-1/2 exp {— [301+ Rage/2] /D}. (3.9)

In Eq. (3.9), 8 is a constant and 3(0)) is an arbitrary zero-mean periodic function,
3(4) + 211) = s(¢). They have to be found by matching p(q, t) (3.7) to the distribution
inside the attraction basin. The function 03(t) in Eq. (3.9) is given by Eq. (3.6) with
i = b, and the factor D"1/2 is singled out for convenience.

The phase of the function 5 is (7) E (77(7), t),

15(1), t) = 9F 111i!) "b(t:t’)/fllel- (3-10)

52

Here,
QF = Lap/[lb E 27171111117,

is the reduced field frequency, 1 D = (2D/ [17,)1/ 2 is the typical diffusion length, and t’

determines the initial value of (75. From Eq. (3.10), ¢(71,t + 1F) = (0(7), t) + 211'.

3.3 Instantaneous and period-average escape rate

3.3.1 General expression for the escape rate

The experimentally accessible instantaneous rate of escape from the metastable state
is characterized by the current j(q,t) from its attraction basin. We assume that
this basin lies for q < q7,(t), i.e., q(,(t) < qb(t). Then the escape current is the rate
of change of the population in the region (——Oo,q], with q lying behind the basin
boundary qb(t). AS explained in the Introduction, of interest is j (q, t) for such q that
Q =2 q -— qb(t) >> [0. Eq. (3.7) is advantageous as it immediately gives such a current.

We consider first j(q, t) not for a fixed q, but for a fixed distance Q = q — qb(t)

from the boundary,

a ab(t)+Q
j(Qat) _' —a _ dqp((I1t)
% ub(t)p(Qb(t)+Q,t)Q. (3-11)

Here we have used the F PE (3.1) and linearized K (qb(t) + Q,t) in Q; we have also
disregarded the diffusion current, which is a good approximation for Q >> 10.

The distribution p(q7,(t) + Q, t) in the expression for the current (3.11) is given
by Eqs. (3.7), (39). For Q >> 17) the term oc 712/D in Eq. (3.9) can be neglected

53

compared to pQ/ D. Changing in Eq. (3.7) to integration over :1: = pQ / D we obtain

10.1) = 111(0801/2/Omd exp[_s(¢,)/D],

(677 = ¢(.’IJD/Q,t)=QFlIl[$I£b(td,t’)]. (3.12)

Here td E td(Q, t) is given by the equation 11.7,(td, t) = lD/2Q.

It follows from Eq, (3.12) that in the whole harmonic region 7' depends 011 the
observation point Q only in terms of the delay time td. This time shows how long it
takes the system to roll down to the point Q, aQtd = —1/[71b(td)Q]. We note that
717,0) can be negative for a part of the period, leading to reversals of the instantaneous
current.

The escape rate W is given by the period—average j(q, t). The averaging can
be done for a given q or a given Q behind the boundary. The result will be the
same, since the period-average value of qu is equal to zero, and therefore the period-
average current is independent of coordinate. It is convenient to do time averaging in
Eq. (3.12) by changing from integration over time to integration over (ad with account
taken of the relation d¢d/dt = Q F717,(t). The result is independent of Q, as expected,

and reads
- 211
W = -g—;ED1/2/ dqfi exp[—s(({>)/D]. (3.13)
0

Eqs. (3.12) and (3.13) provide a complete solution Of the Kramers problem of
escape of a modulated system and reduce it to finding the function 3. It is seen
from Eqs. (3.9), (3.13) that this function has the meaning of the zero-mean periodic
part of the activation energy of escape. Eqs. (3.12) and (3.13) are similar in form
to the expressions for the instantaneous and average escape rates for comparatively
weak modulation obtained in Ref. [92]. For such modulation Is] < R, which allowed

finding .9 explicitly (see Appendix D).

54

3.3.2 Synchronization of escape

Periodic modulation may lead to an exponentially strong time dependence of the
escape probability within a period, that is, to synchronization of escape events. This
effect is determined by the parameter |sm| / D. Here sIn E minga, 5(a)) is the minimal
value of 3 reached for a certain phase (7) = (pm. By construction, sm < 0. We will be
interested in strong synchronization, when the escape current displays sharp narrow
periodic peaks as function of time.

Strong synchronization of escape requires that Ism] >> D (a more precise criterion
will be discussed below). In this case the factor exp[—s((f>d) / D] in Eq. (3.12) for j (q, t)
is a sharp function of (75d. The major contribution to the integral over :1: comes from the
range where s is close to sm and, respectively, the phase did is close to (19m. The time
dependence of j (q, t) can be found by changing in Eq. (3.12) from integration over a:
to integration over 421- Because lsm] >> D, the integrand (dz/add)exp[—x — (s/D)]
is maximal for (75d = 45m — 211k with k = 0,:l:1, . . .. For such 45d and for a given I: the

integrand is equal to

910) = QEIIL'k exp l-J3k — (Sm/Dll ,
where

961305) = (Cb—101115) €XP[(¢m + 211k)/9Fl- (3-14)

The whole integral (3.12) is determined by the sum of gk.

As a function of time, gk is maximal for ark = 1. Within any period of time from
t to t + TF the condition .17, = 1 is met only for one It, and only for one instant of
time tk. For all other k the function 137,: exp(—-a'k) is much smaller provided 51 F g 1.
We note that, if 93k = 1 for a given tk, then it follows from Eqs. (3.3), (3.14) that

21:7,“ = 1 for tk+1 = t7. + TF- This means that, as a function Of time, the escape

55

rate displays sharp periodic peaks with period TF- The shape Of the peaks will be
discussed below.

When ISm] << D, there is no exponential synchronization, and the escape current
smoothly depends 011 time. It happens in particular when modulation is weak or the

modulation frequency is large, Q F >> 1.

3.4 Matching the intrawell and boundary-layer

distributions

3.4.1 Intrawell distribution near the basin boundary

To find 7 (q, t) we match the boundary-layer distribution (3.7) to the tail of the in-
trawell distribution. The matching has to be done close to the basin boundary qb(t),
where Eq. (3.7) applies, but it is convenient to do it well inside the attraction basin,
—[q—qb(t)] >> 17), that is, outside the diffusion layer around q7,(t). Of primary interest
is the case of strong modulation, [sml >> D, since the case of weak to moderately
strong modulation was considered earlier [92].

The intrawell distribution in the region —Q >> I D [Q = q—q7,(t)] can be found, for
example, by solving the F PE (3.1) in the eikonal approximation, with the diffusion

coefficient D being a small parameter,
p(q, t) = e-SW/D, s = so + 031+... (3.15)

(here, in contrast to Ref. [110] we single out the factor D in SI explicitly).

To zeroth order in D, the equation for SO E So(q, t) can be written as

8730 = —H (8(180, q;t) . (3.16)

56

Eq. (3.16) has the form Of a Hamilton-Jacobi equation for the action SO of an auxiliary

conservative system with Hamiltonian [55]
HUD, q; t) = 112 + pK(q, t), 10 = 01,30 (3-17)

The auxiliary system is non-autonomous, its Hamiltonian is a periodic function
of time. Since we are interested in the periodic distribution p(q, t), we need to find a
periodic solution of Eq. (3.16). This can be done using the method of characteristics,

i.e., by studying Hamiltonian trajectories (q(t), p(t)) of the auxiliary system,
(7 = K + 2p, 75 = —p0qK. (3.18)

Eqs. (3.18) have two hyperbolic periodic states, (qu(t), 0) and (q7,(t), 0), where qa(t)
and qb(t) are the metastable state and the basin boundary of the original dissipative
system. We are interested in Hamiltonian trajectories (q(t), p(t)) that belong to the
unstable manifold of the periodic state (qa(t), 0). For such trajectories, action 50(q, t)
is minimal for q = qa(t), and p(q, t) is maximal, respectively. Indeed, a straightforward
calculation based 011 Eq. (3.18) shows that, for q close to qa(t), the momentum on the
unstable manifold is p = [q—qa(t)] /a(2,(t), where (12 is given by Eq. (3.6). Respectively,
SO = [q — qa(t)]2/2ag(t), in agreement with Eqs. (3.5), (3.15) for p(q, t).

To logarithmic accuracy, the escape rate is determined by the probability to reach
the basin boundary q7,(t), i.e., by the action SO (qb(t), t) [94]. The Hamiltonian tra-
jectories of the auxiliary system that form this action belong to the stable manifold
of the periodic state (qb(t),0). The trajectory (qopt(t),popt(t)), which minimizes
So (qb(t), t), approaches qb(t) asymptotically as t —> 00. This is a heteroclinic trajec-
tory of the auxiliary system, an intersection of the unstable and stable manifolds of

the states (q(,(t),0) and ((77,(t),0), respectively [47, 77, 113].

57

In the case of dissipative systems with detailed balance, including systems in
thermal equilibrium, the corresponding manifolds coincide with each other. However,
in nonequilibrium systems this is no longer true. In periodically modulated systems
there is only one heteroclinic trajectory with minimal SO per period. The coordinate
qopt(t) on this trajectory is the most probable escape path (MPEP). This is the
trajectory that the original system is most likely to follow in escape. It is physically
observable and has been seen both in experiments and simulations [94].

Close to qb(t), the Hamiltonian equations (3.18) for q(t),p(t) can be linearized

and solved. On the MPEP

popt(t) = -Qopt(t)/0§(t) = 1310, t')popt(t’). (3-19)

5000,1011) = R — cam/203(1).

where Qopt(t) = (70pt,(t) — (77,(t). The quantity R 2 SO (qopt(t), 01—1 00 is the activation
energy of escape.

Eqs. (3.19) apply to an arbitrary trajectory on the stable manifold of the state
(qb(t), 0) close to this state. The MPEP is just one such trajectory. It is determined
by the condition that it starts at (qa(t), 0) for t —> —00. This condition synchronizes
the trajectory and determines Popt (t) for a given t. It is important that the optimal
paths are periodically repeated in time with period TF- The values of popt(t) form
an infinite series. For neighboring paths they differ by the factor M l;— 1. They can be
also thought of as the values of popt(t + 11177) for the same MPEP and different It,
with popt(t + [$117) ——> 0 for k —+ 00.

From Eqs. (3.19), everywhere on the stable manifold of (qb(t), 0) the action has

the form

81.0.1) = R — Q2/2020)- (3.20)

58

It is parabolic as function of Q.

Due to nonintegrability of the dynamics with Hamiltonian (3.17), the action sur-
face SO(q,t), which gives the intrawell probability distribution (3.15), becomes flat
for small Q — Qopt [47, 77, 113]. It touches the surface S7,(q, t) 011 the MPEP. Away

from the MPEP So(q, t) > Sb(q, t). Therefore the function

pb(q, t) = p(q.t)exp[51(q. t)/Dl (3-21)

is maximal 011 the MPEP.
The prefactor of the eikonal-approximation distribution is given by the term
exp(—Sl), cf. Eq. (3.15). On the MPEP the auxiliary function 2: = exp(2S’1) obeys

the equation [1 16]

(12.: d(z 0.1K) 2 ,

where q = qopt(t), p = popt(t). The initial condition to this equation follows from the

explicit form (3.5) of p(q, t) near the stable state,

z(t) —+ 211Dac2,(t), t ——> —oo. (3.23)
Close to qb(t), from Eq. (3.22)

Z“) = DiZNgU) + Z2P;,,2t(t)l, (3-24)
where 21,2 are constants [93, 112]. This solution was found in Ref. [93, 112], but the

term o< 21 was disregarded.

3.4.2 Matching the exponents and prefactors

We are now in a position to match the functions pb(q, t) (3.21) as given by the eikonal

approximation (3.15), (3.19) and (3.24), and the boundary-layer solution (3.7). In

59

the spirit of the eikonal approximation, matching should be done in the vicinity of
the MPEP, where p7, is maximal.

The eikonal approximation applies when the Optimal escape paths qopt(t + kTF)
(k = 0,i1...) are separated by a large distance compared to the diffusion length
ID. For small noise intensity D the corresponding range incorporates much of the
harmonic region near qb(t), because the width of the latter region Aq is independent
of D and Ag >> I D for small D. Physically, l D characterizes the width of the tube of
paths along which the system moves in escape [94]. In the region —Q g l D the tubes
of escape paths overlap and the eikonal approximation no longer applies.

In contrast, the boundary layer distribution Eq. (3.7) covers the whole harmonic
region IQ] << Aq, including the diffusion-dominated region |Q] ,3 l7). Thus, for small
noise intensity D, the two distributions should overlap in a broad range I D << -Q <<
Aq inside the harmonic region but outside the diffusion-dominated layer.

We first consider the boundary-layer expression (3.7) for pb(q,t) in the region
—Q >> I D and for strong modulation, lam] > D. As we Show, pb(q, t) is maximal
for all times t provided q lies on an appropriate trajectory that belongs to the stable
manifold of qb(t).

Eq. (3.7) is simplified for —Q > ID, because the integral over p can be evaluated
by the steepest descent method. The extremum of the integrand is given by the

condition

ds

—1

— = 0.
£14)

The integrand is maximal if p = —Q/a§(t) and s is minimal for this p, i.e., (12(7), t) =

19030) + Q + 9F]?

05m and s = sm. The relation p = —Q/ 03 (t) holds for any trajectory on the stable
manifold of qb(t), cf. Eq. (3.19). Moreover, since 011 these trajectories p = p(t) =

57,-1“» t’)p(t'), it follows from Eq. (3.10) that (p(p, t) = const. Therefore if the phase

60

(75 = (75m for one instant of time, it will be equal to aim for all times.

The function p7, (3.21) calculated in the eikonal approximation is maximal on the
MPEP. Therefore 707, in the boundary-layer approximation should be maximal on the
MPEP as well. It follows from the above arguments that it will indeed be maximal

on the MPEP provided

$(Popt(t)a t) = (bm- (3.25)

This amounts to choosing the apprOpriate initial phase (the value of t’) in Eq. (3.10).
With this choice, steepest descent integration in Eq. (3.7) gives the boundary-layer

distribution near the MPEP in the form

p(q, t) = 51.0) expl-qu. tVD], (3-25)

- _ _ —1/2
81.0) = ED ”2 [03(1) + 0%» 4:. 100,340]

7

where

~

5 = 8(211D)1/2exp[(R—sm)/D], (3.21)

Siii = [(128 / d¢2l¢m '

It is seen from Eqs. (3.19), (3.26) that, with the right choice of phase, the ex-
ponents of the boundary-layer and eikonal-approximation distributions coincide with
each other along the MPEP. Moreover, the slopes 6q3b(q, t), which are determined
by popt(t), also coincide.

For the matching of the distributions to be complete, the function 87,(t) of the
boundary-layer distribution (3.26) should match on the MPEP the prefactor of the
eikonal-approximation distribution 2'1/2 (3.24). Remarkably, they indeed have the
same form, as functions of time, near qb(t). Therefore the parameters Zl and Zg

allow one to determine 8 and s”, and vice versa, 8 and s” ive Z and Z . With
m 111 g l 2

61

the appropriately defined parameters, not only the exponents but also the prefactors
of the boundary-layer and eikonal distributions fully match each other.

The function popt(t) exponentially decays with time, and therefore the term
oc pggtU) in 87, and 2-1/2 exponentially increases as t —+ 00. This leads to quali-
tatively different forms of the prefactor for different frequencies and strengths of the
modulation. For insfi, << l%/O’g(t), which corresponds to slow and not too strong
modulation (see below), the term oc pgpzt in 87, is small in the whole harmonic region.
Then the constant Zg in z(t) should be small as well. In this case the prefactor is

determined by the constant Z]. This gives

—1/2

5:21 an)

In the opposite case of very strong and/or higher-frequency modulation, where
9%33, ~ Aq2/ag(t), the term 0( pgpzt is much larger than 03(t) in the whole harmonic

region, and the constant 21 in 2(t) can be disregarded. In this case

imp/31;, = 22—1/2. (3.29)

Equation (3.26) describes also the intermediate case where the term cc 7133, in
£b(t) and 2(t) is small in a part of the harmonic region sufficiently far from qb(t), but
becomes large closer to qb while still outside the layer ~ 11). This regime corresponds
to strong synchronization for comparatively weak driving. Here, both parameters 21
and 22 are important and can be used for matching, giving the same result. The
corresponding analysis is provided in Appendix D.

As we will show, to calculate the escape rate in the regime of strong synchro-
nization there is no need in finding the whole function 3((b) in the boundary-layer
distribution, it is sufficient to know only 3%,. Eqs. (3.19), (3.26) then provide the

complete solution of the problem of escape.

62

3.5 Time dependence of the escape rate

3.5.1 Adiabatic limit

Explicit expressions for the escape rate in the regime of strong synchronization can
be obtained for comparatively weak or comparatively slow (adiabatic) modulation,

where 33, ~ Ism] >> D but
51%;, << R. (3.30)

The results for comparatively weak modulation, [3“,] < R, should go over into the
results [92], which were limited to this range, but included the strong-synchronization
region [3",] >> D. This is demonstrated in Appendix D. We show that 3%, found from
Eqs. (3.22), (3.26) by perturbation theory in the modulation amplitude A coincides
with the result of Ref. [92].

Condition (3.30) can be met for large A, where sfi, ~ R, provided the modulation
frequency is small compared to the reciprocal relaxation time, wptr ~ (2 F < 1. In
this adiabatic regime the stable and unstable states qZ‘fflt) are the periodic solutions

of the equation
K(qfi,(t),t) = 0.

In the adiabatic approximation, escape happens very fast compared to the modulation
period. This means that the escape trajectory can be calculated as if the force K (q, t)
did not depend on t explicitly. One can think then of a particle in the instantaneous

potential well U (q, t), cf. Fig. 3.1(a). The intrawell distribution has the form

pat) = 12100301”? exp {— [(10.0- 1103301)] /D}. (3.31)

where 0,72“) = OgU (q, t), with the derivative evaluated for q = q3d(t)

63

The adiabatic MPEP is given by the equation 40p, = ——K (qopt(t), tm). The time
tu'l (i.e., the phase of the modulation (75m = wth) is found from the condition that

the adiabatic barrier height of the potential U (q, t),
AW) -—- (1011010 _ (102%). t). (3.32)

be minimal, dAU / dt = 0 for t = tm. The minimal barrier height gives the activation

energy of escape,
R 2 AU“, E AU(tm)

The value of 34;, can be Obtained by matching the intrawell distribution (3.31)
and the boundary-layer distribution (3.9). The matching is done most easily for
sufficiently large Q = q — qu(t), so that |Q] >> I D and 71b(tm)Q2 >> 12%;, and for
(St = t — tm small compared to the modulation period TF but large compared to the
typical relaxation time 711:1(tm). 111 this range, in Eq. (3.31)

1

U(q, t) _ U(indm, t) 9‘ AUm + 2

.. 1
AUmCSt2 + ille2a

where AU,“ = d2AU/dt2, with the derivative calculated for t :2 tm; here and below
in this subsection for brevity we use 717, for 71b(tm).

TO find the boundary-layer distribution we note that, for the corresponding 6t, Q,
the integral over p in Eq. (3.7) can still be evaluated by the steepest descent method.
However, in contrast to the analysis of Sec. 3.4.2, the phase 0) that provides the
extremum to the integral is not equal to the optimal phase (bm, which corresponds to
the adiabatic MPEP qopt (t). Indeed, the MPEP goes through a given q = Q+q§d (tm)
at the time topt(Q) that differs from tm by the relaxation time ~ f‘b-l' We have chosen
at so that |topt(Q) — tml << |6t|.

A straightforward but somewhat tedious analysis shows that the major term in

64

the phase difference is (b — (0m E (270717, 6t. Then Eq. (3.7) gives

1 1
p(q, t) % (21#3)1/25 exp {—5 [8m + -2- (111.622 + 123.713.9191?” }

By comparing this expression with Eq. (3.31) near qb, tm we obtain

—1/2 _ _s
8 = i47r2Dl‘b/l/ial] ‘3 (AUm ”VD,

311, = AUm/Q%ug (3.33)

[here 71.,- E 717(tm), for i = a, b].

We note that the modulation frequency (1) F drops out from the expression for 3%,,
because 9 F oc wp while AU 0c 0212?. In Appendix E we show that the same expressions
for the distribution parameters can be Obtained by solving equation (3.22) for the

prefactor.

Escape current

The expressions for the coefficients (3.33) allow evaluating the escape current j(q, t)
(3.12) in an explicit form. For 34;, >> D the current has sharp peaks as function of
time, i.e., escape is strongly synchronized. The peak shape can be found by extending

the analysis of Sec. 3.3.2 used to demonstrate the very onset Of synchronization.

For (bd close to (75m —21rk with integer k, the expression (3.12) for (75d can be written

CM 2. 45111 + QF 111(17/171), (3-34)

as). = xoexpewk/np). use = passe/D.

where we used Eq. (3.25) for popt(t) [.177C E :rk(Q, t)]. Expanding in Eq. (3.12) 5075(7)

65

near the minimum and using Eq. (3.33) we obtain

1/2
j(q,t) = Wm Z]: Jk(Q1t)1 Win 2 Mfg—IUD,

2
00 71'
1,, = / tree-=1 exp {—(9/2) [In (:r/xk)]2}, (3.35)
0
where
e = (253;; 71) ~ (2513,1713. (3.36)

The factor Wm in Eq. (3.35) is the Kramers escape rate in the stationary potential
U (q, tm). Because of the modulation, escape current (3.35) is a periodic sequence of
sharp peaks. To show this we note first that, for any given time, the functions
:rk(Q, t) with different k are exponentially different, because Q F < 1. Therefore
J71; are also exponentially different. Only one Jk becomes large within a given period
(t, t+1'p). This happens in a narrow time interval where the corresponding :rk ~ 1, see
below. The periodicity of the current is a consequence of the relation :ck(Q, t + TF) =
xk+1(Q, t), which follows from Eq. (3.19).

The shape of the current peaks is determined by the parameter 6. For 0 << 1, the
typical :rk that contribute to Jk in Eq. (3.35) are given by the condition Oln2 wk g 1.
For t close to tm one can Show from Eq. (3.18) with K = K (q, tm) that popt(t) =

CGXPl_(t _ tm)ubla With 0 N ”binUm) _ (1610111)]. Then from Eq (3°34)

17k“) z eXPi—(t — tk)f‘l'bl1 tk = tm + kTF, (3-37)
with k = 0, 21:1, . . . [we disregard the correction 711:1 ln(QC / D) to tk; it is of the order
of the Suzuki time [117]]. Then the escape current near its maxima becomes

1(q, t) = Wm 21. e-<t-tk>2AUm/ZD, 0 << 1. (3.38)

The current pulses (3.38) have Gaussian form. They are centered at t = tk =

tm+k1'p. The pulse width 6tw is determined by the noise intensity and the modulation

66

frequency, 6tw N (D / AUm)1/ 21F. We note that the condition 19 << 1 means that this
width is much larger than the relaxation time of the system ~ 717:1, cf. Fig. 3.2(a).
Eq. (3.38) corresponds tO the fully adiabatic picture, where the escape rate is given
by the instantaneous barrier height AU (t)

For larger 6, where 0 ~ 1, the pulse Shape is no longer Gaussian. We emphasize
that this happens where the modulation is still slow, Q F < 1, and one would expect
the adiabatic picture to fully apply.

The pulse shape can be found explicitly in the limit 6 >> 1 but 51 F << 1. In this

case Jk can be evaluated by the steepest descent, giving
11.10. t) z 01/6)“th expense. (3.39)

AS a function of time, Jk displays a peak where :ck(t) = 1. The width of the peak
(Stu, ~ 711:1 is now independent of both the noise intensity and the modulation fre-
quency. The peak is strongly asymmetric, because wk depends on time exponentially,
cf. Eq. (3.37).

The evolution of the peak shape with varying parameter 6 is demonstrated in
Fig. 3.2. It is the same as for weak modulation [118], but the dependence of the pa-
rameters on the modulation amplitude is different, for strong modulation. Fig. 3.2 (b)
shows that, with increasing modulation frequency, not only the height of the peaks
decreases, i.e., the modulation of the escape current becomes less pronounced, but

also the width of the pulses with respect to the modulation period rapidly increases.

3.5.2 Nonadiabatic regime

The shape of the escape current peaks can be analyzed also for 9 F g 1, where

the adiabatic approximation does not apply. Our analysis is based on Eqs. (3.12),

67

 

 

 

 

 

 

 

 

 

 

 

o . .
.10 o 10 P104...) 411/5 o 2115 1'1...

Figure 3.2: (a) Pulses of escape current in the adiabatic approximation as functions
of time scaled by the relaxation time. With increasing parameter 6 the pulses change
from Gaussian to strongly asymmetric. (b) The same pulses as functions of time
scaled by the modulation period, (7) = wpt.

(3.26), (3.27). The function pgpzt(t) oc ng(t,t’) in Eq. (3.26) for £7, exponentially
increases in time near qb, therefore the term oc pggt in 57, and 2 becomes dominating
before the MPEP reaches the diffusion region |Q] ~ l D: and in Eqs. (3.12), (3.26)
87, = éD-1/2 (Q Diwsfimgpztun—l/z.

Integration over a: in Eq. (3.12) can be done by the steepest descent method.
Using Eq. (3.25) one can write in Eq. (3.12) ad — aim = 0 F ln[a:D/onpt(t)]. The
major contribution to j (q, t) near its maximum comes from the range l¢d — <me s
(D/Q%sg,)1/2 << 1. Expanding 3(¢d) to the second order in (bd — aim and performing

integration over a we obtain from Eqs. (3.12), (3.26)

. 71(1):;‘1912/ _
1(611t)=————b——“/0216“Do k, (3.40)

9 Feng’u

33k = 550 eXP(27rk/QF)1 $0 = Popt(t)Q/D-

68

 

Note that here popt(t) can be smaller than I D / (f(t).

Eq. (3.40) describes the escape rate in the whole region Diwsfi, >> D; it does
not require the adiabatic approximation. The ratio (ff/M can be obtained from
Eq. (3.29) by solving Eq. (3.22). In the adiabatic limit Eq. (3.40) gives the same
result as Eq. (3.39). The current peaks (3.40) are strongly asymmetric. The current
is maximal, for given Q, when .137. = 1. The width of the current peaks is given by the
reciprocal relaxation time ~ 711:1. The form of the current peaks is totally different
from that of the diffusion current —Dan on the basin boundary Q = 0. This current
was studied in Refs. [93, 112, 54]. It can be easily obtained from Eqs. (3.7), (3.26)
with Q = 0. The regime flies; / D << 1, where the current has the form (3.38), cannot

be studied in the approximation [93, 112] at all.

With increasing 0 F the peaks of 7 (3.40) are smeared out and the escape syn-
chronization is weakened. For 9 F >> 1 the exponentially strong synchronization
disappears. Besides the fact that the width Of the peaks [Lb-1 becomes of the order of

the interpeak distance 1'}: for Q F 2 1, 33, rapidly decreases with tap for large (1 F-

3.5.3 Nonlinear current propagation

In experiments the instantaneous escape rate can be measured as current 7' (q, t) for
a given q well behind the basin boundary qb(t), i.e. q — qb(t) >> 17). When noise is
weak, motion behind the basin boundary is practically noise-free (“deterministic”).
Then the escape current is j (q, t) = K (q, t)p(q, t). The function p(q, t) can be related
to the distribution p(qb(t) + Q, t) in the harmonic region where Q >> 10. The relation
can be obtained from the Fokker-Planck equation in the neglect of the diffusion term,

6771+ 8q[K (q, t) p] = 0, and can be expressed in terms of the trajectories in the absence

69

of noise qdet(t; to). A trajectory Qdet is given by the equation

affldet = K(Qdetatla (3'41)

admit; to) = q, (medic; t0) = (1300) + Q-

The boundary conditions in Eq. (3.41) follow from the fact that qdet(t; t0) arrives at
“observation point” q at time t and starts at point qb(t0) + Q at time to. Eq. (3.41)
gives to as a function of q, t. The distribution behind the boundary is

t
p(q, 15) = 8X1) {-ft (17 [aqKUL Tlldot} 10000). (342)

0

where the derivative 071K is calculated along the trajectory (Idetfti t0), and p0(t0) =
10011100) + Q1150)-

The current 7 (q, t) has a simple form for adiabatic modulation, 9 F << 1. In this
case, as long as q is not too far from qb(tm), one can calculate the trajectory Qdet
for the modulation phase that corresponds to t = tm, where the current is close to

maximum. This gives for to = tad the expression

(1 dq’
tad(q,t) = t —(5tad(q), 61‘“ = / —,——. (3.43)
O 0 0 qb(tm)+Q K (‘1,1 tm)
Eq. (3.43) shows that there iS a simple q-dependent shift between t and tad, which

is equal to the duration of deterministic motion from qb(tm) + Q to the observation
point q. We assume that this shift is small compared to the modulation period TF-
In the adiabatic approximation Eq. (3.42) is further simplified for t close to tm +

.1117. In this case

 

t K m , m
eXP I:_ £0 ‘17. (69K)det] z (”)2“), Zn? t ) z Hb(t111)Q/K((11t1n)a

where we have taken into account that K (qb(tm), tm) = (7711",) cc ”F [in the case of

additive modulation, K (q, t) = K0(q) + F (t), we have K ((77,, t,,,) = 0]. Therefore, we

70

obtain for the current in the adiabatic approximation

11:10.1) = 10.131). (3.44)

Thus, in the adiabatic approximation the current near the peaks does not change
shape but just shifts in phase, with the phase shift described by Eq. (3.43). With
increasing parameter 6 (3.36) the current peaks change from Gaussian, for 0 << 1,
see Eq. (3.38), to strongly asymmetric, see Eq. (3.39). The peaks are located at
t = tk + (itgd, with tk == tm + kTF, Eq. (3.37). We note that in Eq. (3.37) we
disregarded a shift of tk by the Suzuki time ~ 711:1 ln(Q/lD). This shift compensates
the term in (itad (3.43) that logarithmically depends on Q, so that the overall position
of the current peaks (3.38), (3.39), (3.44) is independent of the matching point Q.
In the nonadiabatic case the difference t— to depends not only on the observation
point q, but also on time t. In addition, the factor that relates p(q, t) and p0(t0) in
Eq. (3.42) becomes t-dependent. Therefore, the overall shape of the current changes.
However, the escape current has sharp peaks only where their width is small compared
to the inter-peak distance. From this point of view, the case of slow modulation is
most interesting for studying the shape of escape current and synchronization of

escape as a whole.

3.6 Period-average escape rate
In the range 53, ~ [sml > D, the period-average escape rate (3.13) is
W = Vexp(—R/D), V = flbffDl/2/21n/sfil. (3.45)

The prefactor V can be expressed in terms of 22 using Eq. (3.29), formally giving the

result [93, 112] even where the theory [93, 112] does not apply.

71

The asymptotic technique developed in this Chapter allows obtaining the prefactor
V in several limiting cases. For comparatively weak modulation, D < Isml < R,
Eqs. (3.22), (3.45) give the same result as in Ref. [92], with the scaling V oc (331)-1/ 2 oc
A-l/z. Since the theory [92] covers the whole range Ism] < R, a transition from the

Kramers limit of no modulation to the case of arbitrarily strong modulation is now

fully described.

In the whole range where the adiabatic approximation applies, 9F << 1, from

Eqs. (3.27), (3.33) we obtain

V = (ZWl-3/2l/lal1bl1/201/2WF(AUIII)_l/2 (346)

where 710,7, are calculated for t = tm. Interestingly, V (3.46) is independent of the

modulation frequency.

3.7 Scaling near the bifurcation point

Close to the bifurcational value of the modulation amplitude A = Ac where the
metastable and unstable states qa,b(t) merge, the escape rate displays system-
independent features. As Shown in Chapter 2 (see also Refs. [62, 85]), the activation
energy R of the system scales as R 0( 175, where 17 cc (Ac — A) is the reduced dis-
tance to the bifurcation point along the amplitude axis [see Eq. (3.49) below]. Three
scaling regimes have been identified for R. With increasing modulation frequency 0) F
or decreasing 17, the critical exponent 5 changes from 5 = 3/ 2 for stationary systems
(adiabatic scaling) to 5 = 2 (locally nonadiabatic scaling) and then back to 5 = 3 / 2

(high-frequency scaling). Below we discuss scaling of the prefactor V in these regimes.

72

3.7. 1 Adiabatic scaling

Like in Chapter 2, we start the analysis with the limiting case of slow modulation,
wptr << 1. In this case the adiabatic stable and unstable states (123)“) are given by
the equation K (qfl(t),t) = 0 [cf. Eq. (2.11) of Chapter 2]. The adiabatic critical
amplitude Agd is determined by the condition that the states qgfl)(t) touch each other.
This happens once per period, and we set t 2 k1]: (k = 0, i1, . . .) at this time. We
also set (13%(kTF) = 0 for A = Affd. Expanding the Langevin equation of motion

around this point, we obtain
4 = 01112 + 116A“ — (172(41Ft)? + f(t), (3.47)

This equation has the same form as Eq. (2.12) for the soft mode ql. Here a =
(1/2)83K, fl = BAK, 72 : —(2aw%)-1072K; all derivatives are evaluated at q z
t = 0, A = Aild; 7 is independent of (up; it is assumed, without loss of generality,
that a > 0; (SAad = A —— Agd. The force f (t) is a zero-mean white Gaussian noise,
(f(t)f(t') = 205(t-t')-

The adiabatic approximation applies provided not only tfidwp << 1, but also
aged < 1, where t?d(t) = (1 /2)[(a'7c127:~t)2 — aflMad]-l/2 is the adiabatic relaxation
time, as given by Eq. (2.13) of Chapter 2. The relaxation time strongly depends 011 t
and diverges for A —> Agd and t —> 0. The inequality attfld << 1 is therefore the most
restrictive condition on adiabaticity; it requires that tifd << t7, where t7 = (O’YUJF)-1/2
is a new dynamical time scale [62, 85]. The condition tfid << t7 is equivalent to
WF << pawl/1.

The problem of escape is simplified in the adiabatic regime [30, 31, 56, 57, 62,
85, 109]. For a periodically modulated system, from Eq. (3.47) we obtain [71“] =

717, = 2|(166Aadll/2, AU", = 4w%272|c166Aad|1/2. Then, from Eq. (3.46), the prefactor

73

scales as V O( |(5Aadll/4. We note that, in the adiabatic approximation V decreases as

A approaches the bifurcational value and, because escape is strongly synchronized,

Voc Dl/Z.

3.7 .2 Locally nonadiabatic scaling

As explained in Sec. 2.3.1 of Chapter 2, the critical slowing down of the system motion
makes the adiabatic approximation inapplicable in the region [6AM] /Agd,SQ F1 where
the condition ti‘d << t7 is violated. In this range we rewrite Eq. (3.47) in the form of

Eq. (2.18),
Q=Q2—12+1—n+f(1). (348)

where Q = (mg, 1' = t/t7, Q = (IQ/d1, f(r) = (ywp)_1f(t71'). The control parameter

is defined as
11 = (mm—1m. — A), A... z 421‘. (349)

Here Ail = A?“ + ywp/fi is given by Eq. (2.16). It is, to the first order in (up, the
“true” bifurcation point, which is shifted from Agd because of the slowing down of the
system and the associated with it delayed response. For small driving frequencies,
wptr << 1, where the local expansion (3.47) applies, the shift in the bifurcational
amplitude is linear in frequency. The corrections to A2] of higher orders in (up are
discussed in Sec. 2.3.4 of Chapter 2.

For 17 << 1 the activation energy scales as R o< 172. In this region the most probable
escape path Qopt(1'),Popt('r) corresponding to Eq. (3.48) is given by Eq. (2.36) of

Chapter 2. Using those expressions in Eq. (3.22), we obtain

~ 7'
2(1') 2 411D] (l1'1exp(2‘r2 — 2112). (3.50)
—00

74

Here D = 011/ 2('7(.bja~)"3/ 2D is the intensity of the random scaled force f(1'). This

gives
u = VODI/2I3Al—lwg/4, a4 = A — AC, (3.51)

where

V0 = (32177'1/4Im11/4s'1

= (64e7wF)-1/4|a,21(agKll/S/laAm.

From Eq. (3.51), the prefactor V oc [(5Al”1 sharply increases as the modulation
amplitude approaches Ac. This is qualitatively different from the decrease of V in the
adiabatic approximation. The result agrees with the numerical solution of Eqs. (3.22),
(3.45) for a model system shown in Fig. 3.3. The calculations in a broad range of A

are also confirmed by Monte Carlo simulations, as discussed below.

3.7 .3 High-frequency scaling

For high frequencies, 9 F >> 1, escape is not synchronized by the modulation. The
prefactor in the escape rate is V = | 71,417,]1/2 /211, it is independent of the noise intensity
D. Near the bifurcation point it scales as in stationary systems [30, 57], where
V 0c |(5AI1/2 and R oc |6AI3/2. We note that modulation is necessarily fast very close
to the bifurcation point, because [170,0] —+ 0 for A —> Ac. Therefore the prefactor
always goes to zero for A —> Ac. However, for small (up the corresponding region
of 6A is exponentially narrow [62, 85]. The increase Of V with decreasing Ac —— A in
the locally nonadiabatic region does not contradict this picture because in this region

V oc D1/2, whereas for effectively high-frequency modulation V is independent of D.

75

 

 

 

 

 

 

 

 

 

 

. c .
0‘ BWEEWEEMEEWB -<

r

0 0.15 A 0.3

 

 

Figure 3.3: The prefactor V in the average escape rate W (3.45). The results refer
to the model (3.52) with (up = 0.1 and describe escape in the regime of strong
synchronization, where V O( Dl/z. The solid line for small A shows the scaling
V oc A-1/2. The solid lines for small (5A = A - AC in the main figure and in the inset
Show the scaling (3.51). The dashed line shows the result of the numerical solution
of Eq. (3.22). The squares and crosses show the results of Monte Carlo simulations

for R/ D = 5 and R/ D = 6, respectively.

3.8 Results for a model system

To illustrate the findings, we consider a simple model system, a Brownian particle in
a cubic potential subject to sinusoidal modulation. The Langevin equation is of the

form

q = K(q, t) + f(t), K = q2 — 1/4 + Acos(wpt), (3.52)

with f (t) being white Gaussian noise of intensity D.

76

 

3.8.1 The adiabatic regime

The adiabatic stable and unstable states of the system (3.52) in the absence of
noise are qgfj)(t) = IF [1 /4 — Acos(wpt)]1/2, and the adiabatic critical amplitude is
Agd = 1/4. The adiabatic barrier height is AU(t) = (4/3) [1/4 — Acos(wpt)]3/2. Its
minimum AUm = (4/3)(1/4 — A)3/2 is reached for tm = 1111:, with k = 0,:l:1,. . ..

2 .11

The reduced curvature QFsm of the function 3(0)) in the boundary-layer distribu-

tion at (75m = wptm is given by Eq. (3.33),
053;; = (1/2)Aw]2p(1/4 — A)—1/2. (3.53)

Therefore the condition of strong but Slow modulation, 12%;, << D, which must hold

for the pulses of the escape current to be of Gaussian shape, takes the form
1.1% << D(1/4 —- A)1/2/A.

It becomes more and more restrictive for the modulation frequency as the modulation
amplitude A approaches the adiabatic bifurcational value 1/ 4.

The prefactor V of the period—average escape rate in the adiabatic limit for suf-
ficiently strong modulation is given by Eq. (3.46). For our model it has a simple

explicit form
V = (2113/2)_1D1/2(1/4 — A)1/4A-1/2. (3.54)

As expected, V oc A‘l/2 for small amplitude, whereas close to the adiabatic bifurca-

tion point V oc (A?d -— A)1/4.

3.8.2 Locally nonadiabatic regime near the bifurcation point

AS explained in Section 3.7, sufficiently close to the bifurcation point the adiabatic

approximation breaks down. As a result, the bifurcation point AC shifts away from

77

 

 

0.04 . .

 

 

 

 

RID=4

V .

3 U 0/ c1 O

f X X x x
0.02] [

RID=5
0 . r
0.6 0.69 A 0.78

Figure 3.4: The prefactor V in the average escape rate W (3.45) close to the bifurcation
point A 2 Ac. The results refer to the model (3.52) with 01F = 1. The squares and
crosses show the results of Monte Carlo simulations for R/ D = 4 and R/ D = 5,
respectively. The solid line shows the asymptotics V = | [10,717,]1/2 / 211 0c (Ac — A)1/2.

A2“ (to higher amplitude, in our case). Close to Ac the pulses of escape current
become strongly asymmetric, even though the modulation frequency is small. The
scaling of the prefactor in the period-average escape rate also changes dramatically,
from decreasing (as in the adiabatic approximation) to increasing for A ——> AC. From

Eqs. (3.51), (3.52)
V = (64x/21r7)‘1/4D1/2|(5Al‘1w2/4. (3.55)

The results on the prefactor for the discussed model system in the range 10 7151].”) <<
1 are shown in Figure 3.3 (do) is the relaxation time in the absence of modulation;
for the model (3.52) if“) = 1). They refer to the modulation frequency (by = 0.1.
The dependence of V/\/_D on the modulation amplitude A is shown in the main part
of the figure. The solid line for small A represents Eq. (3.54). The solid line close
to the bifurcational amplitude Ac 2 0.29 is given by Eq. (3.55). The dashed line

for intermediate values of A is given by Eq. (3.45) with (S/ «34;, (3.29) evaluated by

78

numerically integrating Eq. (3.22). It is also well described by Eq. (3.54) for A < 0.2.
The analytical results agree with the results of simulations represented by squares and
crosses. The inset shows in more detail the locally nonadiabatic scaling V cc [6241]"1

in the region near Ac.

The simulations have been done using the standard second-order integration
scheme [107] for stochastic differential equations. The period-average escape rate
was found as a reciprocal of the average dwell time of particles leaving the attraction
basin. For each set of parameter values we accumulated ~ 105 escape events. The
prefactor of the escape rate V was evaluated as V = Wexp(R/ D). The values of acti-
vation energy R were Obtained independently by solving the appropriate instantonic
problem. We checked previously [62, 85] that these values agree extremely well with
Monte Carlo simulations. For each value of A the noise intensity D was adjusted so
as to keep R/D fixed at R/D = 5 (squares) and R/D = 6 (crosses).

The results on the prefactor for higher modulation frequency are shown in
Fig. 3.8.1. They were obtained in the same way as for lower frequency. It is seen from
Fig. 3.8.1 that for the used model, already for wptfio) = 1 the amplitude dependence
of the prefactor differs very significantly from the result of the adiabatic approxima-

tion. In particular, the prefactor displays the scaling behavior V oc (Ac — A)”2

near
the bifurcation point. It is independent of the noise intensity and is well described

by the expression V = [flaflbll/ 2 /21r.

3.9 Conclusions

The results of this Chapter and the previous work allow us to draw a general scheme

of the dependence of the rate of activated escape on the modulation parameters. This

79

 

 

  
    

Locally
nonadiabatic

 

 

 

A
I .
. Nonexponentlal
1 synchronization
.. .. I
1’ Vs Asymmetric :
\
Gaussian “ pulses ' _' __ ..
pUiSBS'fi" “ ____.="—"“
l" " - "~—..:_: _, —— " T 1 Weak driving
\ 0
0 Log-linear wFt( )
r

Figure 3.5: Different regions of escape behavior in modulated overdamped systems

(0)

depending on the modulation frequency (by and amplitude A; tr is the relaxation
time in the absence of modulation. The smeared boundaries between the regions
are shown by dashed lines. The bold solid line indicates the bifurcational amplitude
where the metastable state disappears. The shaded region below it indicates the range
where the activation energy of escape R g D. The transition between the regions of

exponentially strong and nonexponential synchronization occurs for wptgo) ~ 1.
scheme is sketched in Fig. 3.5.

The weak-driving region corresponds to the case where the modulation-induced
change of the activation energy of escape is small compared to the noise intensity D.
In this region the major effect of modulation is weak “heating” of the system, which
is quadratic in the modulation amplitude (for underdamped systems, the effective
temperature depends on energy [2, 6, 86, 87, 88]). Escape is not synchronized by
the modulation. The width of this region along the amplitude axis is o< D for low
frequencies and becomes o< wpD1/2 for large Qp ~ wptr.

Synchronization emerges once the magnitude of the oscillations of the “instanta-

neous” activation energy [.9111] becomes much larger than D. There is a broad region

80

of modulation amplitudes where [am] oc A and the logarithm of the period-average
escape rate W is linear in A, too [92]. This log-linear region in Fig. 3.5 is bounded on
the large-A side by the condition A /Ac << 1, where Ac is the bifurcational value of A.
Strong synchronization occurs for small frequencies, 9 F << 1. Here, the escape cur—
rent has peaks with width much smaller than the modulation period. The prefactor

V in the period-average escape rate scales as (D /A)1/ 2.

Synchronization persists for higher modulation amplitudes. The shape of the
peaks of escape current is Gaussian for (lief; < D, and their width is ~
(D/ 34;)1/217. For higher frequencies, the peaks become strongly asymmetric and
non-Gaussian, with width ~ tr. In the log-linear region the boundary between the
two types of peaks is 01F oc A‘1/2.

For high modulation frequencies, [sm] becomes small and exponentially strong
synchronization of escape disappears. The escape current is still modulated in time,
of course, but generally it does not have a shape of sharp narrow peaks even for small

noise intensity.

Of special interest is the bifurcation region, because there the dynamics and fluctu-
ations display system-independent features. The region is determined by the condition
IAC -— A] < Ac, as shown in Fig. 3.5. In this region, in the adiabatic approximation
the boundary of the range where escape current peaks are Gaussian has the form
0F 0( IAC -— All/4.

Close to Ac, where (Ac — A) /ACQ F ~ 1, the adiabaticity is broken. This condition
and the condition 9p << 1 determine the boundary of the locally nonadiabatic re-
gion. Inside this region the escape current has the form of asymmetric narrow peaks.
Special scaling is displayed for (Ac — A) /ACQ F << 1. Here, the activation energy

of escape scales with the distance to the bifurcation amplitude A; as (Ac — A)2,

81

whereas the prefactor in the period-average escape rate is o< (AC — A)—1. Because of
the slowing down near the bifurcation point, the locally nonadiabatic behavior and
synchronization of escape disappear for small Ac — A, which determines the high-A
region boundary. This boundary is very close to Ac for small Up, but for higher
{2 F the region of locally nonadiabatic behavior shrinks and ultimately disappears.
Outside this region on the high—01F side, for A close to AC the prefactor V scales as
0 OC (.4, — A)1/2.

In conclusion, we have Obtained a general solution of the problem of noise—induced
escape in periodically modulated overdamped systems. For small WF the pulses of
escape current are exponentially sharp and change with increasing 0) F from Gaussian
to strongly asymmetric. For large WF exponential current modulation disappears.
The prefactor V in the period-average escape rate is a strongly nonmonotonic function
of the modulation amplitude A for low frequencies. It first drOps with increasing A
to V oc (D/A)1/2 [92], then varies with A smoothly [54, 93, 112], and then sharply
increases, V at D1/2/(Ac — A) near the bifurcation amplitude AC. We found three
scaling regimes near Ac, where V oc (Ac — A)C with 5 = 1/4, ——1, or 1 / 2. The widths

of the corresponding scaling ranges strongly depend on the modulation frequency.

82

Chapter 4

Pathways of activated escape in

periodically modulated systems

4. 1 Introduction

The theory of activated escape should answer two closely related questions: what
is the escape rate and how does the system move during escape? Fluctuational
trajectories leading to escape or interstate switching are of significant interest and
have been extensively studied in recent years using various numerical techniques, see
Refs. [119, 120, 121, 122] and papers cited therein. Even though escape is a random
event, the probabilities of following different paths are strongly different. Therefore
most likely the system follows a certain pathway, i.e., its trajectory is close to the
most probable escape path (MPEP), see Ref. [95] and references therein.

The distribution of escape trajectories can be characterized by the prehistory
probability distribution (PPD), a quantity accessible to direct experimental measure-
ments. It is obtained by recording trajectories of the system that lead to escape and

superimposing the trajectories that, after the system has escaped, pass through a

83

small vicinity of a point q f (for a certain modulation phase, in the case of a period-
ically modulated system). The point qf is chosen in the phase space sufliciently far
behind the boundary of the basin of attraction to the initially occupied metastable
state. Formally, the PPD ph(q, thf, t f) is the probability density for the system to
have passed through a point q at an instant t provided it had been fluctuating about

the metastable state for a long time and passed q f at a later time t f, if > t [80].

The PPD p7,(q, t|q f, t f) should peak at q lying on the optimal fluctuational path
that leads to q f. Therefore it “maps out” optimal paths. For stationary systems
this has been directly confirmed by extensive simulations [123, 124, 125] and also in
laser experiments [81]. 111 such systems escape can occur at any time, with equal

probability, therefore the tube of paths around the Optimal escape path is broad.

In this Chapter (see also Ref. [126]) we study escape pathways in periodically
modulated systems. Modulation synchronizes escape events. This can be easily un-
derstood for escape from a Slowly modulated potential well. Here, escape is most
likely to occur when the instantaneous potential barrier AU (t) is at its lowest, once
per period, cf. Fig. 4.1(a). This corresponds to exponentially strong synchronization.
Strong synchronization persists even where the modulation is not slow and the adi-
abatic picture in which the escape rate is determined by the instantaneous barrier
height does not apply [110, 111, 112, 127, 128]. As a result of escape synchronization,

there is one most probable escape path per period.

In turn, as we show, the prehistory probability distribution for escape pathways
displays a Sharp narrow peak, which is centered at the MPEP. Moreover, the PPD
may display several narrow peaks in the (q, t) space. For strong escape synchroniza-
tion and not too low modulation frequencies, the width of the peaks is determined by

the typical diffusion length l D = (2Dt,-)1/2, where D is the noise intensity (D = kBT

84

 

for thermal noise) and tr is the relaxation time of the system. This is qualitatively dif-
ferent from the Shape of the PPD in stationary systems [81]. Of course, in modulated
systems along with the final point q f through which the system passes one should
fix the modulation phase when the passage happens. It is given by the passage time

t f(mod 1p), where 1p is the modulation period.

The occurrence Of a narrow peak of the PPD can be understood from the qualita—
tive picture of motion in escape. This picture is sketched in Fig. 4.1 for a system with
one dynamical variable q. Escape from a static potential well corresponds to going
over the barrier top qb from the vicinity Of the potential minimum qa, see Fig. 4.1(a).
Similarly, a modulated system escapes when it goes over the periodic basin bound-
ary qb(t) from a periodic metastable state qa(t), see Fig. 4.1(b) (here and below we
assume that the noise correlation time is small; see Ref. [51] for a more general case).
Let us suppose that the escaped particle is found at time t f at a point q f sufficiently
far behind qb(t). A typical trajectory to this point displays four distinct sections with

different types of motion shown schematically by letters A through D.

We discuss the motion backward in time from t f. Immediately adjacent to q f
is section A of the trajectory in Fig. 4.1. Here, for small noise intensity the system
moves close to the noise—free trajectory from the vicinity of qb(t) to q f. In the case of
a static potential this corresponds to sliding down the potential slope from qb to q f.
In the diffusion region of width ~ I D around qb(t), section B in Fig. 4.1, the motion is
mostly diffusion about qb(t), because noise-free motion with respect to qb(t) is slow (in
the case of a static potential the potential is locally flat at qb). Section C is the motion
from the attractor to the basin boundary. The corresponding trajectory is close to the
most probable escape path. This motion is a result of the large fluctuation that has

led to escape detected at q f. For strongly synchronized escape, there is one MPEP

85

 

 

 

 

 

 

 

Figure 4.1: Activated motion leading to detection of an escaped particle at point q f
at time t f (schematically). Panels (a) and (b) illustrate escape from a static and
a periodically modulated potential well, respectively. In the latter case qa(t) and
qb(t) are the periodic stable state and the basin boundary. The trajectories qégkt)
(n = 1,... , —2) are the periodically repeated most probable escape paths. The four
major stages of motion A, B, C, and D on the way to qf are discussed in the text.

86

per modulation period, as shown in Fig. 4.1(b). It approaches qb(t) asymptotically
as t —> 00. If the system was observed behind qb(tf) at time t f, it has most likely
arrived to the vicinity of qb(t) along the MPEP that approached qb(t) before t f, but
not too much in advance, as shown in Fig. 4.1(b). Finally, well before t f, in the region

D, the system was fluctuating about the attractor.

The above picture suggests that the PPD will display a peak along the trajectory
singled out in Fig. 4.1. It also explains why this peak should be narrow: in contrast
to stationary systems, which can arrive to the vicinity of qb at any time, periodically
modulated systems approach qb(t) only once per period. By changing qf, t f we can
switch between neighboring MPEP’S, and therefore there is a possibility for the PPD

to have two and potentially even more peaks inside the attraction basin.

As mentioned above, we are interested in the regime where escape events are
strongly synchronized by the modulation. It means that the probability density to
find the escaped system at a point q f behind the basin boundary, cf. Fig. 4.1, displays
sharp peaks as a function of the observation time t f. These peaks are periodically
repeated in time. They were studied in Ref. [111]. We will choose the time t f close to
the maximum of the peak. This choice is justified, because the corresponding PPD
characterizes the most probable escape trajectories. Otherwise the PPD would be
formed by trajectories with exponentially smaller probabilities that are very rarely

followed in escape.

Interestingly, the condition that the tubes of escape trajectories be narrow in
time and space requires that the modulation frequency WF = 211/1'F be within a
range limited both from below and from above. To understand the lower limit we
note that, as mentioned above, for Slow modulation escape occurs every period around

the time tm + k1'p where the height of the instantaneous potential barrier AU (t) is

87

at its minimum (k = 0, i1, . . .). The typical width of the time window for escape
At is given by the condition [AU (tm i At) — AU (tm)| g D, which leads to At =
[D/AU(t,,,)]1/2 ~ w;1[D/AU(tm)]1/2. If At >> trln[AU(tm)/D], the PPD has the
same shape as if the system were escaping out of a stationary potential well of height
AU (tm). In this case the peak of the PPD inside the attraction basin is broad and

asymmetric, its width is independent of the noise intensity [81].

In the opposite limit Of high-frequency modulation, wptr >> 1, escape of an over-
damped system is not synchronized. The dynamics is characterized by the coordinates
averaged over modulation period. The PPD over such coordinates is described by the

theory for stationary systems, and the PPD peak inside the attraction basin is broad.

In the intermediate range of frequencies not only are the PPD peaks narrow but,
as mentioned above, the PPD may display several peaks inside the attraction basin.
This happens because the motion of the system near the basin boundary is slow.
Therefore if the system is observed behind the boundary qb(t) at a given time t f, it
could have arrived to the boundary along one of a few periodically repeated optimal
escape paths, fluctuated about qb(t) for some time, and then made a transition to
(qf, if) over time ~ tr.

The multiple-peak structure of the PPD is a specific feature of periodically mod-
ulated systems far from equilibrium, i.e. away from both the adiabatic limit of slow
modulation and the limit of fast modulation. It is illustrated in Fig. 4.2 for a model
system. Two peaks of the PPD inside the attraction basin are clearly resolved in
this figure. Their shape is well described by the asymptotic theory developed in this

Chapter.

In Sec. 4.2 below we discuss the dynamics of a periodically modulated system and

the equations for the most probable escape path. In Sec. 4.3 we obtain a general

88

 

 

F 3 _1

Figure 4.2: The prehistory probability density (PPD) ph(q,t|qf,tf) and its contour
plot for a noise-driven overdamped system with equation of motion 17 = q2 — 0.25 +
A cos wpt+ f (t), where f (t) is white noise of intensity D. The parameters are A = 0.7,
(up = 2, D = 0.01, qf = 0.8, t7 = (Tp/2)(mod1'p), where TF = 21r/wp is the
modulation period. The shadowing (color code on line) corresponds to the 4 regions
of the height of the distribution separated by the values 717, = 0.5, 2, 7.

89

 

expression for the PPD near the basin boundary and relate it to the distribution
that describes the quasiperiodic current away from the attraction basin, which gives
the escape rate. The PPD is simplified in the adiabatic limit where the modulation
period is small compared to the relaxation time. In Sec. 4.4 it is shown that two
types of behavior may be displayed near qb(t) in this case, depending on the ratio of
the two small parameters that characterize the dynamics. The analysis is extended
to the case of nonadiabatic driving in Sec. 4.5. In Sec. 4.6 we obtain the central result
of the Chapter, the PPD inside the attraction basin. We Show that it has the shape
of a sum of diffusion-broadened Gaussian peaks centered at the periodically repeated
most probable escape paths. In Sec. 4.7 we describe the results of simulations of a
model system and compare them to the analytical results. Sec. 4.8 provides a brief

summary of the results.

4.2 Escape Of a periodically modulated system

We investigate the prehistory probability distribution for an overdamped system char—
acterized by one dynamical variable q. The system is in a periodically modulated
potential U (q, t) = U (q,t + 1F). Fluctuations are induced by an external noise f (t)

The motion is described by the Langevin equation

(1 = K(at) + f(t). (4.1)

Here, K (q, t) E BQU(q, t) is the regular periodic force. We consider the simplest case
where f (t) is zero-mean white Gaussian noise with correlator (f (t) f (t’ )) = 2D6(t—t’).

In the absence of noise the system (4.1) has a periodic metastable state (attractor)
qa,(t) = qa(t+1'p) and a periodic boundary qb(t) = qb(t+1p) of the basin of attraction

to qa(t), see Fig. 4.1(b). The states qa(t) and qb(t) are, respectively, the stable and

90

unstable periodic solutions of the equation (7 = K (q, t). For concreteness we assume
that qa(t) < 13(0-

The PPD p7,(q, t|q f, t f) as defined in the Introduction is the conditional probability
density (with respect to q) of passing through a point (q, t) in the coordinate-time
space on the way from the attractor to a point (q f, t f). It has the form [80]

(1(1), tflq, t)p(q, tlIIim tin)
F(intfl‘lixiatin)

 

ph(qvthf7tf) : 1 (4'2)

where p(q1,t1|q2,t2) is the probability density (with respect to ql) of a transition
from (q2,t2) to (q1,t1), with t1 > t2.

We assume that the noise intensity D is small. Then the period-average escape
rate is W oc exp(—R/ D), where R is the activation energy of escape, R >> D [127,
112, 128, 110, 111]. The condition on D is that W"—1 largely exceeds the relaxation
time of the system tr and the modulation period TF- Eq. (4.2) gives the distribution

of escape paths in a broad time range
W—l >> tf - tin > t - tin >> tr, TF, (4.3)

where the population of the attraction basin practically does not change. For t—t;ll >>
tr the initial state qin, which is close to the attractor qa(ti,,), gets forgotten. Then
the right-hand side of Eq. (4.2) becomes independent of tin-

The functions p(qithmitin) = P(q,t) and P(Qf,tf|qm,tin) = Pf‘Ifatf) give the
time—periodic probability density to find the system in states q and q f, respectively.
To study escape pathways we calculate the PPD for q f outside the attraction basin,
(1 f > Qb(tf)-

The transition probability density p(q, th’, t’) is a solution of the Fokker-Planck

equation (FPE)
ap = — ,,[1((q, t)p] + 1953,, (4.4)

91

 

with the initial condition p(q, t' Iq’, t’) = 6(q —q’). Even for a 1D system, this equation
does not have a known explicit solution except for the case of K linear in q. To analyze
the PPD we will have to find approximate solutions in different regions and match

them.

In the following subsections we discuss the dynamics of the system prior and

during escape.

4.2.1 Dynamics near the periodic states

Following Sec. 3.2.1 of Chapter 3, we briefly review the dynamics close to the periodic
states q7(t), i = a, b. In the absence of noise it is described by the linear equations
(Sq == 71.,-(t)6q with 6g 2 q — q,:(t) and 717(1) given by Eq. (3.2). Time evolution of the
deviation 6q frOm q.,-(t) is given by Eq. (3.3).

Because of the periodicity of 71,:(t), the Floquet multipliers 111,- (i = a, 1)), Eq. (3.4),
are independent of time, with Ma < 1 and M7, > 1. The relaxation time of the system
can be chosen as tr 2 [Lb—1 ~ [ital—1.

Weak noise leads to fluctuations about the attractor, which have Gaussian distri-
bution near the maximum in the time range (4.3). Horn Eq. (4.4) this distribution

has the form Of Eq. (3.5), which we rewrite here as

pa(q, t) Z G (q - Qa(t);Ua(t))2 (4-5)
with
G(.1:; a) = (211D(12)_1/2 exp(—a:2/2D02). (4.6)

The variance (1,2,(t) is periodic in time. It is given by Eq. (3.6) with i = a.

92

4.2.2 Most probable escape paths

Along with small fluctuations, there also happen occasional large deviations from
qa(t), including escape from the attraction basin. In a broad parameter range escape
events are exponentially strongly synchronized with the modulation, see Ref. [111] and
papers cited therein. Because large fluctuations have small probabilities, in escape
the system closely follows the trajectory that is least improbable among all possible
escape trajectories. AS mentioned in the Introduction, it is usually called the most
probable escape path, qopt(t). Inside the attraction basin, escape trajectories lie
within periodically repeated narrow tubes. The tubes have a width ~ I D and are
centered at the MPEP’S (133“) E (Ioptft + nTF), n = 0, i1, . . ..

As discussed in Sec. 3.4, the MPEP’s provide a solution to the variational problem
of maximizing the probability of a fluctuation in which the system moves from qa(t)
to qb(t). This problem can be mapped onto the problem of dynamics of an auxiliary
Hamiltonian system. The latter is described by the Wentzell-Freidlin Hamiltonian
(3.17). Its equations of motion have the form (3.18).

The MPEP’S correspond to the minimal action heteroclinic Hamiltonian trajec-
tories (qopt(t), popt(t)) [127, 129]. They start for t —> —00 at the periodic hyperbolic
state (qa(t),p = 0) of the auxiliary system and for t ——> 00 approach its another
periodic hyperbolic state, (qb(t), p = 0).

Well outside the diffusion regions around the periodic states qa,7,(t) the motion
along the MPEP is fast. It is seen from Eq. (3.18) that the system moves between
these regions over time tr. Close to qb(t) the system is slowed down, and in the
region [q — qb(t)| g l D the motion is dominated by diffusion. The duration of staying
in the vicinity of qb(t) can be obtained by linearizing the equation of motion (4.1)

near qb and is given by the Suzuki time t S ~ jab—1 ln(Aq/lD) [117, 130], where Aq =

93

 

mint |q7,(t) — qa(t)| is the typical distance between the periodic states.

Periodically modulated systems are advantageous as they allow one to observe, via
the prehistory distribution, both the fast motion along the MPEP and the slow motion
near the unstable state. As we show below, the peak of the PPD does not display
broadening due to diffusion near qb, as does the PPD in the absence of modulation

[31].

4.3 Prehistory Probability Distribution near the
basin boundary

We will calculate the PPD in the regime of strong synchronization of escape [110, 111].
In this regime the probability distribution p(q, t) of finding a particle behind the basin
boundary has the form Of sharp periodic pulses as a function oft. It is most interesting
to find the PPD for a final point (q f, t f) on the (q, t) plane near the center of such a
pulse.

We will assume that the point (q f, t f) is sufficiently far from the diffusion-
dominated layer (region B in Fig. 4.1) around the basin boundary, so that the distance
to the boundary is Q f = q f — q7,(tf) > ID. This condition is usually realized in ex—
periments, where the position of a particle detector is chosen so as to ensure that the
detected particles have practically no chance to return to the attraction basin. At
the same time we assume for convenience that the final point is still in the harmonic
region, Q f < Aq, in which case the motion between qb and q f can be described by
linearizing equation (4.1) in Q = q — qb(t).

We will start the analysis of the PPD 777, (q, thf, t f) with the case where not only

q f but also the point (q, t) through which the trajectory of interest has passed is also

94

in the harmonic region around the basin boundary q7,(t).

4.3.1 Transition probability density

As seen from Eq. (4.2), finding the PPD requires calculating the transition probability

density p(q f, t flq, t). The Fokker-Planck equation for p(q f, t flq, t) can be linearized

near qb(t),
01,1) = —ub(tf)6e,(0fp) + 005,12. (41)
Where P = P(qf.tf|q,t) and Qf = (If - Qb(tf)~
The solution of Eq. (4.7) can be sought in the form of a Gaussian distribution
p = G (07 — 0(516071) . (4.8)

with C(IE; a) given by Eq. (4.6). Substituting Eq. (4.8) into Eq. (4.7) we obtain

(152 ~2 dQ~ ”
cl—tf. — 271b(tf)0 + 2, 8?; — Hbftle- (4'9)

The initial conditions for these equations follow from the condition p(qf,t|q,t) =
6(qf ——q). They have the form (”72(t) = 0 and Q(t) = Q = q — qb(t). Then the solution

of Eqs. (4.9) is
‘1
62(tf) = 2/ dTK72)(tf,7-),
t

Q(tf) = QKbUf. tl- (4-10)

Finally, using the function
2 —2 ~2 ti -2
af(tf,t) = ”b (tf,t)(1 (if) = 2ft d115,) (1,t), (4.11)

we can write the distribution in the form

p = e;1(1f,t)(; (Q — Qfe;1(tf,t);ef(tf,t)). (4.12)

95

 

The function Q f’fb— 1(t f, t) has a simple meaning. Consider the noise-free trajec-
tory that passes through the point Q f at time t f. This trajectory should have passed
through the point anb‘1(tf,t) at time t. As expected, the transition probability
(4.12) is maximal for Q coinciding with this point.

The function 012., Eq. (4.11), is simply related to the function 030%), Eq. (3.6),

introduced earlier,

03(1) — e;2(1f,1)e§(1f) = 0301”). (4.13)

We will broadly use this relation in what follows.

4.3.2 General expression for the PPD near qb(t)

From Eq. (4.2), the PPD is determined by the product of the transition probability
(4.12) and the ratio of the quasiperiodic distributions p(q, t) / p(q f, t f). The distribu-

tion p(q,t) close to qb(t) was found in Sec. 3.2.1. It has the form

p(q, t) =/() (402(1), Q, t), Q = q — (130),

1 192030)

9,00,62,11) '—' WfieXp{—D[ 2 +pQ+S(¢)] }

1
(7)(p, t) = QF 1n [p__/€7,(t,t )
#le

 

] i 9F = WF/flb- (4-14)

Here 8 and t’ are constants, 5(0)) 2 3(0) + 211') is a zero-mean 21r-periodic function,
and 9p is the dimensionless modulation frequency.

The function 3(0)) in Eq. (4.14) plays the role of an instantaneous modulation-
induced change of the activation energy. In the regime of strong synchronization of
escape the minimal value sm of 3(0)) satisfies the condition [Sm] > D. The minima
Of 3(0)) lie on the optimal escape paths (see Chapter 3), p = p230) E popt(t + n1'p).

Here n = O, :i:1, . . . enumerates periodically repeated MPEP’S, see Fig. 4.1(b); we set

96

0 0
(45011125301): 00310153310».
Near the basin boundary qb(t) the Optimal paths satisfy the linearized equations

(3.18) and evolve in time as

p040) = 6,710,103.01. 00310) = —a£<t>popt(t).

QOPtU) = Gent“) _ ‘Ib(t)1 lQoptl << A(I. (4.15)

Eq. (4.15) describes how a given optimal path approaches (q7,(t), p = 0) for t —+
00. The parameter t' is determined by matching to one of the periodically repeated
trajectories (3.18) that start from (qa(t),p = 0) for t ——> —oo.

Expanding the function 3(0)) around its minima at the periodically repeated

MPEP’s, to second order in (0(7), t—) 00(7):; (t), t) we Obtain the probability distribu-

tion (4.14) as a. sum of contributions from the MPEP’S,

88 —sm/D

p’(p,Q,t)= \/— Zexp[-1()” (1),Qt)/D]

)2 (12 t 928 ”

2 2 p030)

rm) = pQ +

 

(4.16)

Here 311, E ((123 / d¢2hn is the curvature of the function 3 at the minimum. For strong

synchronization, where [311,] > D, we have 34;, >> D as well, which is a consequence of

5(0)) being a zero-mean periodic function. The quantity 3” can be found [110111],
along with the constant 8 exp(—sm / D), by matching the periodic distribution (4.14)
to the distribution well inside the attraction basin.

It follows from Eq. (4.15) that near the basin boundary

£530“): P( )(t‘l'kTF): M; “ 12“”(t) (4.17)

opt popt

With account taken Of the relation A47, = exp(27r/QF), this leads to the expression

 

1(n+k)= T(") + 27erF3ii11n (111)) + 2791025111, (4'18)
popt(t)

97

Combining Eqs. (4.12) and (4.16) we Obtain for the PPD near the basin boundary,

1)).(q,t|qf,tf)

= nb-1(tf,t)G (Q —- an;1(tf,t); (U(q,t))

Zn IOOO dp exp {—7.00 (I), Q) tl/D] (4 19)

X . .
Enfo‘” dpexp [—r<">(p,0f.tf)/D]

 

This expression gives the general form of the PPD near the basin boundary. We note
that, even though the equations of motion near q7,(t) can be linearized, the PPD is

generally non-Gaussian. This distortion is an important feature Of escape dynamics.

4.4 Adiabatic regime near the basin boundary

we start the analysis with the case of slow modulation, 9 F << 1, where the motion
can be described in the adiabatic approximation. As we show below, in the adiabatic
regime and for strong synchronization, 34;, > D, only one term contributes to each
of the sums in Eq. (4.19). The shape of the PPD in this case is determined by the
parameter 6, Eq. (3.36), introduced in Sec. 3.5.

We call 6 the distortion parameter. This is because, as shown in Sec. 3.5, for
6 << 1 the escape current has a form of Gaussian peaks, whereas for larger 6 the
peaks of the current become non-Gaussian. Formally, 6 is equal to the ratio of two
small parameters, the squared reduced modulation frequency 1.1) F / [17, = 51p and the
reduced noise intensity D/sfil.

The physical meaning of 6 can be understood in the following way. In the adia-
batic picture one usually thinks of escape as occurring in the instantaneous potential
U (q,t). Most likely it happens once per period at the time tm when the barrier

height AU(t) = U(qb(t),t) — U(qu(t),t) is minimal, cf. Fig. 4.1. As explained in

98

 

the Introduction, the typical width of the time window for escape At is determined
by the condition [dzAU/dt2]m(At)2 = D, where the subscript m indicates that the
derivative 18 evaluated for t— — tm. The parameter s” in the adiabatic approximation,

as given by Eq. (3.33), is
Siii “ —ld2AU/dt2lm/QF/fb(t ml~ ~id2AU/dt2]m/w%.

Therefore the parameter 6 ~ [(At)27'1g]_1 is the squared ratio of the relaxation time
of the system tr = [1;1 to At. It shows whether the system moves fast enough to
escape while the barrier remains at its minimum or the barrier noticeably changes
during escape leading to a delay of escape with respect to tm and a distortion of the
tube of escape trajectories.

Because the PPD evolves over time ~ tr, of interest is the time range It — t f] g
t, << 1p. Moreover, both t and t f should be close to tm. Then the instantaneous
relaxation rate 71b(t) can be approximated by its value 717,”, E 717,(tm). The functions

11.7, and (I; become

“b111(tf1t) = exp [Hbmuf _ ti] 1
0%",(tf, t) = of“, {1 — exp [—2717,m(t f —-1)]} , (4.20)
where

2 2 ._ 2
00(t) K)" 0b(tm) : me : l/flbm.

4.4.1 Weak distortion, 6 << 1

We consider first the limit of weak distortion, 6 << 1 (more precisely, the weak dis-
tortion condition has the form 6ln2(sg, / D) << 1, see below). In this limit one can

think of motion in the fully adiabatic way, assuming that it occurs in a quasistatic

99

potential U (q, tm). The periodic distribution behind the attraction basin p(qf,t),
which is proportional to the instantaneous escape rate, has a form of periodic in time
Gaussian pulses, with width ~ (D/sgf/zrp [111].

For small 6 expression (4.19) for the PPD can be simplified. We will start with the
analysis Of the denominator in this expression. If there were no term oc (ti-‘83, / D = 6
in 10‘) (p, Q f, t f) / D, the typical values of p contributing to the integral over p would
be ~ D / Q f. For such 7) there may be only one n for which the term 6 ln2 [p/pgg[(tf)]
in 1("l/D is small, whereas for all other 11 it is oc sis/D [cf. Eq. (4.18)], making
the integrand exponentially small. A similar argument applies to the numerator
in Eq. (4.19), except that, depending on Q, the typical values Of p are Of order of
(D/l'l'bm)1/21D/Q) or ~u3mQ-

Keeping only the leading term in the sums in the numerator and denominator of

Eq. (4.19) and disregarding corrections cc 6, we Obtain

ph<§latl9fitfl : C (Q _ angn11(tfat); Uf111(tfat))

_1 2
x anbmaf’t) exp [—Q—2—] erfc -—Q-— . (4.21)

2 2D 2
( (21)me me ‘ / 217me

In deriving this expression we also took into account that the final point q f is suffi-

 

ciently far from the diffusion-dominated region behind the basin boundary, Q f > ID,
and disregarded corrections ~ lg/fo. We note that the terms cc 6 in rm) in the
numerator and denominator in Eq. (4.19) have a logarithmic factor. This factor may
become large in the weak-noise limit. One can show that for the corrections cc 6 to
be small it is necessary that 6ln2(sfil / D) < 1. This condition is equivalent to the
inequality At >> t 5, where At is the characteristic time window within which the
barrier height is practically constant and t 3 is the Suzuki, see Eq. (4.22) below.

Equation (4.21) for the PPD is further simplified for such times that the point

100

Q fab— 1 (t f, t) is far behind the basin boundary compared to ID. This is the point that
the noise-free trajectory arriving at Q f at time t f passes at time t. For such t, the
PPD as a function of Q has the form of a Gaussian peak with variance D0§(tf, t).
This means that the trajectories arriving to the point Q 7 are close to the noise-
free trajectory. The tube of these trajectories is diffusion-broadened, with width
o< [D(tf — t)]1/2 for small tf — t.

Because for 6 << 1 escape as a whole occurs in the quasistatic potential U (q, tm),
the full calculation of the PPD described in Section 4.6 leads to the same result as
in the case of escape in a stationary potential studied earlier [81]. With increasing
t f —-t the peak of the PPD (4.21) crosses the basin boundary and enters the attraction
basin. This is the Slowest part of the PPD evolution. Its duration is determined by

the Suzuki time
13 =t,1n(s;;,/D). (4.22)

The peak is sharply broadened in this region. Deep on the intrawell side, —Q >> lD,
the width of the peak becomes independent of D, i.e. parametrically larger than the
diffusion-limited width ~ lD. As t f — t increases further, the PPD peak approaches
the attractor and narrows down, with the width becoming again diffusion-limited.

This part of the evolution takes ~ tr.

4.4.2 Strong distortion, 6 >> 1

The shape of the PPD near the maximum changes dramatically in the range where
the modulation is still dynamically Slow, OF << 1, but the distortion parameter 6
becomes large, 6 >> 1. Here, the shape of the potential barrier for escape changes as
the particle crosses the diffusion region around the basin boundary. As a consequence,

the pulses of escape current become strongly asymmetric, see Sec. 3.5. This leads to

101

a strong change of the PPD compared to the picture based on the quasistatic barrier
that was discussed before.

For 6 >> 1 in an important range of Q, t and Q f, t f the integrands in the numerator
and denominator in Eq. (4.19) have sharp extrema as functions of p for p = pg]: (a
more precise condition is specified below). Integration over p can be done by the

steepest descent method. It gives the PPD in the form of a Gaussian distribution

over Q with time—dependent center Qn0 and variance D0?

phv((I1tiqf)tf) = G (Q _ Qn0;0f(tfi t)) 9 (423)

where

0.. :- 0..(t|qf.tf)=0§,’,:l(0+(0f—02Z1(tf))n;1<t7.t) (4.24)

and no E 110(q f, t f). Here we have taken into account that the major contribution to
each of the sums in Eq. (4.19) comes from one term, with 11 = no.

It is clear from the analysis of the denominator in Eq. (4.19) that the value of no
is determined by the condition that pggg)(t f)Q f / D ~ 1 for the noth optimal path.
The very existence of such n0 follows from the fact that we consider t f for which the
probability density p(q f, t f) is close to its maximum over t f; this maximum is reached
once per period for p£g[(tf)Qf/D = 1 [110, 111]. The terms with [n — no] 2 1 in the

denominator in Eq. (4.19) are
oc [Ugo—n exp (—1W;lo_n) << 1

where we have used M7, = exp(21r/QF) >> 1 (this estimate is written for

71
105.3211 00 f/D = 1).
A simple qualitative argument shows that the same no gives a major contribution

to the sum in the numerator of Eq. (4.19) in the adiabatic regime. Indeed, the terms

102

with different n correspond to t changing by an integer number of the modulation
periods 7p, whereas in its central part (section C in Fig. 4.1) an optimal escape
trajectory lasts for the time small compared to T1? for adiabatic modulation. Therefore
It — t fl << TF. The formal condition for this approximation to be true is that the
Suzuki time t 5 (4.22) is small compared to the modulation period.

The maximum of the Gaussian PPD peak (4.23) lies at Qn0(t|q f, t f). This func-

(n)

opt’ which is located inside the attraction basin,

tion is a sum of the optimal path Q
and the decaying in time term or Q f — Qggutf), which is determined by the final
point (q f, t f) and is located outside the attraction basin. Its time dependence is par-

ticularly simple in the case where t f corresponds to the maximum of the distribution
F((Ifitf), i-e-, Pg:(t),)(tf) = D/Qf,

——D cub"‘(tf_t). (4.25)

'n t ,t z e—“mef'O __
Q 0( le f) Qf umef

The first term in Q7101 Eq. (4.25), decreases with increasing t f — t, which describes

 

 

 

 

 

 

 

 

 

1 1
on 02
01 _f_
Q1 ID 0:",
0~ :
Q, 0.1 0.5,
I
D
—=0.2 «
-1, or
f . . . . . . . o . . s . - . .
o 1 2 3 4 5 o 1 2 3 4 5
11mm) ubmflrt)

Figure 4.3: The reduced position of the maximum (left) and the reduced width (right)
of the PPD (4.23) in the adiabatic regime as a function of the reduced time pbm(tf—t).

approaching the basin boundary backward in time. The second term, on the other

hand, increases with t f — t; this term describes the motion, backward in time, from

103

the boundary to the interior of the attraction basin. The motion at the boundary
is initiated by noise, and therefore this term is or D. For sufficiently long t f —— t

(”0) (t). The overall

the distribution maximum Qno approaches the optimal path QOpt

behavior of Qno is shown in Fig. 4.3. It is seen that the motion is slowed down near
the basin boundary, and the slowing down strongly depends on the noise intensity, in

agreement with Eq. (4.25).

The width of the Gaussian PPD peak (4.23) is diffusion—limited; its time depen-
dence is given by Eq. (4.20) and is also simple. It is shown in the right panel of

Fig. 4.3.

The condition that the integrands over p in Eq. (4.19) are maximal for pg?)
imposes a limitation on Q where the explicit expression for the PPD (4.23) applies.
For Q close to the maximum of the distribution, |Q — Qnol 3 l0, this condition takes
the form 6 >> ng'g‘g) (t) / l DI- Therefore expression (4.23) describes the distribution not
only in the whole range between (Q f, t f) and the basin boundary, but also throughout

the diffusion region around the basin boundary and a region that goes deeper into

the attraction basin.

4.5 Nonadiabatic regime near the basin boundary

The PPD can display a qualitatively different behavior for nonadiabatic modulation,
0 F ~ 1. With increasing t f —t the PPD inside the attraction basin can split into sev-
eral peaks. Such splitting occurs already close to the basin boundary and is described
by the general expression (4.19). Formally, the PPD has well-resolved multiple peaks

when several terms in the numerator of Eq. (4.19) are of the same order of magnitude.

In the nonadiabatic regime necessarily 0 >> 1. Therefore integration over p in

104

Eq. (4.19) can be done by the steepest descent method. The integrands are maximal
(n)

opt“ Each term in the numerator gives a Gaussian distribution over Q

for p = p
centered at Q" E Qn(t|qf, tf), Eq. (4.24), with variance D0%(tf, t), Eq. (4.11). This is
similar to the adiabatic case (4.23), except that now the PPD is a sum of appropriately

weighted Gaussian peaks,

BnG - n; t,t
Ph(qatl(1fatf) = E" (g; in of( f D. (4.26)

 

The weighting factors 8,, are

8,, E Bn(qf, tf) = xn exp(—:1:n),

a. = mum, tf) = pgzkthf/D. (4.27)

If we keep only one term in the numerator and denominator in Eq. (4.26), with
the same n = no given by the condition that Bn is maximal for n = no, Eq. (4.26)
goes over into Eq. (4.23). However, in the nonadiabatic case there are regimes where
the PPD as a function of (Q,t) may display several peaks. Eq. (4.26) gives their
shapes near the maxima.

The number of peaks of ph depends on Q,t, and Qp, as well as the final point
(qf,tf). As before, we choose the final point so that the probability density be-
hind the basin boundary p(q f, t f) is close to its maximum over t f, which occurs for

an(q f, t f) ~ 1. The amplitudes of different peaks are
3,,0 +1.: = anMb‘k exp (—xn01w,;’~‘) , (4.28)

with Mb = exp(27r/QF) being the Floquet multiplier, Eq. (3.4). In practice, in
the whole range of modulation parameters where escape is strongly synchronized the

factor Mb is comparatively large. Therefore for 33710 ~ 1 the coefficients 3710+ k rapidly

105

decay with increasing |kl, and only few peaks of the PPD can be simultaneously

observed, primarily with non-negative k = 0, 1,. . ..

The other condition for observing several peaks is that the distance between them
exceed their width, that is in particular IQ”O — Qu0+1l >> ID. For Q % Q-n0 >> l D,
i.e., when the PPD maximum is well behind the basin boundary with respect to the
attraction basin, we have lQno — Qn0+1| ~ |Q£:2)(t)| ~ l2DKb(tf,t)/Qf << 11), where

we have used that Q f >> I D and that nb(tf, t) ~ 1 when the PPD maximum is behind

the boundary. Therefore in this region the PPD has indeed only one peak.

For longer t f — t when Kb(tf, t) >> 1 and the peak positions Qno, Qno+1 are well
inside the basin of attraction, the distance between them exceeds ID. In this range
the corresponding peaks of p}, can be resolved. The ratio of their amplitudes is given
by the factor Alb, as seen from Eq. (4.28). This is in good agreement with the results
of numerical simulations for a Specific model shown in Fig. 4.2, as will be discussed

below.

We note that for 9 F >> 1 synchronization of escape by modulation becomes
exponentially weak. Although the system is far from the adiabatic limit, dynamics
of the period-average coordinate is similar to dynamics in the case of a stationary
system. Even though the factor Mb becomes of order I, of interest is the PPD with
respect to the period-averaged coordinate, and this PPD does not display multiple

peaks.

106

 

4.6 The PPD inside the attraction basin

We assume throughout this Section that escape is strongly synchronized and 0 >> 1.

To find the PPD for (q, t) deep inside the attraction basin we will use the equation

Ph(q1tl(1fatf) = [ck/mm,th',t’)ph(q’,t'qu.tf)- (4-29)

It applies for t < t’ < t f, as follows from the definition (4.2), and can be obtained [81]
from the Chapman-Kolmogorov equation for the transition probability p(qf, t flq, t).

As we show, t' in Eq. (4.29) can be chosen in such a way that only a narrow range
of q’ contributes to the integral. The corresponding q’ are close but not too close to
qb(t’). In this range both factors in the integrand, and hence the integral as a whole,
can be explicitly calculated. In particular, the function p},(q',t’|qf,tf) is given by
Eq. (4.26).

The function p},(q, th’ , t’) is the PPD inside the attraction basin. For a periodically
modulated system it was found earlier [131]. Because in escape the system is likely
to move close to one of the periodically repeated most probable escape paths (15);): (t),
we are interested in ph(q, th’, t’) for both (q, t) and (q’, t’) lying close to such a path.
If both (q,t) and (q’,t’) are close to an nth path (15,30), the corresponding PPD

p51") (q,t|q’ ,t' ) is Gaussian [131],

14:001. th’, t') = C(q - qn(tl(1', t'); 0n(t', t)) (430)

Here qn(t|q’, t’) is the value of the coordinate at time t on the optimal path that leads
to the point (q’, t’). This optimal path is described by Eq. (3.18).

The coordinate qn(t|q’, t') as a function of time is close to qézlu). We can seek it
(n

qopZ(t) + 6qn(t|q’, t’). To the lowest order in the deviation of

in the form qn(th’, t’) =

(q’, t’) from (101)

Opt (t'), the function 6g" can be found from linearized equations (3.18).

107

This gives

661." = Vn(t)6qn, (4.31)
I
Vu(t)= ath+—0(1q(K2) -
2 q(n)(t)
opt
The boundary conditions for (Sqn are 6q,,(t’lq’,t’) = q' — qégzu') and

limt_._OC. dqn(t|q',t') = 0; the latter condition simply means that the optimal path
approaches the attractor qa(t) as t —> —00.
It follows from Eq. (4.31) that function flnU) = (mu/6% satisfies a first-order

(Riccati) equation,
Bu + 1872; = Vn(t)~ (4'32)

For q',q(7 )(t') both close to the basin boundary qb(t’), with account taken of the

l
opt

boundary conditions for 61]., the solution of Eq. (4.31) can be written in terms of [in

64.0qu t’) = (Q’ — 6251,3103) mat’), (4.33)
where
n,,(t, t’) = exp[ [1' 2175741)] . (4.34)

We now discuss the asymptotic behavior of fin“). For not too large t’ — t the
function Vn(t) in Eqs. (4.31), (4.32) should be calculated for qéfikt) z qb(t). For such
Vn(t), the solution 6qn(t) of Eq. (4.31) can either exponentially increase or decrease
with increasing 1." —t. Of interest to us is the decreasing solution, which will ultimately

go to zero for t’ — t —+ 00. A simple calculation shows that for this solution

6.0) z MW) 5 [5’ququ (15.220) 2 (11.0)-

For much larger 15’ —t the optimal path qégut) approaches the attractor qa(t), and

then the function Vn(t) in Eqs. (4.31), (4.32) should be calculated for 9632.“) z qa(t).

108

 

One can show that the solution 6qn(t) —-> 0 for t —> —00 corresponds to

flu“) z #0.“) + 20;2(t)’ (18:20:) ‘3 (10“):

where 03(t) is the variance of the periodic distribution about the attractor given by
Eq. (3.6).

To the lowest order in q’ — q(")(t’), the variance of the prehistory probability

opt

distribution (4.30) is also expressed in terms of the function 67, [131],

t,
0,2,(t’,t) = 2 / dTrt;2(T,t). (4.35)
t

Using Eqs. (4.32), (4.34), and (4.35) one can show that a 72,,(t’ t —> —oo) = 03(t).
For t —> ——00 the distribution pg")(q,t|q’ , t’) goes over into the periodic Gaussian
distribution pa(q, t), Eq. (4.5), centered at the periodic attractor.

It follows from Eqs. (4.30)-(4.33) that, for q close to qopt(t), the PPD p(n) (q, th’, t')
displays a diffusion-broadened Gaussian peak as a function of q’ with maximum close
to q(7l:t n) t(t ’.) The displacement of the maximum over q’ from qOpt )(t’) is o< [q— qopt(t)]
On the other hand, the function ph(q’,t'|qf,tf) as given by Eq. (4.26) is a sum of
Gaussian distributions centered at the optimal escape paths. Therefore integration
over q’ in Eq. (4.29) can be done by the steepest descent method. The extreme values
of q’ are also close to the optimal escape paths.

Different peaks of the PPD ph(q’ , t’ lq f, t f), Eq. (4.26), correspond to the MPEP’S
shifted by an integer number of modulation periods. For each of these peaks one
should use in Eq. (4.29) the PPD ph(q, th' t’ ) given by pg”) with the appropriate n.

The result of integration over q’ describes the peaks of the PPD ph(q, th f, t f) for

q close to gm)

Opt(t) with appropriate n. These peaks are Gaussian near the maxima,

Ph(q,t|(1f.tf)= in 3110 ((I— qopt( t-l ngglUf) K’fil(tfat);0'n(tfat))-
Ean
(4.36)

109

Here B” E 8,, q ,t are given by Eq. 4.27). In obtaining this equation we used
f f

the relations

Kb(tf,t’)rtn(t’, t) = 14,,(tf,t),

2 — 2
a,.<t’, t) + 03(tf,t’>n..2<t’. t) = am, a

which in turn follow from the relation 6,,(t) z pb(t) in the harmonic region near the
basin boundary. As a consequence, the result of integration over q’ in Eq. (4.29) is
independent of the matching time t’.

Equation (4.36) is the central result of this Chapter. It gives, in the explicit form,
the PPD of finding the system at a position q inside the attraction basin at a moment
t, provided the system has been observed at a point q f outside the attraction basin
at a time t f > t. It shows that, in the regime of strong synchronization, the peaks
are Gaussian. They are centered at the most probable escape paths and are diffusion
broadened throughout the attraction basin. This is in dramatic contrast with the
PPD in the absence of synchronization, where the PPD peak inside the attraction
basin is strongly asymmetric and its width is independent of the noise intensity D
[81].

It follows from Eq. (4.36) that the PPD may have multiple peaks inside the attrac-
tion basin. They can be observed only for strongly nonadiabatic modulation, where
the Suzuki time is t 3 R T}? . In this case the system stays in the diffusion layer around
the basin boundary qb(t) long enough to accumulate influxes from several periodically
repeated most probable escape paths qéght). On the other hand, the modulation pe-
riod TF should not be too short, because the exponential synchronization of escape

would be lost. Since synchronization loss occurs for t,- > TF, the PPD displays well

resolved multiple peaks only in a limited parameter range. The limitation is more re-

110

strictive to the considered case where the final point (q f, t f) is close to the maximum
of the distribution behind the barrier. If this condition is not imposed, the peaks are
well resolved in a broader range, but measuring the PPD becomes more complicated
on the whole. An example is discussed in the next Section.

An important feature of the distribution (4.36) is weak dependence of the shape
of the Gaussian peaks inside the attraction basin on the final point (q f, t f), 'which is a
consequence of the smallness of the factor 551 (if, t). This shows that the PPD reveals
the actual structure of the tubes of the paths followed in escape inside the attraction
basin. In contrast, the relative amplitudes of the PPD peaks 3,, are sensitive to the

choice of the point (q f, t f).

4.7 Results for a model system

In this section, we present the results of numerical simulations of the PPD for a
simple model system and compare them with the analytical predictions. We consider
a Brownian particle in a sinusoidally modulated potential of the form of a cubic

parabola. The Langevin equation of motion has the form of Eq. (4.1) with
2 1 .
K(q,t) = q — 4 + Acos(wpt). (4.37)

The dynamics was simulated using the second-order integration scheme for
stochastic differential equations [107]. The system was initially prepared in the vicin-
ity of the metastable state qa(t). The final point (qf,tf) was chosen behind the
basin boundary qb(t). The PPD was calculated as a normalized probability dis-
tribution of paths q(t) arriving at the point q f for a particular modulation phase
(bf = wth(IIlOd 27r).

An example of the full PPD is shown in Fig. 4.2. The point (qf,tf) is chosen

111

so that in escape the system is likely to pass near it (t f is determined mod 71:),
that is the quasistationary distribution p(q f, t f) is close to its maximum over t f for
a given qf behind the basin boundary [the parameter pg:?)(tf)[qf — qb(t f)] / D is
equal to 1.2, whereas the maximum of p(qf,tf) is expected where this parameter
is equal to 1, [111]]. For the chosen modulation parameters and noise intensity the
calculated activation energy of escape is R z 0.0910 and R/ D m 9.1; the ratio

of the modulation frequency to the relaxation rate Qp z 2.24. We accumulated

~ 105 escape trajectories that arrive into a small area centered at (qf,tf), with

width (Sq = 0.02, 6t = 0.02TF.

It is seen from Fig. 4.2 that, in the regime of strong synchronization and for
strongly non-Gaussian pulses of escape current, the peaks of the PPD are narrow
both behind the basin boundary and inside the attraction basin. Moreover, for the
chosen parameter values two distinct peaks of the PPD are well resolved inside the
attraction basin. They correspond to the two paths the system is most likely to follow

in escape.

The positions of the PPD peaks on (q,t) plane are shown in Fig. 4.4 with full
squares where there is one peak, and with crosses where two peaks are well resolved.
They are compared with the periodically repeated optimal escape paths calculated by
solving numerically the variational equations (3.18) for the model (4.1), (4.37). Such
paths start from the periodic attractor qa(t) for t —+ —00 and approach the basin
boundary qb(t) for t ——+ 00, which are also shown in the figure. It is seen that, as
expected from our analysis, the PPD maxima lie nearly on top of the most probable
escape paths. The small deviation is due to diffusion broadening and associated small
asymmetry of the PPD peaks for the noise intensity used in the simulations. Fig. 4.4

demonstrates that studying the PPD provides a direct way of observing most probable

112

 

(cw

 

 

 

 

Figure 4.4: The positions of the maxima of the PPD ph(q, t|q f, t f) in Fig. 4.2, which
show the most probable paths followed by the system in escape. The data of simula-
tions are shown by full squares where the PPD has one peak and by crosses where two
peaks are well resolved. Solid lines show periodically repeated most probable escape

paths qég]: (t) for the model (4.1), (4.37). Dashed lines show the basin boundary qb(t)
and the attractor qa(t).

escape paths.

The observed shape of the PPD peaks is compared with the theory in Fig. 4.5,
which shows the cross-sections of the PPD at several instants of time counted off from
the final time t f. For small t f —t the system is behind the basin boundary and moves
close to the noise-free trajectory leading to (q f, t f) . The top left panel refers to the
case where the system is localized close to the boundary. Here and for smaller t f — t

the PPD has a single sharp peak.

For earlier time (larger t f — t) the system could either be moving towards the
basin boundary along the most probable escape path or could have been fluctuat-
ing about the basin boundary after it had arrived to its vicinity along the previous

MPEP (shifted by 7p). As we showed analytically, the probability of staying near

113

 

the boundary is smaller, but the PPD may still display two peaks. This is seen in
the right top panel. The main peak corresponds to motion along the MPEP, which
is close to qb(t) for the chosen time. The higher-q shoulder corresponds to the poorly

resolved (for the chosen time) peak for fluctuations about the basin boundary.

 

 

'(= t1 - 0.3‘EF

pp, 1 l \
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.5: Cross-sections of the PPD ph(q, t|q f, t f) shown in Fig. 4.2 as functions of
the coordinate q for the time t f — t = 0.317, 0.571;, 7p, 27F. The results refer to the
model system (4.1), (4.37) with A = 0.7, to}: = 2, D = 0.01, qf = 0.8, tf = 0.577;.
The point (Qf, t f) is close to the expected maximum of the distribution behind the
basin boundary. Solid lines show the expression (4.36). Squares show the results of

simulations.

For still larger but not too large t f — t the escaping system should have been

114

moving towards the basin boundary. In the present case it most likely followed one
of the two periodically repeated MPEPs, with different probabilities. Well-resolved
peaks of the PPD in this range are seen in the left lower panel of Fig. 4.5. For
large t f — t compared to the relaxation time and TF, the system should have been
fluctuating about the attractor qa(t). The PPD in this case should have a single peak,

which is seen in the right lower panel.

It is seen from Fig. 4.5 that the results of simulations agree with the analytical
results. Not surprisingly, the observed peaks are broader than the asymptotic theory
predicts. This is a consequence of the relatively large noise intensity used in the

simulations.

4.8 Conclusions

In this Chapter we have studied the prehistory probability distribution for activated .
escape in periodically modulated systems. We have shown that the PPD can display
one or several narrow peaks within the basin of attraction to the metastable state.
The peaks are located at the periodically repeated most probable escape paths. They
are Gaussian near the maxima and are diffusion broadened, see Eq. (4.36).

The number of peaks that can be observed depends on the parameters of the
system. The multipeak structure is best resolved in a limited parameter range. On
the one hand, the modulation should not be too slow, so that the system does not
follow it adiabatically. On the other hand, it should not be too fast, so that escape
events are strongly synchronized and the system dynamics is not described by period-

averaged coordinates.

Most of the results refer to the case where the final point (qf,tf) is chosen so

115

that the escaped system has a high probability density of passing through this point.
Such choice simplifies the experimental observation of the PPD. For the corresponding
(q f, t f) the amplitudes of different peaks of the PPD differ from each other signif-
icantly, as seen from Eq. (4.28). These amplitudes are sensitive to the choice of
(q f, t f). In contrast, the positions and shapes of the PPD peaks very weakly depend
on (q f, t f). This shows that the PPD provides a means for studying the distribution
of trajectories leading to escape.

We have performed extensive numerical simulations of escape of a Brownian par-
ticle from a model modulated metastable potential. The simulations confirmed the
possibility to clearly observe most probable escape paths. They are in a good quali-
tative and quantitative agreement with the analytical theory.

Observing most probable escape paths is interesting from several points of view.
First, periodically modulated systems are far from thermal equilibrium. In contrast
to the case of systems in equilibrium, most probable escape paths in nonequilibrium
systems have no immediate relation to dynamical trajectories in the absence of noise
and display interesting and often counterintuitive behavior [132]. Therefore observing
escape paths has broad implications. Second, understanding the dynamics of a system
in escape enables efficient control of this dynamics and the escape probability by
applying a control field in the right place and at the right time.

In conclusion, the results of this Chapter suggest a way of direct observation of
most probable escape paths, in space and time. They also describe, qualitatively and

quantitatively, the distribution of the trajectories followed in escape.

116

Chapter 5

Conclusions

While it is generally understood how escape happens in equilibrium, it is still largely
an open question for systems away from equilibrium, such as those modulated by
time-dependent fields. In this thesis we have established previously unknown universal
features of activated escape in nonequilibrium systems. In particular we investigated
scaling of the activation energy and the pre—exponential factor of the escape rate near
a bifurcation point, time dependence of the escape current, and dynamics of escape
pathways in periodically modulated systems.

In equilibrium systems, the rate of noise-induced activated escape from a
metastable state is W oc exp(—R/ D), where R is the activation energy, and D is
the noise intensity. For thermal fluctuations D = kBT and the activation energy
R is given by the height of the free energy barrier. When the system approaches a
saddle-node bifurcation point (spinodal point) where'the energy barrier disappears,
the activation energy scales as R oc |A — Aclg, with 5 = 3/ 2. Here A is the control
parameter, e.g. the strength of the applied DC field, and Ac is the critical value of
A. This scaling is related to the universality of critical dynamics near the bifurcation

point, where the effective potential becomes locally cubic.

117

In periodically modulated systems, the period-averaged escape rate has a similar
form, 17V— o< exp(-—R/ D), but the activation energy R is not equal to the barrier height.
The most interesting and unexpected scaling behavior is displayed for sufficiently
slow modulation, where dynamics away from the critical point can be well described
in the adiabatic approximation. Due to the critical slowing down, the adiabatic
approximation necessarily breaks down close to A = Ac. As we have shown, this
breakdown of adiabaticity occurs in a universal way. Sufficiently far from Ac the
adiabatic scaling with 5 = 3/ 2 is observed. For smaller [AC — A] there emerges a new
dynamical time scale and a new, locally nonadiabatic scaling of the activation energy
with 5 = 2. For even smaller lAc — A] the exponent goes back to £ = 3/2. Thus,
scaling of the activation energy in periodically modulated systems involves crossovers
between three scaling exponents E = 3/ 2, 2, 3/ 2 instead of a single exponent 5 = 3/ 2
in the equilibrium case. The widths of the three regions with different scaling depend
on the modulation frequency. The new dynamical scaling with g = 2 is a consequence

of nonadiabatic retardation effects and has no analogue in equilibrium systems.

The full escape rate in equilibrium is independent of time and has the form
W = uexp(—R/ D), where the prefactor u is given by the generalized “attempt fre-
quency” in the metastable potential well. In periodically modulated systems escape
rate W(t) is a periodic function of time. It is given by the current j (q, t) away from
the metastable state, which also depends on the position q where it is measured. It
is physically meaningful to measure the current sufficiently far behind the boundary

of the basin of attraction to the metastable state.

We calculate the escape current using the eikonal approximation and methods
of nonlinear Hamiltonian dynamics and matched expansions. We identify a single

parameter 0 that determines the shape of the periodic pulses of the escape current

118

as functions of time. It depends on both the strength and the frequency of the
modulation and is given by the ratio of two small parameters, the scaled modulation
frequency and the scaled noise intensity. In the limit of adiabatically slow modulation
or small |A —- Ac] we have 0 << 1, and the current pulses have Gaussian shape with the
width much smaller than the modulation period. For 0 >> 1 the pulses are strongly

non-Gaussian and asymmetric.

The method developed for the analysis of the instantaneous escape rate made it
possible to find the prefactor u of the period-averaged escape rate and analyze it close
to the bifurcation point A 2 Ac where the periodic metastable state disappears. We
have found that the prefactor displays three regions of system-independent scaling
corresponding to the scaling regions of the activation energy, with V oc [A — ACIC.
The scaling exponents are C = 1 / 4, —1, 1/2. The exponent C = 1/4 can be observed
for sufficiently slow modulation not too close to the bifurcation point. The negative
exponent C = —1 is a consequence of the onset of local nonadiabaticity, which leads to
the exponent C = 2 for the activation energy. The exponent C = 1 / 2 can be observed

for fast modulation or very close to the bifurcation point.

One of important questions regarding fluctuating systems is how a fluctuation
evolves in time. For instance, if we find a system at a phase-space point far away
from equilibrium it might be of interest to know how the system was moving on the
way to this point. This knowledge gives one the power to control fluctuations in
the system, e.g. to increase or suppress the probability to visit certain areas of the
phase space by applying external fields. Control of infrequent events such as escape
from a metastable state is facilitated by the fact that, in fluctuations leading to such
events the system is most likely to follow a certain dynamical path. Fluctuational

trajectories form a narrow tube centered at this path. We have developed a formalism

119

for studying these tubes in periodically modulated systems. We found, and confirmed
by detailed simulations that, in contrast to stationary systems, periodically modulated
systems can exhibit several tubes of trajectories leading to escape at a given time.
The tubes have specific shapes and intensities, which we have described analytically.
The spatial and temporal localization of the trajectories suggests the ways of optimal

and selective control of escape events.

120

APPENDICES

121

Appendix A

Variational equations for the

escape problem

We will consider here the optimal trajectory that the system is most likely to fol-
low in escape, (lopt (t), and the random force fopt (t) that drives the system during
this motion. The trajectories qopt(t),fopt (t) provide the absolute minimum to the

functional ’R[q, f] (2.9). The variational equations have the form

/ dt’f-(t — t’) f(t’) -— Mt) = o, (A.1)
A = —V(>. - K), (.142)
q — K(q; A, t) — f(t) = 0 (4.3)

(here and below we use the hat to indicate a matrix; V .=_ B/aq).
From Eqs. (2.7) and (A.1) it follows that the optimal noise realization is expressed
in terms of the Lagrange multiplier A and the matrix of the noise correlation functions

(5 (2.3) as

fopt = D‘1 / dt’¢(t —- t’)).(t'). (A4)

122

Therefore, for the optimal path, the activation energy functional (2.6) is

72 = éD‘l // dtdt’A(t)- an — t’)).(t’) (11.5)

(note that the noise intensity D drops out from R, because the correlation functions
«,0,- j (t) are proportional to the noise intensity themselves).

From the structure of the functional (A.5) (integration over time goes from —00
to 00 and the integral is nonnegative) and from the fact that the system is initially
in the vicinity of a stable state one immediately derives the boundary condition (2.8)
for t —+ —00. The arguments generalize to periodically driven systems the analysis of
Ref. [51] for stationary systems.

Close to a periodic stable state qa(t) time evolution of Mt) can be described
using the matrix fla(t) = (am/6(1))“, where the derivatives are evaluated for the
state qa(t). Because of the periodicity of qa,(t), the matrix [La(t) is also periodic in

time. It determines the monodromy matrix

.. t+TF
Ala“) = Tt exp [ft dtl fia(t1)] 1

where Tt is the operator of chronological ordering (cf. Appendix C). The matrix
Ma shows how the distance between a point q and the cycle qa(t) varies over the

modulation period in the absence of noise for small |q —— qa(t)|:

(10+ TF) - qa(t) = 1171a(t)[Q(t) - qa(t)l

From the condition that qa(t) is a stable state, the eigenvalues of the matrix [flu
are all less then 1 in absolute value: in this case the distance between q and qa(t)
decreases with increasing time.

It is seen from Eq. (A.2) that the monodromy matrix for A is the inverse transposed

of 11:1“. Therefore its eigenvalues are all larger than 1 in absolute value. Hence, if the

123

system is in the stable state qa(t) for t ——* —00, then A(t) -> O for t —+ —00, and from
(A.1) f(t) —» 0, too.

For the periodic saddle qb(t) on the boundary of the attraction basin, one of
the eigenvalues of the corresponding monodromy matrix exceeds 1 in absolute value.
It is such a saddle-type boundary that can merge with an attractor at a saddle-
node bifurcation we are interested in. Respectively, one eigenvalue of the matrix
that describes time evolution of A is < 1. If A is pointing along the corresponding
eigenvector, it will decay as t —> 00. Then f (t) will decay, too, from Eq. (A.1). This
means that the system may asymptotically approach a saddle-type state. Note that
there are no optimal paths that would go from one stable state to another, because
the condition A —> 0 for t —> 00 is not satisfied there. This explains the boundary

condition (2.8) for t —> 00.

Because the function K is periodic in time, Eqs. (A.1)—(A.3) with boundary con-
ditions (2.8) have a periodic set of solutions. If q(t),f(t),A(t) is a solution, then
q(t + 7F), f (t + 7F), A(t + 7F) is a solution, too. These solutions are heteroclinic
orbits: they connect the states qa(t),f = A = 0 and qb(t),f = A = 0, which are
also solutions of Eqs. (A.1)—(A.3). Generally, only one heteroclinic orbit per period

provides the minimum to the functional R (2.9).

A.1 Escape in systems with a slow variable

Eqs. (A.1)—(A.3) are largely simplified if one of the motions in the system is slow and
all other variables follow this motion adiabatically, i.e., their relaxation time do) is
much smaller than the relaxation time of the slow variable tr, at least for a part of

the modulation period near a bifurcation point. We will assume that slow motion

124

is described by ql; the variable q1 itself may be a periodic function modulated by a
slowly varying factor Q1, as in the case discussed in Sec. 2.3.3. In this case we will
be interested primarily in the factor Q1.

(0)

Over a time ~ tr , the variables qi>1 reach their equilibrium values q;(q1,t) for

given Q1, t. They are determined by the equations

(If = K2:(611,61§>1;A.t)+ fi (2' > 1) (A-G)

In the absence of noise, f, = 0, the solutions of these equations are periodic for given
Q1. As we will see below, the terms f,- here are small and give small corrections.

It is important that the periodic solutions qg>l with fi>1 = 0 be stable. From
this condition and Eq. (A.2) it follows immediately that Ai>1 = 0, otherwise the
components Ai>1 would exponentially grow in time, leading to the onset of a large
force that would drive the system away from the state (A6) with given Q1.

The nonzero component of the Lagrange multiplier A1 is determined by the com-
ponent of the optimal force f1. The latter should overcome the restoring force on the
slow variable Q1 and drive it from the stable to the unstable state. But the slowness
of Q1 means that the restoring force is small, and therefore the force f1 should be
small, too. It is smaller than the force that would be needed to overcome the restoring
force for the fast variables by at least a factor ~ t(0) /tr. Therefore A1 is small as well.

For given A1(t) we can find all components fi>1 from Eq. (A.4). They are all
oc l/tr, and therefore to lowest order in t9) /tr they can be disregarded in the solution
of Eq. (A6) for q,->1.

The problem of escape is therefore reduced to a one-dimensional problem for the
variable q1, the force f1, and the Lagrange multiplier A1. In Eq. (A3) for ql, the

functions qi>1 should be replaced by q£>l calculated for f,- = 0.

125

 

Further simplification occurs if the noise spectrum is smooth. The analysis here
is different for the cases of slow and fast modulation, i.e. whether wptr is small or
large. The case wptr >> 1 is discussed in Appendix C. Here and in Appendix B we
consider the case wptr << 1. The major effect of noise on the slow variable ql comes
from the noise spectral components at frequencies a) g 1/ tr. If the noise spectrum is
flat for such frequencies, the noise can be assumed to be white on the ”slow” time
scale. In other words, the correlation function cpl 1(t) can be replaced by 2D5(t) [in
this situation it is convenient to choose D from this condition rather than to define
it by Eq. (2.4)]. Then 51(1) = $60).

For a 1D system driven by white noise of intensity D, Eqs. (A.1)—(A.3) have a

solution

f1(t) = 2/\1(t) = ()1 - K1. (A7)

R: i/dt ((31 —K1)2

This reduces the variational problem of finding the optimal path to the known for-

mulation for white-noise driven systems [55].

126

Appendix B

Reduced equation of motion for

slow driving

In this Appendix we derive simplified equations of motion for the case of slow driving
where the relaxation time tr << 7'}? and the motion can be described in the adiabatic
approximation. We will consider the vicinity of the adiabatic bifurcation point q =
0,t = 0,A = A2“. A convenient basis for q is provided by the set of the right
eigenvectors of the matrix [4 == (Blfi/qu), where the derivatives are evaluated at the

adiabatic bifurcation point. In this basis the equation of motion (2.1) has the form

. 1 ,
(Ii % #141 + 5 Z]. k I‘ig’k‘lj‘lk + Ki;A(5Aad + Kwt

1
+ “Ki;ttt2 + -Ki;thjt + fi(.t) (13.1)
2 .7

Here Ki;jk = 62Ki/8qj6qk, K2111 = 6K1; / 6A, etc, with all derivatives calculated at
the adiabatic bifurcation point, and ($24”d = A — Afid. Since the function K depends
on t only in terms of the modulation phase 4') = wpt, we have Kid oc cap. The
expansion in t in (B1) is, in fact, an expansion in wpt.

Because the eigenvalue p1 is equal to zero, for small [q] relaxation of (11 is much

127

 

 

(0)

slower than relaxation of qi>1. In the absence of noise, over the relaxation time tr
the variables q,->1 approach their quasistationary values for given ql and d) = wpt.
They can be obtained from the equations of motion (8.1) for qi>1 in which q,- and

f,(t) are disregarded,
(12' z -u,-"1 [Kfitt + Kmlqg + [fa/“Vlad + . . .] (i > 1). (B2)

Here, the major term is linear in wpt. The full expression is a series in q1, wpt, and
(SA, and the omitted terms are of higher order in these variables. Because of the noise,
q.,->1 will additionally perform small fluctuations with amplitude oc D1/2.

The dynamics of the slow variable ql on times exceeding 40) is given by Eq. (B.1)
with 2' = 1, in which qi>1 are replaced with their quasistationary values. To lowest

order in ql, 6AM], and wpt only the linear in t term should be kept in Eq. (B.2). This

gives

41 = aqf + (MAM — 072(wFtl2 + f1(t). (13.3)

:%Kl;11118 = K1;A1
with
2
7 = 1-111WF:;(Kltt+Zid>lflilfljfllK1¢jKiflKfit

—2Zj>1.1K1,,-,K,) . (8.4)

Note that the coefficient 7 is independent of cup.

Eq. (B.3) reduces the multi—dimensional problem of random motion near a bifurca-
tion point to a one-dimensional problem. In the case of a one-dimensional overdamped
system driven by an additive periodic force it was derived earlier in Refs. [133, 134].
Besides the noise term, it differs from the equation of motion for stationary systems

in the vicinity of a saddle—node bifurcation point [84] in that it has a term or (wpt)2.

128

Depending on the sign of (Mad, Eq. (B.3) has either two adiabatic solutions
((1113:; = =Fsgn(a)[—(fi/a)64ad + (7....)211/2 (8.5)

or none. For concreteness, we assume that the adiabatic solutions exist for (Mad < 0,
i.e. 06 > 0. The solutions are even functions of time. They touch each other at t = 0
for 6A”d = 0. We assume that the periodic adiabatic states quij) exist for all times
provided dAad < 0.

The term K1;tt in Eq. (B.1) has to be equal to zero, otherwise the bifurcation
point will be far from (5Aad = t = 0. On the other hand, the equation of motion (B.3)
may contain the term quwpt, where C is a sum of K131 and appropriately weighted
products K1;1,-K.,:;t. This term can be eliminated by a linear transformation q1 ——+
q1 + CwFt/2a and renormalizations 6Aad —> (SA‘B‘d + Cup/2016, 72 —+ 72 + (C/2a)2.
The renormalized 72 should be positive, if the stable and unstable adiabatic periodic
states touch each other only for 6Aad = t = 0 and only once per period.

The term oc qlwpt does not arise in the important case where the modulation is
performed by an additive periodic force F(t), see Eq. (2.2). Here, the adiabatic states
qflflba) correspond to the minimum and maximum of the potential U0(q) — F(t) - q,
cf. Fig. 2.1 (a). They merge first with increasing modulation amplitude A when the
field component [F1] is at its maximum over t. This means that BtK = BgtK = 0 at
the bifurcation point.

As explained in Sec. 2.3, the typical relaxation time near the bifurcation point
does not exceed t1 = (aywp)-1/2. If the correlation time of the noise f1 (t) is much

tifd

less than t), and the power spectrum of the noise does not have singular features

for high frequencies, then the dimensionless noise

~

f(T)—__-('YWF)_1f1(t) (T = Wt) (36)

129

1.-

is effectively 6-correlated as a function of the “slow” time T, with (f (T)

2D6(T). From Eqs. (2.3), (B6) the effective noise intensity is

D = la/4I1/2(1wp)‘3/2 / «1mm.

—00

130

f

(0)) =

Appendix C

Reduced equation of motion for

fast driving

In this Section we consider the case where modulation near the bifurcation point is
effectively fast, so that wptr >> 1. Here, throughout the modulation cycle the stable
and unstable states Qu,b(t) stay close to each other and to the critical cycle qc(t) into
which they merge at the bifurcation point A 2 Ac. Therefore the equation of motion
(2.1) can be expanded in 6q = q—qc(t), A —Ac, leading to Eq. (2.22). The expansion
coefficients are periodic in time.

It is convenient to start the analysis by simplifying the part of Eq. (2.22)

561 = (15% it = (10+ TF),

that describes motion in the linear approximation in 6Q. We introduce the matrix

1%(t, ti) such that

t
k.(t,t,) = Tt EXP (ft dthtll) . (C-ll

where Tt is the operator of chronological ordering. This matrix satisfies the equation

131

01%(15, t,)/0t = [1(t)k(t, ti) and gives the monodromy matrix NI,
Me) -=- 1171(1 + TF) = R:(t + 7F, t).

The eigenvalues 1111,, of the matrix M determine the evolution of 6q(t) in linear
approximation. Over the period TF, the coefficients of the expansion of 6q(t) in the
right eigenvectors eu(t) of If! change in My times (we use Greek letters to enumerate
eigenvalues and eigenvectors; they should be distinguished from the vector compo—
nents, like q,). The eigenvalues My are independent of time because of periodicity of
MU), 11(t). They are simply related to the Floquet exponents for the periodic state
qc(t).

At the saddle-node bifurcation, where stable and saddle—type states coalesce, one
of the eigenvalues (for example, Ml) becomes equal to 1, whereas |MV>1| < 1. This
means that the system is attracted to qc(t) in all directions except for the critical
direction e1(t); the distance from qc(t) along e1(t) does not change over the period,
in linear approximation. In what follows we choose e1 to be real.

For small 6A of an appropriate sign, the state qc(t) splits along el (t) into a stable
and an unstable state. The system approaches the vicinity of these states along the
directions eu>1 over a short time TFmax[1/]ln|1lfu>1|]] ~ t[.0). In contrast, the
motion along e1 is slow.

The el-component of 6q is the soft mode. We are interested in its dynamics
on times long compared to do) ,TF. The analysis is simplified by the fact that, for

t — ti >> [TF/ 1n |MV>1|], the matrix k(t,ti) projects any vector 6q(ti) on the vector

e1(t). In particular,
k(t,ti)el(t1) % K110. t061(0- (C?)

This is a consequence of the transitive property 1%(t, ti) = k(t, t’)F;(t’, ti) and the fact

132

that, for an arbitrary vector 6q, we have NI”(t)6q ——> Ce1(t) for n —> 00. The

function [$11 in Eq. (G2) is given by the expression

K110. ti) = é1(1) ° (<0. ti)91(ti)-

Here, él is the left eigenvector of the matrix M, which corresponds to the eigenvalue
1111 = I, and we use normalization é1(t) - e1(t) = 1. The matrix element 511(t,ti) is
periodic, n11(t + TF1 ti) = n11(t,ti).

Equation of motion for n11(t, ti) for large t — ti follows from Eq. (C.1),

gt—[K11(titi)el(t)l = ”11(t)K.11(t,ti)el(t),

#110) = é1(t) - M0610) (03)

Close to the bifurcation point, the component of 6q along the vector e1(t) has
a slowly varying factor. In contrast, the components of 6q along the vectors eu>1
are “fast”. Over time Ztso) they reach quasiperiodic values for a given value of
the slow component, and then fluctuate with amplitude o< 01/2. From (2.22), the

quasiperiodic values are quadratic in the slow component and therefore small. As a

consequence, the slow motion is indeed one-dimensional,

5010) *5 Q1(t)r€11(t. t091(1) (C4)

The instant ti here is arbitrary; Q1(t) contains a multiplicative factor that depends
on ti (but 6q(t) is independent of ti). The time t, drops out of all final expressions,

see Sec. IV.
The equation for Q1(t) is obtained by substituting Eq. (C.4) into Eq. (2.22) and

then multiplying Eq. (2.22) by the vector é1(t) from the left. This gives
K110, ti)Q1 = K(Q1,t)+ é1004(1). (C5)
2
K(QM) = %K¥1(t.ti)Q¥(el(t) ' V) K1+ 6A(0.4K1).

133

where K1 = él - K.

In the absence of noise, the solution of Eq. (C .5) is a sum of smooth and oscillating
parts, Q1(t) = Qsm(t) + QOSC. The term Qsm remains nearly constant on the time
scale TF, whereas Q080 ~ prOSC. It is seen from Eq. (OS) that QOSC o< 6A. The term
Qsm is much larger. An equation for Qsm can be obtained by averaging Eq. (C.5)

over time. It has the form
625’“ = a’(QS’“)2 + 6’ 6A + f’(t). (G6)

The coefficients (1’, I)" in Eq. (C6) are given by the expressions

cr’ é<fl11(t,ti)(el(t)' V)2K1>t’

13' = (1.1-11(1, awn/0.4),, (C.7)

where <>t means period-average centered at time t,
t+TF/2
<g>t : TF1] dt, Q(tIa ti) (C.8)
t—TF/2
The result of the averaging (C8) is independent of t for time-periodic g, as in the

case of the coefficients a', 6’, and therefore a’, B’ are independent of t.

The function f’ (t) in Eq. (C6) is a random force,

f(t) = 4:110. totem) . f(t). (0.9)

Eq. (C.6) has the same form as the equation for the soft mode in the adiabatic
limit (2.12) in the absence of the term oc (wpt)2. For a’fi' 6A < 0 the system has
a stable and an unstable stationary solution Q3"; These solutions are given by an

equation similar to Eq. (2.19),

2‘32: = rsgnm’x—a’ Wool/2 ((3.10)

134

(in what follows without lost of generality we set (1’ > 0).

Typical values of Qsm are oc |6All/2, as seen from Eq. (C.10). They largely exceed
the amplitude of the fast variables in (Sq, which are all oc 6A, in the neglect of noise.
The relaxation time of Qsm is tr = |2onf,mI—1 or IdAl‘l/Z. It is much larger than TF
close to the bifurcation point. The condition wptr >> 1 was the major approximation
made in the derivation of Eqs. (C5), (C6), besides the condition of the weak noise.

A transformation from QS’“,t to reduced variables
Q ___ all/2Qsm’ T = all/2t (0'11)

allows us to write Eq. (C6) in the compact form (2.23). The random force f(7) =

f'(a’1/2t) is effectively 6-correlated. From Eqs. (2.3), (C.9), its intensity is

an» . m
D = [GI/4ll/2/ dtl (Kl—11(t+t1.ti)’€1-11(t+t2ati)

xé1(t+t1)-¢(t1—t2)é1(t+t2))t. (C.12)

Here, 95 is the matrix of the noise correlation functions (2.3).

As a result of the period-averaging over t, in Eq. (C.12) the integrand becomes a
function of t1 — t2, and therefore the integral over t1 is independent of t2. Still, it
depends on ti, but this dependence will drop out of the final expressions for observable

quantities, in particular the activation energy of escape.

135

 

Appendix D

Distribution matching for

dynamically weak modulation

In this Appendix we study, using our general approach, the case of moderately weak
modulation and compare the results to the previous work [92]. Following Ref. [92] we
assume that the force K (q, t) = —U0q + F (t), where U0(q) is the metastable potential
in the absence of modulation [in this Appendix we use the notation fq E 69 f]. The
force F (t + TF) = F (t) is dynamically weak. This means that it weakly disturbs the
motion in the absence of fluctuations. Yet it may strongly change the escape rate,
because there it is compared with the small noise intensity.

The distribution p inside the attraction basin can be found by calculating the
action S(q, t) in Eq. (3.15) to the first order in F. From Eqs. (3.16), (3.17), S can be
written in a standard way as an integral of the Lagrangian L,

t 1
sum = [Goddamn L = ,— [q — M4012. (m)
The linear in F correction to S can be obtained by integrating the term o< F along

the optimal escape path q0(t) in the absence of driving, ()0 = p0 = U0q(q0). This

136

gives

501.15) = U001) — U0(qa0) + 8((1, t),

t
sat) -—— — / drqo<r)F(r>, (D2)

—00

where the optimal path is chosen so that q0(t) = q, and qao is the stable state qa(t)
in the limit F = 0, with U0q(qao) = 0; similarly, we use below (Ibo as the basin
boundary for F = 0, with U0q(q1,0) = 0. The quantity x(t) = —q0(t) determines
the field-induced change of the logarithm of the escape rate, and therefore was called
logarithmic susceptibility.

Eq. (D.2) allows one to match the intrawell distribution p(q,t) :2

[nag /27rD|1/2 exp[—S (q, t) / D] to the boundary-layer distribution (3.7)-(3.9) for

—Q = q... — ., >> p(q,t)/4.011”. 1.. (133)

[here ”,0 E —U0qq(q,-0) with z' = a,b]. In the range (D.3) the integral over p (3.7)
can be evaluated by the steepest descent, giving p(q, t) z E (27r/2b0)1/ 2 exp[—s(¢) / D],
with ()3 = (wp/pbo)ln(—Q/lp) + wp(t -— t'). The exponent of this boundary-layer

distribution coincides with -—s(q, t) / D in the range (D.3) if we set [92]
s<¢> = Z ammuemt
n

aw) = — / cleanest. (13.4)

—00

Here Fn are Fourier components of F (t) Matching the prefactors in the intrawell

and boundary-layer distributions gives
1
5 = ‘Z—WlflaOHbOll/z expl-AUo/Dl- (D5)

We emphasize that the matching has been done not only far from the diffusion region,
but also in the range (D.3), that is, much further away from the boundary than

diffusion length ID, in the regime of strong synchronization, Isml >> D.

137

We will show now that the alternative approach of Section 3.4 gives the same
result in the case of strong synchronization, Isml > D. To do this we have to solve
the equation for the prefactor (3.22) to the first order in F, which in turn requires
finding the first-order corrections to the optimal path q1, p1. Linearizing the first of

Eqs. (3.18), one obtains
41 = KOQQI + F(I?) + 2P1- (D6)

Here K0(1(q) E —U0qq(q), and the derivative is evaluated for the zeroth-order optimal
path, q = q0(t). The correction to the momentum, p1 E BS/Bq — p0, from Eq. (D.2)
is

, 1 t . -
P1 = —K0q(11 — F(10+ m _OOdT(10(T)F(T)- (D7)

To obtain Eq. (D.7), we used the fact that q0(t) = q, and therefore qu(7')/0q =
q0(T)/qo(t). We also took into account that p1(t) —> 0 for t ——+ —00.

In the absence of driving the prefactor is constant, and the function 2(t) from
Eq. (3.22) is determined by the initial condition (3.23), z 2: 20 = 27rD/I/1aol. Let

2: = 20 + 21, where zl is a correction or F. The linearized Eq. (3.22) is

£751 — 21(0qu = ZZOKqu(Cfl - P1) (US)
From Eqs. (D.6)-(D.8),
. 2 t . '
z. = 53% [00 d1 (10(7)F(T)lK0q(qo(t)) — K0q((10(7))l- (139)

To find the parameters 3]], and E of the boundary-layer distribution (3.26), (3.27) we
need to find zl close to (Ibo: i.e. for t —> 00. It follows from the results of Sec. 3.5

that, for strong modulation, in this range

2 z 2,1,002240—20), (D.10)

138

where we used that p0 = (1'0, ['20 = —p.1,0p0. Before we compare this expression with
Eq. (D.9) we note that the condition that s, as given by Eq. (D4), is minimal for

a) = (15m corresponds to

[00 dtq0(t)F(t) = 0.

--—00

This condition describes synchronization of the most probable escape path by the
modulation. In the absence of modulation q0(t — tc) is an MPEP for any tc. Modula-
tion lifts the time degeneracy, only one MPEP per period provides a minimum to s.
As a consequence, the first integral in Eq. (D.9) goes to zero for t —+ 00. Then from

Eqs. (D.9), (D.10) we obtain, taking into account that 40 = —KO,

271' 0°
22 = —-———/ (It ()0(t)F(t). (D.11)
|#a0#b0| —00

It follows from Eq. (D.4) that 22 = 27rw%.s;{,/ [flag/11,0]. Taking into account
Eqs. (3.29), (3.27), we obtain the same result for the prefactor in the boundary-layer
distribution as Eq. (D.5). We note that this result refers to the case of compar-
atively strong modulation, [Sm] > D, and is obtained by matching the intrawell
and boundary—layer distributions not in the region (D3) [92], but closer to the basin

boundary, where Isml >> usz.

139

 

 

Appendix E

Nonadiabatic corrections for slow

modulation

In this Appendix we show an alternative approach to the analysis of adiabatically
slow modulation. By treating time variation of the modulation as a perturbation, we
find the most probable escape path (MPEP) and also solve Eq. (3.22) for the auxiliary
function 2 and obtain the constant 32, cf. Eq. (3.24). This provides an alternative
way of finding the parameters of the boundary-layer distribution and also gives an

insight into the actual dynamics of escape for slow modulation.

The small parameter of the slow-modulation theory is the reduced frequency,
wptr << 1. The typical duration of escape is the relaxation time ~ tr, it is much
smaller than the modulation period T}? = 27r/wp. Escape is most likely to happen
once per period, for the modulation phase am = wptm (which, as we show, corre-
sponds to the minimal barrier height; tr is the adiabatic relaxation time for t = tm).

For t close to tm we can expand the force in the form

K((Lt) = K001) + F1(<1.t) + 17201.0. (E-l)

140

 

where K001) E K(q, tm). F101. t) E Kt(€1.tm)(t _— tm). F2(q,t) E (1/2)Ktt(q.tm)(t -
tm)2 (here and below ft E at f, fq E 8g f) Thus, K0 represents a stationary force,
F1 and F2 are the time-dependent corrections of first and second order in wptr.

To zeroth order in wptr, the positions of the stable state qa and basin boundary qb
of the system are the solutions (IaO and qu of the adiabatic equation K0(q) = 0 with
1200 < 0 and 121,0 > 0, respectively, where 12,-0 = K0q(q,:0) (2' = a, b). The parameters
12,-0 characterize the relaxation rate of the system. We emphasize that, in contrast to
Appendix D where K0 referred to the system in the absence of modulation, here K0
and the parameters 9a,!» #21,!) are calculated for strong modulation but at a specific
instant of time tm.

Because of the time-dependent terms in K, the positions q, (2' = a, b) acquire
corrections q,” of the first and second order in wptr, so that q,- = (120 + qn + gig.

They can be found from the equation q, = K (q), t) and have the form

F F .
(121__ -—;t'+'—l 1 z=a9b1 (E.2)
£0 #20 ,
(120

 

 

and

 

 

5K0qu12t 3F1tF1qt F212
4m: - + —

_ 5 4 3
2 ”2'0 ”2'0 “1'0
2K F F 3F F F
_ qu4 1t 1 + 11:); 1 __ % (E3)
#10 “2'0 “1'0
1K F2 F F F
__2._04_§_1_+_1<I§_1___2 , iza,b
#10 ”2'0 “20 q“,

We now consider corrections to the MPEP. For K given by Eq. (E.1) it is natural
to seek the solution of Eqs. (3.18) in the form q 2 go + q1 + (12, p = 190 + p1 + m. To

zeroth order in w F the MPEP is given by the equation

()0 = P0 = —K0((10)-

141

 

The first-order corrections q1,p1 satisfy Eq. (D.6) with F = F1 and with the boundary

conditions q1 —+ (Jul at t —+ —00 and q1 —+ (1111 at t —> 00. In the second order
. 1 2
Q2 = K0q‘12 + EKOQQQ1 + Fiqcn + F2 + 2P2 (34)

with q2 —» qag at t —> —00 and q2 —> qbg at t ——> 00. Here and below the functions
K0, F13 and their derivatives are calculated along the zeroth-order MPEP q0(t).

From the expression for the action (D.1) with the force (E.1) we obtain, similar
to Eq. (D7),

1 t ,
P1 = —K0q(11 - F1+ .—- quoFlr~
‘10“) —00

Using this expression and Eq. (3.18), we further obtain

. 1 t .
€11 — P1 = -.—-/ quoFlr- (E5)
(10“) —00

The momentum p1 ——> 0 at t -—> :lzoo, therefore Eq. (E5) requires that

1 t , (falv t —-> -001
—. (t) / dTQOFIT —* (E16)
(10 —OO (fbli t _, 00
where gum are given by Eq. (E2).
For t —+ —00 the condition (E6) is met, hence the choice of the lower limit of

integration. To consider the t —> +00 limit we write the left-hand side of Eq. (E6)

8.8

—. deOFl — —. / qu F1 °
qua) -0. T «mm 1 0 T

For t —> 00 the second term goes to 4,1. Therefore to satisfy the condition (E.6) we

must have

00 (Ibo
/ mom: / qu17-(q)=0- (B7)
(1

‘00 00

In changing the integration variable we used that F It does not depend on t explicitly.

142

 

 

The condition (E7) is equivalent to the requirement that the adiabatic barrier

height

9b“)
AU(t) = [q (t) dq{Ko(<1) + F1(q.t)+ F2<q.t)1 (13.8)

be minimal, dAU/dt = 0, to the first order in (Up.

We note that the condition (B.7) is invariant with respect to time shift of the
MPEP q0(t) —+ q0(t — tc), with an arbitrary tc. However, it does specify the modu-
lation phase (12m = wptm when escape is most likely to occur.

The term in the action of second order in wptr has the form

1 t ,
52m. t) = — / dag. — Koqql — F1)2

4—00

t
. . 1
'1'] (17-(10012 — [(OqCI2 " §K0qq9¥ " qu‘ll _ F2)-

—00

Using this equation to calculate p2 = (2,82, we obtain from Eq. (E.4)

. 1 t .

(12 — P2 = .—-/ 6” (1071(7), (39)
90“) —00

”(11) = qutQI + F21 + (41 - P1)(K0qq611 + Fig)-

Using Eq. (E.3), one can check that (E9) satisfies the boundary condition q2 —>
qaz, p2 —+ 0 for t ——> —-00. To satisfy the boundary condition at t —> +00 we must

require that

00
/ qu0u(7') = 0. (E.10)
—CD
The integral (E.10) depends on the position tc of the “center” of the MPEP
q0(t — to) and thus specifies this position. Eqs. (B.7) and (E.10) fully determine both
the phase of the modulation where escape is most likely to occur and the MPEP as
a function of time.

We are now in a position to develop a perturbation theory for the function z(t),

which should be calculated along the MPEP from Eq. (3.22). The parameter of

143

 

interest Zg is determined by the asymptotic behavior of z for t —> 00: this is the
coefficient at the diverging term Dp62(t) in z(t), cf. Eq. (3.24). As we will see,
22 or 1.0%., and therefore we need to find 2 to second order in wptr. Respectively, we
seek z in the form 2 = :30 + 21 + 22, with zj 0< [wptrp (j = 0,1,2).

The initial conditions for zj follow from Eq. (3.23). The function 03(t) in
Eq. (3.23) is a periodic solution of the equation ('73 = 212003, + 2, which can be
easily solved by perturbation theory in wptr. To zeroth order (03)) = 1/ [1200], and

20 = 2WD/luaol-

From Eq. (3.22), the equation for 21 has the form

51 - 2Koqi'l = 2ZOlKquUh - P1) + qut]
This equation differs from Eq. (D.8) by the term F lqt. which allows for q—dependence

of the perturbation force. The left-hand side can be written as (1/q8)d(z’1qg)/dt,

which immediately gives

t
2143 = 220/ deglKquMi -P1)+F1q~rl

—00

= 2Zo<iothiq(q'1 - P1) + Fitl- (E-ll)
Here we have taken into account the initial condition for zl for t ——1 —00 and also

used the relation

£011 — P1) = Kqul - P1) + Flt. (153-12)
which follows from Eq. (E.5). Taking into account that 41,101,112 do not diverge for
t —+ 00, we see from Eq. (E.11) that 210343 ——> 0 as t —+ +00. Therefore 21 does not
contain a term or exp(2ub0t) 0c p62“) for t —> 00, and as a consequence it does not
contribute to Zg.

In the second order, from Eq. (3.22) we have
252 — 2K0qi'2 = 12(t), (13.13)

144

 

 

where

”(t) = 2:5Oll{0qq((f2 _ P2) + (1(0qqq91 + quq)(‘fl _ P1)
+F1qqt(11 + F2112] + 221(K0qqq1 + qu)

+2z1[1\’0qq((}1 - P1) + qutl- (314)

The left-hand side of Eq. (153.13) is (1 /q8)d(i:2q(2)) /dt, and therefore
2 t 2
222(t)q0(t) = / quO'v(T). (E.15)
—00
A cumbersome calculation, which involves integration by parts using conditions (B.7),

(E.10) and Eqs. (E2), (E4), (E5), (E.9), shows that, as t —+ +00, the integral in the

right hand side of Eq. (E.15) tends to a constant

 

00 F 2 F 2
C = 220 [_/ (ITQOFZtt + l “(QM)” _ l 1f((I(lO)1 . (E16)
—00 “b0 #00

Eq. (E.16) is the central result of this Appendix. It follows from this equation

and the fact that the singular part of z behaves as exp(2/2b0t) for t —> 00 that

C
Z=D-ll' zt2t= .
2 1330 ()I)0() 2%00

 

On the other hand, from Eqs. (E8), (E.16) we obtain

AU(tm) = _C'_ _ CluaO]

2250 4WD .

Therefore

Z2 = 27TAU(t111)/]/‘00/1b0]v

which coincides with the result of Eqs. (3.29), (3.33).

145

 

BIBLIOGRAPHY

I46

 

Bibliography

[1] A. Einstein, Annal. Phys. 33(16), 1275 (1910).

[2] M. H. Devoret, D. Esteve, J. M. Ma1tinis, A. Cleland, and J. Clarke, Phys.
Rev. B 36(1), 58 (1987).

[3] J. M. Martinis and R. L. Kautz, Phys. Rev. Lett. 63(14), 1507 (1989).

[4] V. Lefevre—Seguin, E. Turlot, C. Urbina, D. Esteve, and M. Devoret, Phys. Rev.
B 46(9) (1992).

[5] R. L. Kautz, Rep. Prog. Phys. 59(8), 935 (1996).

[6] E. Turlot, S. Linkwitz, D. Esteve, C. Urbina, M. H. Devoret, and H. Grabert,
Chem. Phys. 23508), 47 (1998).

[7] I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. F runzio,
and M. H. Devoret, Phys. Rev. Lett. 93(20), 207002 (2004).

[8] F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman, Phys.
Rev. Lett. 89(22), 226802 (2002).

[9] I. N. Krivorotov, N. C. Emley, A. G. F. Garcia, J. C. Sankey, S. I. Kiselev,
D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 93(16), 166603 (2004).

[10] S. Urazhdin, N. O. Birge, W. P. Pratt, and J. Bass, J. Appl. Phys. 95(11), 7429
(2004).

[11] M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brun-
ner, and R. Hovel, Appl. Phys. Lett. 77(22), 3514 (2000).

[12] B. Nagler, M. Peeters, J. Albert, G. Verschaffelt, K. Panajotov, H. Thienpont,
I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, Phys.
Rev. A 68(1), 013813 (2003).

[13] M. Sondermann, T. Ackemann, S. Balle, J. Mulet, and K. Panajotov, Opt.
Commun. 235(4-6), 421 (2004).

[14] R. Landauer, J. Appl. Phys. 33(7), 2209 (1962).

147

 

 

[15] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. Lett. 58(12),
1256 (1987).

[16] G. Pilling, D. H. Cobden, P. L. Mchen, C. I. Duruoz, and J. S. Harris, Jr.,
Surface Science 361-362, 652 (1996).

[17] M. Rogozia, S. W. Teitsworth, H. T. Grahn, and K. H. Ploog, Phys. Rev. B
x 65(20), 205303 (2002).

[18] J. A. Bonetti, D. S. Caplan, D. J. Van Harlingen, and M. B. Weissman, Phys.
Rev. Lett. 93(8), 087002 (2004).

[19] N. Birge and B. Golding, Tunneling of defects in metals, in: Mesoscopz'c Quan-
tum Phenomena, (Nova Press, New York, 2002).

[20] X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, Nature
421(6922), 496 (2003).

[21] J. S. Aldridge and A. N. Cleland, Phys. Rev. Lett. 94(15), 156403 (2005).

[22] R. L. Badzey, G. Zolfagharkhani, A. Gaidarzhy, and P. Mohanty, Appl. Phys.
Lett. 86(2), 023106 (2005).

[23] M. Blencowe, Phys. Rep. 395(3), 159 (2004).
[24] L. I. McCann, M. Dykman, and B. Golding, Nature 402(6763), 785 (1999).
[25] A. Simon and A. Libchaber, Phys. Rev. Lett. 68(23), 3375 (1992).

[26] P. T. Korda, M. B. Taylor, and D. G. Grier, Phys. Rev. Lett. 89(12), 128301
(2002).

[27] S. H. Lee and D. G. Grier, Phys. Rev. E 71(6), 060102 (2005).

[28] Focus issue: unsolved problems of noise and fluctuations, Chaos, Vol. 11, no. 3
(2001).

[29] H. Kramers, Physica (Utrecht) 7, 284 (1940).

[30] J. Kurkijarvi, Phys. Rev. B 6, 832 (1972).

[31] R. Victora, Phys. Rev. Lett. 63, 457 (1989).

[32] C. B. Muratov and E. Vanden—Eijnden, J. Stat. Phys. 114(3-4), 605 (2004).
[33] A. Garg, Phys. Rev. B 51(21), 15161 (1995).

[34] T. Fulton and L. Dunkelberge, Phys. Rev. B 9, 4760 (1974).

[35] J. Martinis, M. Devoret, and J. Clarke, Phys. Rev. B 35, 4682 (1987).

148

 

[36] F. Sharifi, J. L. Gavilano, and D. J. V. Harlingen, Phys. Rev. Lett. 61, 742
(1988).

[37] 3. Han, J. Lapointe, and J. E. Lukens, Phys. Rev. Lett. 63, 1712 (1989).

[38] S.-X. Li, Y. Yu, Y. Zhang, W. Qiu, S. Han, and Z. Wang, Phys. Rev. Lett. 89,
098301(2002)

[39] W. Wernsdorfer, K. Hasselbach, A. Benoit, B. Barbara, B. Doudin, J. Meier,
J. P. Ansermet, and D. Mailly, Phys. Rev. B 55(17), 11552 (1997).

[40] W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. De-
moncy, A. Loiseau, H. Pascard, and D. Mailly, Phys. Rev. Lett. 78(1791),
(1997)

[41] R. Koch, G. Grinstein, G. Keefe, Y. Lu, P. Trouilloud, and W. Gallagher, Phys.
Rev. Lett. 84, 5419 (2000).

[42] S. Ingvarsson, G. Xiao, S. S. P. Parkin, W. J. Gallagher, G. Grinstein, and
R. H. Koch, Phys. Rev. Lett. 85(15), 3289 (2000).

[43] W. Wernsdorfer, Adv. Chem. Phys, Vol 118 118, 99 (2001).

[44] A. D. Ventcel’ and M. I. Freidlin, Uspehi Mat. Nauk 25(1 (151)), 3 (1970).
[45] M. I. Dykman and M. A. Krivoglaz, Zh. Eksper. Teor. Fiz. 77(1), 60 (1979).
[46] R. Graham, Phys. Lett. A 80(5—6), 351 (1980).

[47] R. Graham, Phys. Lett. A 103(5), 255 (1984).

[48] J. Luciani and A. Verga, Europhys. Lett. 4(3), 255 (1987).

[49] A. J. Bray and A. J. McKane, Phys. Rev. Lett. 62(5), 493 (1989).

[50] A. J. McKane, H. C. Luckock, and A. J. Bray, Phys. Rev. A 41(2), 644 (1990).
[51] M. I. Dykman, Phys. Rev. A 42(4), 2020 (1990).

[52] R. S. Maier and D. L. Stein, Phys. Rev. E 48(2), 931 (1993).

[53] M. I. Dykman, D. G. Luchinsky, R. Manella, P. V. E. McClintock, H. E. Short,
N. D. Stein, and N. G. Stocks, Phys. Lett. A 193(1), 61 (1994).

[54] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 87(27), 270601 (2001).

[55] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems
(Springer-Verlag, 1998).

[56] O. A. Tretiakov, T. Grarnespacher, and K. A. Matveev, Phys. Rev. B 67, 073303
(2003).

149

 

[57] M. I. Dykman and M. A. Krivoglaz, Physica A 104(3), 480 (1980).
[58] E. Knobloch and K. A. Wiesenfeld, J. Stat. Phys. 33(3), 611 (1983).
[59] A. P. Dmitriev and M. I. Dyakonov, JETP Lett. 44(2), 84 (1986).
[60] B. Golding and J. R. Kruse, unpublished work.

[61] R. Graham, Europhys. Lett. 3(3), 259 (1987).

[62] M. I. Dykman, B. Golding, and D. Ryvkine, Phys. Rev. Lett. 92(8), 080602
(2004).

[63] M. I. Dykman, I. B. Schwartz, and M. Shapiro, Phys. Rev. E 72(2), 021102
(2005)

[64] I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, L. Frunzio, M. Metcalfe, C. Rigetti,
and M. H. Devoret, cond-mat/0507248 (2005).

[65] C. Stambaugh and H. B. Chan, Phys. Rev. B 73, 172302 (2006).

[66] K. Kim, M. Heo, K. Lee, H. Ha, K. Jang, H. N011, and W. Jhe, Phys. Rev. A
72, 053402 (2005).

[67] A. F0rster and A. Mikhailov, Phys. Lett. A 126, 459 (1988).

[68] L. Onsager and S. Machlup, Phys. Rev. 91(6), 1505 (1953).

[69] D. G. Luchinsky, J. Phys. A 30(16), L577 (1997).

[70] J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95(1 - 2), 333 (1999).

[71] G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80(5—6), 931 (1995).

[72] J. Kurchan, J. Phys. A: Math. and General 31(16), 3719 (1998).

[73] G. E. Crooks, Phys. Rev. E 60(3), 2721 (1999).

[74] D. J. Evans and D. J. Searles, Adv. Phys. 51(7), 1529 (2002), ISSN 0001-8732.
[75] R. van Zon and E. Cohen, Phys. Rev. E 69(51), 056121 (2004).

[76] M. I. Dykman, M. M. Millonas, and V. N. Smelyanskiy, Phys. Lett. A 195(1),
53 (1994).

[77] R. Graham, D. Roekaerts, and T. Tel, Phys. Rev. A 31(5), 3364 (1985).
[78] H. R. Jauslin, Physica A 144(1), 179 (1987).

[79] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 85(7), 1358 (2000).

150

 

[80] M. I. Dykman, P. V. E. McClintock, V. N. Smelyanski, N. D. Stein, and N. G.
Stocks, Phys. Rev. Lett. 68(18), 2718 (1992).

[81] J. Hales, A. Zhukov, R. Roy, and M. I. Dykman, Phys. Rev. Lett. 85(1), 78
(2000)

[82] E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. Buhrman, and D. C.
Ralph, Phys. Rev. Lett. 89, 196801 (2002).

[83] L. Landau and E. Lifshitz, Statistical Physics Part 1. 3rd edition revised by E. M.
Lifshitz and LP. Pitaevskii (Pergamon, Oxford, 1980).

[84] J. Guckenheimer and P. Holmes, Nonlinear Oscillators, Dynamical Systems and
Bifurcations of Vector Fields (Springer-Verlag, New York, 1987).

[85] D. Ryvkine, M. I. Dykman, and B. Golding, Phys. Rev. E 69(6), 061102 (2004).
[86] A. Larkin and Y. Ovchinnikov, J. Low Temp. Phys. 63, 317 (1986).

[87] B. Ivlev and V. Mel’nikov, Phys. Lett. A 116, 427 (1986).

[88] S. Linkwitz and H. Grabert, Phys. Rev. B 44(21), 11901 (1991).

[89] I. Dayan, M. Gitterman, and G. Weiss, Phys. Rev. A 46, 757 (1992).

[90] R. Mantegna and B. Spagnolo, Phys. Rev. Lett. 76, 563 (1996).

[91] M. Bier, Phys. Rev. E 59, 6422 (1999).

[92] V. N. Smelyanskiy, M. I. Dykman, and B. Golding, Phys. Rev. Lett. 82(16),
3193(1999)

[93] J. Lehmann, P. Reimann, and P. Hanggi, Phys. Rev. E 62(5), 6282 (2000).

[94] M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky,
R. Mannella, and P. V. E. McClintock, Chaos 11(3), 587 (2001).

[95] S. M. Soskin, R. Mannella, and P. V. E. McClintock, Phys. Rep. 373, 247
(2003).

[96] M. V. Fistul, A. E. Miroshnichenko, and S. Flach, Phys. Rev. B 68(15), 153107
(2003)

[97] M. I. Dykman, D. G. Luchinsky, R. Mannella, P. V. E. McClintock, N. D. Stein,
and N. G. Stocks, Nuovo Cimento 17(7-8), 661 (1995).

[98] K. Wiesenfeld and F. Jaramillo, Chaos 1998 8, 539 (1998).
[99] F. Julicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69, 1269 (1997).
[100] P. Reimann, Phys. Rep. 361, 57 (2002).

151

 

l l I [I]. 111 7

 

 

[101] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
(McGraw—Hill, New-York, 1965).

[102] J. S. Langer, Ann. Phys. 41, 108 (1967).
[103] S. Coleman, Phys. Rev. D 15, 2929 (1977).

[104] E. Benoit, ed., Dynamic Bifurcations, vol. 1493 of Lecture Notes on Mathematics
(Springer-Verlag, Berlin, Luminy, 1990).

[105] P. Mandel and T. Erneux, Opt. Commun. 44, 55 (1982).

[106] L. Landau and E. Lifshitz, Mechanics (Pergamon, London, 1976).

[107] R. Mannella, Intl. J. Mod. Phys. C 13, 1177 (2002).

[108] C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69(16), 2318 (1992).

[109] M. Bier, Phys. Rev. E 71, 011108 (2005).

[110] M. I. Dykman and D. Ryvkine, Phys. Rev. Lett. 94(7), 070602 (2005).

[111] D. Ryvkine and M. I. Dykman, Phys. Rev. E 72(1), 011110 (pages 15) (2005).
[112] J. Lehmann, P. Reimann, and P. Hanggi, Phys. Rev. Lett. 84(8), 1639 (2000).
[113] R. Graham and T. Tel, J. Stat. Phys. 35(5-6), 729 (1984).

[114] M. Berry, Adv. Phys. 25 (1976).

[115] D. Ludwig, SIAM Rev. 17, 605 (1975).

[116] D. Ryter and P. Jordan, Phys. Rev. Lett. 104A, 193 (1984).

[117] M. Suzuki, J. Stat. Phys. 16(6), 477 (1977).

[118] V. N. Smelyanskiy, M. I. Dykman, H. Rabitz, B. E. Vugmeister, S. L. Bernasek,
and A. B. Bocarsly, J. Chem. Phys. 110(23), 11488 (1999).

[119] I. B. Schwartz, L. Billings, and E. M. Bollt, Phys. Rev. E 70(4), 046220 (2004).
[120] D. Moroni, T. S. van Erp, and P. G. Bolhuis, Physica A 340(1-3), 395 (2004).
[121] D. M. Zuckerman, J. Phys. Chem. B 108(16), 5127 (2004).

[122] W. N. E, W. Q. Ren, and E. Vanden-Eijnden, Phys. Rev. B 66(5), 052301
(2002)

[123] D. G. Luchinsky and P. V. E. McClintock, Nature 389(6650), 463 (1997).

[124] M. Morillo, J. M. Casado, and J. Gomez-Ordonez, Phys. Rev. E 55(2), 1521
(1997)

152

 

 

[125] M. Arrayas, J. M. Casado, J. G. Ordonez, P. V. E. McClintock, M. Morillo,
and N. D. Stein, Phys. Rev. Lett. 80(11), 2273 (1998).

[126] D. Ryvkine and M. I. Dykman, Phys. Rev. E 73, 061109 (2006).

[127] V. N. Smelyanskiy, M. I. Dykman, H. Rabitz, and B. E. Vugrneister, Phys. Rev.
Lett. 79(17), 3113 (1997).

[128] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 86(18), 3942 (2001).
[129] R. Graham and T. Tel, Phys. Rev. Lett. 52(1), 9 (1984).
[130] M. Suzuki, J. Stat. Phys. 16(1), 11 (1977).

[131] M. I. Dykman, D. G. Luchinsky, P. V. E. McClintock, and V. N. Smelyanskiy,
Phys. Rev. Lett. 77(26), 5229 (1996).

[132] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 71(12), 1783 (1993).
[133] N. Berglund and B. Gentz, J. Phys. A 35, 2057 (2002).

[134] M. Dykman, B. Golding, J. Kruse, L. McCann, and D. Ryvkine, cond-
mat/0204621 (2002).

153

 

   

1[17111111711711[[1711