stochastic dynamics
Post on 09-Feb-2018
230 Views
Preview:
TRANSCRIPT
-
7/22/2019 Stochastic Dynamics
1/161
Effective stochastic dynamics in deterministic
systems: model problems and applications
by
Gil Ariel
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
September 2006
Eric Vanden-EijndenAdvisor
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
2/161
UMI Number: 3234111
Copyright 2006 by
Ariel, Gil
All rights reserved.
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
UMIUMI Microform 3234111
Copyright 2006 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
3/161
Gil Ariel
All Rights Reserved, 2006
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
4/161produced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
5/161
to Orit, Guy, Mia,
and who ever may join us next
iv
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
6/161
Acknowledgements
First and foremost, I would like to thank my advisor, Eric Vanden-
Eijnden. Working with Eric, who has a rare skill to find the interesting
mathem atics in physical problems has been a rewarding experience. His
guidance is evident in every aspect of this work.
I would also like to thank the faculty at the Courant institute, in partic
ular Jonathan Goodman, Robert Kohn, Charles Newman, Ragu Varadhan
and Lai-Sang Young for always being willing to explain, talk and advise.
I am also grateful to my fellow students: An drea Barreiro, M atthias
Heymann, Jose Koiller, Stan Mintchev, Junyoep Park, Haiping Shen and
Paul Wright for making my time at Courant pleasant and helping with all
those little things we know we should know, but are nevertheless not quite
sure of.
Finally, I would like to than k my family. My wife, Orit, who followed
me to New York and now to Texas, and with great love built our new home.
Our son Guy, who grew to become a little boy and provided ample enjoy
able distractions, and last but not least Mia, who came just in time to be
included.
v
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
7/161
Contents
Dedicat ion ........................................................................................................ iv
Acknowledgements ....................................................................................... v
List of Figures ................................................................................................. ix
List of T a b le s ................................................................................................. xi
1 Introd uction 1
2 The Kac-Zw anzig m odel 13
2.1 Introduction ......................................................................................... 13
2.2 Th e Kac-Zwanzig model: Formulation
and elementary propert ies ............................................................... 15
2.3 A stro ng limit th e o re m ...................................................................... 19
2.3.1 Introduction ............................................................................. 19
2.3.2 Main results and ex am ple s .................................................. 21
vi
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
8/161
2.3.3 Proof of the th e o re m ............................................................. 27
2.3.4 Conclusion and generalizations ......................................... 49
2.4 Testing Transition State Theory on Kac-Zwanzig Model . . . . 53
2.4.1 Introduction .............................................................................. 53
2.4.2 The m od el ................................................................................. 58
2.4.3 M etastability in K ac -Z w an zig ............................................. 61
2.4.4 Numerical ex pe rim en ts .......................................................... 65
2.4.5 Tran sition rate s for the limiting dynamics ......................... 70
2.4.6 Transition state th e o ry .......................................................... 82
2.4.7 Local dynamics around the hyperbolic point:
Why does VTS T works while naive TS T does not? . . 93
2.4.8 Concluding re m ar ks ................................................................. 96
3 Accelerated simulation of a heavy particle in a gas of elastic
spheres 101
3.1 Introduction ............................................................................................. 101
3.2 The m od el ................................................................................................. 105
3.3 Limiting dynam ics ................................................................................ 109
3.3.1 Ap proximate Markov nature of colloid-gascollisions . . 112
3.3.2 The rate of momentum tr a n s fe r............................................ 114
vii
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
9/161
3.3.3 Th e limiting equations of m o t i o n .........................................115
3.3.4 Improved a c c u ra c y ................................................................... 120
3.4 Th e numerical scheme ......................................................................... 121
3.4.1 The basic algorithm ................................................................ 123
3.4.2 Higher order accuracy ............................................................. 127
3.4.3 Some technical a s p e c ts ............................................................. 128
3.4.4 Pra ctical considerations ......................................................... 130
3.5 Numerical e x p erim en ts ..........................................................................131
3.5.1 Simulation m e th o d ................................................................... 131
3.5.2 Simulation re su lts ....................................................................... 133
3.6 Conclusion ................................................................................................. 134
Bibliography 138
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
10/161
List of Figures
2.1 Comparison of the rand om noise, /y(f) (solid line), and the
limiting one (f) (dashed line). The frequencies are draw n
from a one sided exponential distribution and N ~ 2000. The
bath in itial cond itions are cano nical .............................................
2.2 A semi-log plot of the distrib ution of the waiting times between
transitions, P [ t s > s], for the case 7 = 10. In figure (a), the
sharp jump near the origin is due to rapid re-crossings of the
{xo = 0} plane. In figure (b), the statistic s of trans ition times
confirms the quasi-Markov hypothesis. The slope of the curve
is 1.1-10-4, the same as the average tran sition rate. The graph
diverges from the linear fit near the origin ..................................
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
11/161
2.3 A semi-log plot of the joint dis tribu tion of successive waiting
time, -P[tqV + > s], for the case 7 = 10. As expected, th e
slope of the curve is 5 10- 5 which is about half the average
transition rate ........................................................................................... 1 0 0
3.1 A car icature of a colloid moving with velocity v. The large
sphere collides with all particles whose center of mass is within
a distance of R + r from its center. Hence, the colloid collides
with all particles in a volume of vdtn(R + r) 2 ................................. 108
3.2 Th e figure depic ts a situatio n in which subsequent collisions
be tween the colloid an d a part icula r gas pa rt icle carries some
memory: a slow moving particle approaches the colloid (a).
After this collision, the particle bounces off with almost the
same speed, only in the oppo site direction (b). After a second
collision with a fast moving particle (c), the colloid catches up
with the first particle and collides again (d) ..................................... I l l
x
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
12/161
List of Tables
2.1 Com parison between tran sition rates with two different can
didate metastable sets.............................................................................. 67
2.2 Com parison between TS T predictions and simulation results
for small and large values of 7 ................................................................90
3.1 Sim ulation results for a system of iV = 40000 gas partic les
and inverse temperature ( 3 = 1 . The table compares the av
erage kinetic energy and diffusivity of the colloid as obtained
using our newly suggested, accelerated method, with values
predicted by th e lim iting OU process................................................. 134
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
13/161
Chapter 1
Introduction
Deterministic dynamical systems, in particular, Hamiltonian systems, often
display very com plicated chaotic behav ior when th e number of degrees of free
dom in the systems is large. It is not surprising th a t the evo lution of certain
observables in these systems can be approximated by a stochastic process.
Results in this direction abound in the literature [29, 36, 42, 47, 77]. A typ
ical example is the extraction of conformational dynamics of biomolecules,
which is stochastic in nature even though the underling dynamics is Hamil
ton ian [31, 61]. In this con text, th e development of simple models in which
the emergence of the stochastic behavior out of the deterministic dynamics
can be analyzed rigorously is of importance.
1
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
14/161
In this thesis we consider two different model examples in which a single
particle is coupled to a larger, man y pa rt icle system which plays th e role of
a heat ba th. The dynamics in both cases is deterministic. As a first step we
prove th a t taking a part icula r lim it, and un der appro priate in itia l conditions,
the dynamics of the single particle of interest is given by a solution to a
stochastic equation. Next, we apply the knowledge of the limiting equation
for the study of some properties of the deterministic dynamics.
The first example, presented in Chapter 2, considers a variant of the Kac-
Zwanzig model [24, 25, 76]. In th is model a single dis tinguish ed partic le is
moving in a given external potential and is coupled to N harmonic oscilla
tors via linear springs. The system is one-dimensional. The oscillators have
random frequencies that are chosen independently from a probability distri
bu tio n. In it ia l co nd itions are chosen rando mly from an inva rian t measure.
In Section 2.3 we prove that under certain sufficient conditions on the model
pa rameters , and in any fin ite time segment, th e traje cto ry of th e partic le
converges strongly (in L 2) to the solution of an effective stochastic equa
tion. The limiting equation has the form of a generalized Langevin equation
(GLE) and satisfies a fluctuation-dissipation relation. Strong convergence is
proved by map ping the in itial co nd itions of the fin ite dimensional system to
2
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
15/161
the probability space on which the random noise term of the GLE is defined.
Bo th canonical and micro-canonical ensembles are considered. We also prove
convergence for a more general class of initial conditions that are absolutely
continuous with respect to the above two invariant measures.
Once the form of the limiting equation is established, it is used for study
ing properties of the d ynamics a t large, but finite system size. In the second
half of the chapter, the model is used as a benchmark problem for a rigorous
analysis of transition state theory (TST) and variational TST (VTST).
In the particular case in which the distinguished particle moves in a
double-well potential the dynamics is bistable. Amid the complexity of
individual trajectories, it is found that that most trajectories remain con
fined for very long periods of time in well separated regions of phase-space
whose boundary are loosely determined by the double-well potential, and
only switch from one region to ano ther occasionally. The confinement is due
to the presence of dynam ical bottlenecks between these regions. The system
is then said to display m etastability, and th e regions in which the trajec tories
remain confined are referred to as meta stable sets. Example of systems dis
playing m eta stability abound in natu re , with examples arising from physics
(phase transitions), chemistry (chemical reactions, conformation changes of
3
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
16/161
bio-molecules), biology (switches in popula tio n dynamics), and many others.
In these systems, it is worth verifying whether the dynamics can be approx
imated by a Markov chain over the state-space of the metastable sets with
appropriate ra te constants. The m ain question of interest then becomes the
determination of these rate constants. This question is nontrivial because it
amounts to understanding the pathways by which the transitions between
the metastable sets occur. These pathways are usually non-trivial and com
plicated .
It is here that our knowledge of the limiting dynamics can be applied
by ca lculat ing th e ex ac t transit io n ra te s for th e lim iting equation. This is
done by spectral decomposition of the backward operator characterizing the
limiting stochastic equation. Eigenvalues and eigenfunctions are found using
the matched asymptotics method [60].
One of the earliest attempts to determine transition pathways and rate
con stan ts is tran sitio n sta te theo ry (TST ) [22, 35, 75]. Under the sole as
sumption that the dynamics of the system is ergodic with respect to some
known equilibrium distribution, TST gives the exact average frequency at
which trajectories cross a given hypersurface or hyperplane which separates
two metastable sets of interest. This average frequency can be used as a
4
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
17/161
first approximation for the frequency of transition between the metastable
sets. Unfortunately, it was recognized early on th a t this approx ima tion can
be qui te po or , for not every crossing of th e dividing surface correspond s to
a transition between the m etastable sets. Indeed, the trajectories can cross
the dividing surface many times in the course of one transition . As a result,
the TST prediction for the frequency of transition always overestimates the
ac tua l frequency, sometimes grossly so. One way to minimize this problem is
to use the freedom in the choice of dividing surface. The best prediction for
the frequency from TST is the one corresponding to the dividing surface with
minimum crossing frequency. In variational transitio n state theory (VTST )
[35, 51, 65, 70] one minimizes the TST rate over some class of candidate
separating hyperplanes.
Unfortunately, VTST (just like TST) is an uncontrolled approximation,
for it only provides an upper bound on the transition frequency between
the m etastab le sets. In general, one does not know how sharp th is boun d is.
Other assumptions beyond TST and VTST are usually difficult to assess too.
Are successive transitions between the metastable sets well approximated by
Poisson events (i.e. statistically indepen dent and with exponentially dis
tributed waiting times) as required for the approximation of the dynamics
5
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
18/161
by a Markov chain to hold? How does th is pro perty depend s on th e defin ition
of the m etastable sets? Etc.
The questions described above are first studied here through computer
simulations. M etastable sets are identified and it is shown tha t the dynam ics
is well app rox imated by a two-s tate Markov process over these sets. We also
examine the various assumptions required for a consistent use of TST in this
system.
We then turn to evaluate the transition rate using TST and VTST and
compare them to both numerical results and to transition rates obtained
from the limiting equations. We find th a t the VTST approxim ation for
the transition rate is exact, rather than just an upper bound, which is not
generally the case. We explain why the m ethod works for the c urrent model,
draw necessary conditions for the applicability of VT ST in the way used here,
and discuss the nature of spurious recrossings of the TST plane in light of
our findings.
The motivation for studying TST in the context of the Kac-Zwanzig
model is its simplicity, which allows a detailed , rigorous analysis. We have
shown that in this model, VTST given the exact rates of crossing between
the two me tastable sets. Unfortunately, th e general case is not as simple. In
6
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
19/161
part icular, th e only reason why hy perplanes, ra th er th an more complicated
hypersurfaces were sufficient is the quadratic nature of the interaction with
the bath . This is obviously an artifact of this particu lar model. Because
the transition region is not localized at a saddle point, the VTST dividing
surface is in general a more complicated surface than the lift-up in phase
space of a hyperplane in configu ration space. It has been proved [64] tha t
VTST is always exact when minimizing over hypersurfaces in phase space.
The question of whether minimization over hypersurfaces in configuration
space (position only, not momentum) is sufficient is still open, and is of
much practical importance.
The theoretica l pa rt of proving stong convergence to the limiting equa tion
poses some inte rest ing op en problems and generalizations as well. The ra te
at which the solution of the finite system converges to the solution of the
limiting one depends exponentially on the time length considered. Hence,
it seems like stron g convergence is too strong . It would be interesting
to prove a weak type of convergence (in dis tribu tion) for all times. Such a
statement is not trivial since there is no clear separation between the time
scale for the dynamics of the distinguished particle and that of the bath.
The second example considered in this thesis, presented in Chapter 3,
7
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
20/161
studies a system of elastic spheres. The dy namical theory of Brownian motion
has been the subject of extensive research both from the theoretical and
the num erical points of view. A particularly simplified model is th a t of
spheres interacting only through elastic collisions. In this model, a large
heavy spherical particle (colloid) is placed in a gas of smaller lighter ones.
Due to the collective effect of numerous collisions between colloid and gas,
the d isplacement an d velocity of the colloid seem erratic an d stochastic. It
has been proved [15, 34, 62, 63] that in the limit of an infinitely massive
colloid, and under specific scaling of time and space, the dynamics of the
colloid can be described by an Ornstein-Uhlenbeck (OU) process
t
dx(y) = v( t )dt
( i . i )
dv(t) = av{t)dt + V D d W ( t ) ,K
where x( t) and v(t) denote the position and velocity of the colloid, respec
tively, W(t) denotes the Wiener process, a > 0 is the viscosity and D > 0 is
the diffusion coefficient. Althoug h the limiting eq uation is known, the rate
of convergence is still undeterm ined. As a result, comparison between the
dynamics predicted by the limiting OU process and numerical simulations is
of importance.
Different attempts for simulating hard sphere systems with disparate
8
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
21/161
masses can be found in the litera ture [2, 39], all of which amou nt to advancing
the system collision by collision [1, 27]. Th e main draw back of this metho d is
that due to the disparity between the masses of the two constituents, it takes
an extremely large number of colloid-gas collisions to produce a significant
change in the colloid position or velocity. In add ition, one also has to resolve
all collisions between gas partic les themselves. A simple estimation shows
that the required simulation time is proportional to the ratio between the
colloid and gas particle masses.
In the second chapter we present an alternative method for simulating
the system in three dimensions. Although it is stochastic in natu re, we
prove th a t th is method ap proximates th e dynamics of th e colloid with in a
calculated error. The efficiency of the me thod is indepen dent of the ra tio
be tween th e two masses, which const itutes a big improvem ent over th e full
simulation method . The a lgorithm is based on the heterogeneous multiscale
meth od [16, 18] and is co nstructed along the lines of [18, 23, 69]. Th e basic
idea is to calculate the drift, av, and diffusion terms, DdW/dt , on the fly
by simulating th e gas for a sh ort tim e segment At . Using these values the
colloid is advanced by a big time step As according to the limiting equation
(1 . 1 ).
9
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
22/161
The key property of the system used in this Chapter is the gap between
the timescale on which the gas and the colloid evolve. Because the colloid
is so much heavier than gas particles, the gas will reach equilibrium before
x(s) and v(s) change considerably. This mo tivates an approximation of the
stochastic differential equation 1.1 by a forward Euler scheme with step size
As
(x i+i = xt+ ViAs
( 1.2 )
vi+i = Vi - aViAs + y/DAs{,i,
where are indepen dent random variables with normal distribu tion in M3.
The idea behind th e algorithm is to evaluate a and D in each step by making
a short simulation of the gas with fixed v V{. Each Euler step consists of
three parts:
1. Simulate the gas for a time segment of length A t while keeping the
velocity of the colloid is fixed.
2. Use the statistics o btained from pa rt 1 for evaluating a and D.
3. Move the colloid according to the forward Euler approximated equa
tions of motion (1 .2 ).
Denoting the ratio between the colloid and gas particle masses by E 1, it is
10
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
23/161
shown that ifA tis 0 (1 ), i.e., independ ent of E, while As is proportio nal to E,
then the variance in the error introduced by this approximation is bounded.
Hence, the simulation is accelerated by a factor E. This is the m ain result of
Chapter 3.
The numerical experiments presented here are benchmark examples, sim
ple enough for demonstra ting th e ad vantag es of th e method. First , we calcu
lated the viscosity and diffusion coefficients of the colloid in the case E = 100.
Th e results deviate substan tially from values obtained by Du rr et. al. [15]
in the limit E >oo are found. The source of the errors are co rrections due
to correlations and the finite size of gas particles, not taken into account by
the limiting equation. A second example, in which E 104, demonstrates
that when E is large enough, the dynamics of the colloid is well described by
the limiting stochastic equation.
The new simulation method suggests numerous applications and general
izations in which the colloid dynamics is more complicated, and the limiting
equation is more difficult to ob tain or may no t be known. For instance, th e
case in which the colloid is not a sphere but an ellipsoid poses an enormous
com putation al challenge. The an gular mom entum of the colloid has to be
taken into account and calculating collision times is much more complicated.
11
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
24/161
It is also interesting to investigate more thoroughly a system in which the
gas is in a dense regime. In this case the correlation function C{t\ 12) does
not decay exponentially, and a larger time step A t may be required.
Another interesting generalization is to the case of two or more colloids.
When the separation between two colloids is a few times the diameter of gas
pa rticles , th e collective effect of colloid-gas collisions is to pu sh th e colloids
closer toge ther [7, 45]. In other words, th e gas induces an effective close
range attra ction . Because the rang e of attra ctio n depend s on the size of gas
particles, it vanishes in th e lim it E > oo. This implies th a t th e lim iting
equations of motion of the two colloids is the same as for a single one (except
for elastic collisions between th e colloids). However, with a fixed bu t large E,
the dynamics is completely different. We expect to find th a t und er app ropri
ate scaling, the two colloids become effectively trapped in a metastable state
keeping them close for a long time. Hence, the limiting rate for the dynamics
may be the escape rate o ut of this state. This rate does not depend on E
expo nentially since fluctu ation s do not vanish even in the limit E oo.
12
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
25/161
Chapter 2
The Kac-Zwanzig model
2.1 Introduction
In this Chapter we consider a variant of the Kac-Zwanzig model [24, 25, 76].
In this model a single distinguished particle is moving in a given external
potential and is coupled to N free one dimensional free particles via lin
ear springs. The pa rticles oscillate around th e distinguished one and have
random frequencies that are chosen independently from a probability distri
bution. In itia l co nd itions are chosen rand omly from an inva rian t measure.
In Section 2.2 we introduce the model in detail and derive the Hamiltonian
equation of motion.
13
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
26/161
The re st of the chap ter is devided into two topics. In the first, Section 2.3,
we prove that with appropriate initial conditions, and in any finite time
segment, the trajectory of the particle converges strongly (in L2) to the
solution of an effective stochastic equation. The limiting equa tion has the
form of a generalized Langevin equation (GLE) and satisfies a fluctuation-
dissipation relation. Strong convergence is proved by map ping the initial
conditions of the finite dimensional system to the probability space on which
the rando m noise term of the GLE is defined. Bo th canonical and micro-
canonical ensembles are considered. We also prove convergence for a more
general class of initial conditions that are absolutely continuous with respect
to the above two invariant measures.
The second topic is an application of the limiting equation of motion for
the distiguished particle in a particu lar choice of model param eters. In Sec
tion 2.4 we use the Kaz-Zwanzig model as a b enchm ark problem for a rigorous
analysis of transition state theory (TST) and variational TST (VTST).
In a par ticula r case, in which the distinguished particle moves in a double
well potential the dynamics is bistable. We begin by studying metastab ility
in Kac-Zwanzig throu gh num erical simulations. In particula r, we show th at
by an appro priate choice of metastable sets th e dynamics of th e pa rtic le may
14
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
27/161
be ap proximately described by a two state s Markov process. We th en use
our knowledge of the limiting equation to calculate transition rates for the
limiting equations. This is done by spectral decom position of the backward
ope rator characterizing the limiting stochastic equation. Eigenvalues and
eigenfunctions are found using matched a symptotics [60]. The numerical
and limiting results will serve in our analysis of the various assumptions
underlying TST , VSTS and th e predictions of these theories. We find th at
the VTST approximation for the transition rate is exact, rather than just an
upp er bound , which is not generally the case. We explain why the m ethod
works for the current model, draw necessary conditions for the applicability
of VT ST in the way used here, and discuss the n atu re of spurious recrossings
of the TST plane in light of our findings.
2.2 The Kac-Zwanzig model: Formulation
and elementary properties
The Kac-Zwanzig model is a system describing the evolution of a distin
guished particle with unit mass and position x 0, moving in an external po
tential V (x q) [24, 25, 58, 76]. Th e pa rticle is coupled by a harm onic p otentia l
15
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
28/161
to a bath of N particles of mass m* > 0 with positions Zj, i = 1 , . . ., N. The
system is described by the Hamiltonian
# ( x , p) = i p i + v ( x 0) + 2 ^ - + Y l ^ X i ~ x ^ ^ t2-1)i = l 1 t = l
where, for short hand we use the vector notation x = (x0, X \ , - , x ^ ) , P =
(po.Pi, ,Pn)i and Pi is the momentum associated with X{. The coupling
constant between the distinguished particle and each oscillator in the bath
is 7/ N > 0. Th e scaling with N emphasizes the fact that the interaction is
weak. The equations of mo tion derived from this Ham iltonian are:
N
Xq = - V ' ( x q) - ~ - X i )
- 1 (2 .2)
X i = U J i { x0 - X i )
where
J - z k - (2'3)
We also assume that the frequencies {(^1) 1=1,.,.,jv are independent and iden
tically distrib uted (i.i.d.) rando m variables whose distribu tion is absolutely
continuous to the Lebesgue measure on M.
Here we proceed informally and postpone the mathematically rigorous
analysis to Section 2.3. Similar arguments can be found in [1 1 , 29, 32, 58].
In the derivation presented here we assume that initial conditions for the
16
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
29/161
bath are dis tr ib uted according to th e canonica l ensemble, i.e., a t t = 0 bath
pa rticles are dis tr ib uted according to th eir equilib rium Gibbs measure which
is given by
Xi (0) = A f ( x 0(Q), N/ /3at i)
(2.4)
P i ( 0 ) = N ( Q , m i / P ) ,
where a 2)denotes the Gaussian distribution w ith mean m and variance
-
7/22/2019 Stochastic Dynamics
30/161
In the limit N > oo, th e strong law of large numbers implies th a t for any
fixed t, R N(t) will converge to its average R(t)
R( t) = lim RnU) = 7 lim -J- V '' cosuit = yElcoswf]. (2.9)N00 N>OON
In order to ev aluate the rate of convergence, we calculate the second moment
of Rn. Breaking all double sums into the diagonal and off-diagonal pa rts
yields
E[RN( ti)RN(t2)] E[.Rjv(i)]E[i?/v(f2)] = ^ ( i v ) (2-10)
We conclude that the limiting equation describing the dynamics of the dis
tinguished particle has the form of a Generalized Langevin equation
rf 1x 0+ V'{ xQ) + 7 J R( t - T)x0(T)dT = (2 .1 1 )
where ^(t) is a G aussian process with zero mean a nd covariance function R( t) .
The equality between the memory kernel R(t) and the covariance function
of the random noise is a fluctuation-dissipation relation.
18
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
31/161
2.3 A strong limit theorem
2.3.1 Introduction
Deterministic dynamical systems with a large number of degrees of freedom
typically display chaotic behavior and it is not surprising that the evolution
of certain observables in these systems can be approximated by a stochastic
process. Results in th is direction abound in th e li tera tu re . Most of these
results, however, are of weak convergence-type, i.e. it is shown th a t the
evolution of the observables tends to that of a stochastic process in some dis
tribu tiona l sense. It is more surprising th a t th e evolution of some observables
in deterministic dynamical systems converges pathwise to that of a stochas
tic process, since this requires relating the probability space on which this
stochastic process is defined to the pa ram eters in the system. The present
chapter offers a result in this direction.
Consider
N
m0x0+ f ( x Q)+ rriiUjf(xo - Xi) = 0
< R is a C 1 function, and {u i ,mi} i=i...at
are positive param eters. Eq uation (2.12) is the well-known system introduced
19
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
32/161
by Ford, Kac, Mazur and Zwanzig [24, 25, 76] as a toy model to investigate
several issues in nonequilibrium sta tistic al mechanics. Kac-Zwanzig model
has been intensely investigated, both in the physics and mathematical liter
atures [11, 24, 25, 29, 32, 38, 40, 41, 42, 57, 58, 74, 76]. In pa rt icular it is
known that, with appropriate choices of /(), {w,, n and initial con
ditions, in the limit as N oo the traje cto ry of the distinguished particle,
{x0 (t),x 0(t )} , can be approximated by the solution of a integro-differential
equation with random coefficients, namely
m 0Xo + f { X Q) + J R ( t - T ) X 0(T)dT = (2.13)where 0 > 0 is a parameter, R : [0, oo) t>R is a ce rta in memory kernel and
: [0, oo) t>R is a Gaussian ran dom fu nct ion w ith mean zero and covariance
/?(). Th e typical results, however, are weak convergence theorem s. The
pu rpose of th is cha pter is to offer a strong er convergence re su lt, namely the
pa thwise convergence of {xo(t), x 0(t )} toward { X 0(t),Ao(t)} as N>oo.
The dynamics of a Hamiltonian system is given by a system of determin
istic differential equations. In many particle systems, solving these equations
is either impossible or imprac tical, bo th an alytically and numerically. How
ever, it is often the case that one is not interested in the dynamics of the
full system, b ut ra the r only in a small part of it. The rest of the system is
20
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
33/161
referred to as a bath . It is then reasonable to look for effective equations
that describe the behavior of the smaller, interesting part, in the presence of
the ba th. We prove tha t if the b ath initial conditions are distributed accord
ing to the microcanonical invariant measure (and assuming some sufficient
conditions on the model parameters), then the solution of the equations of
motion for a finite N converges strongly (in L 2), in the limit N >oo, to the
solution of an effective stochastic integro-differential equation describing the
dynamics of the distinguished particle.
2.3.2 M ain results and exam ples
We study a variation of the well-known Ford, Kac, Mazur an d Zwanzig model
[24, 25, 76], in which a single - distinguished - one dimensiona l partic le is
coupled to a ba th ofNone dimensional free particles via harmo nic po tentials.
Similar models were previously studied from several different aspects [11, 24,
25, 29, 32, 38, 40, 41, 42, 57, 58, 74, 76]. As already n oted above, the model
is described by the Hamiltonian
where xo, Po and m 0 denote the position, momentum and mass of the dis
tinguished particle we are interested in describing, and ay, px and m, denote
^ { x i - x o ) 2, (2.14)
t=i
N
21
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
34/161
the position, momentum and mass of the i-th bath oscillator, i = 1 . . . N .
For shorthand we use the notation x = (xq,Xi, . . . , xn ) and similarly for
writing vectors in MN+1. The disting uished particle is placed in an extern al
potential V{xq). Some restrictions on this potential that guaranty existence
of solutions are specified later. The coupling between th e pa rticle and each
oscillator is taken as harmonic, with spring constant 7/ N > 0. Th e scaling
with N emphasizes the fact that the interaction between the particle and
each one of the oscillators is weak. The equa tions of mo tion derived from
this Hamiltonian are (2.12) where
For fixed N , the Hamiltonian dynamics preserves the total energy and the
dynamics is restricted to an energy shell {H = E}. Since the to tal energy is
an extensive variable, we take E = Eq+ N//3 , where Eq= p%/(2mo) + V(x q)
is the initial energy of the distinguished particle in the absence of the bath.
The parameter 0 plays the role of an inverse tempera ture. We therefore make
the following assumptions regarding initial conditions:
Assumption la: The initial conditions of the distinguished particle
are fixed to (xo,Po).
22
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
35/161
and for the bath,
Assumption lb: The initial conditions of the bath distributed ac
cording to the microcanonical equilibrium measure.
For fixed N this measure is given by
dtiNtE(x,P) = | ^ )r (2'16)
where | | is the s tanda rd Euc lidean norm in R2iV+2, dcr(x,p) denotes a surface
element (Lebesgue measure) on H (x , p) = E, and Z E is a normalization
constant.
Louivilles Theorem implies th at the Ham iltonian dynamics preserves vol
ume in phase space. Hence, the equilibrium distributio n of the m icrocanon
ical ensemble is invariant under the dynamics (2 .1 2 ).
We also make the following assumptions regarding the model parameters:
Assum ption 2: The bat h frequencies, {wj}i=i...Ar are independen t, iden
tically distrib uted (i.i.d.) rando m variables, with probability density
function (PDF) p(u>) with respect to (w.r.t.) the Lebesgue measure
on [0, oo). In add ition , p(u>) is strictly positive in its support and the
number of spectral gaps is finite, i.e., the support is a finite union of
segments.
23
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
36/161
The assumption on the finite number of spectral gaps can be relaxed at the
cost of a slower convergence rate. Th is generalization is discussed in Section
2.3.3. In order to avoid confusion, exp ecta tions w.r.t. initial cond itions are
denoted Eo[-], while expec tations w.r.t. the frequencies are deno ted =
J[-]p(uj)duj. Assumption 2 can be relaxed to admit a more general case in
which both the coupling coefficient 7 and the PDF p(cu) depend on N , but
converge to a b ound ed integrable function. This general set up is discussed
in Section 2.3.4.
Our goal is to derive a stochastic equation that approximates the dy
namics of the distinguished particle for large N . The effective equation has
the form of a generalized Langevin equation (2.13), where R(t) = Ew[cos ut\
is a memo ry kernel and f() is a rando m noise. It is a Gaussian process
with zero mean and covariance given by a fluctuation-dissipation relation
Eo,u[(ti)(t2)] = R ( h 2)- If V'{%q) is uniformly Lipshit, then the solution
of (2.13) is well defined.
Denote by (xN(t),pN (t)) the full solution of the 2N+ 2 equations of mo
tion (2 .1 2 ), and by (xq (t),p^ ()) its projec tion on the coordinates describing
the d istinguished particle. Th e solution of the effective stochastic equation
(2.13) is denoted (X(t), P(t)). We prove the following strong convergence
24
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
37/161
theorem which is the main result of this chapter
Theorem: Suppose assumptions 1 and 2 hold. Then, fo r any T ). For instance, if the tail of p(uj) is polynomial, C u~ q ) < D u ~ q, then I = (q l ) / (3 q1). I f the tail of p(u>) is exponential or
better, then Z= 1/3 . In the case of macrocanonical bath init ial condit ions,
the random noise (t) is a Gaussian process with zero mean and covariance
m t i ) s ( t2 ) = R { t i - t 2).
Note th a t th is rate is only an upper bo un d. Improved bo un ds may be ob
tain ed for specific cases. Th e pro of of strong convergence requires identifi
cation of a map between the probability space on which the random initial
conditions are defined, and the probability space on which the effective ran
dom noise is defined. Path -wise convergence of Xq (t) to xq(Z) is obtained
using the Gronwall inequality.
In particular, the theorem implies that the probability that trajectories
of the distinguished particle given by the finite system (2 .1 2 ) are far from
25
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
38/161
that of the effective one (2.13) can be made arbitrarily small, i.e.,
lim PN oo
sup (t) X()| > eo< t < T
= 0 . (2.18)
Our results improve on previous works in several aspects. Fir st, we con
sider a wide range of model parameters rather than a particular choice of
PDF p(u>). Second, our initial conditions are drawn from the microcanon-
ical invariant measure, and not the canonical one. Since the H amiltonian
dynamics conserves energy, the microcanonical ensemble is the appropriate
one for this model. Finally, we prove a stron g type of convergence rath er
than convergence in distribution or probability. The proof is constructive in
the sense that it shows explicitly how the random initial conditions blend
into the noise. W ith arb itrarily large probability, every choice of (x(0), p(0))
that satisfies #(x (0), p(0)) = Eq + N/ (3 corresponds to a continuous func
tion () such that the solution of ( 2 .1 2 ) with initial conditions (x(0),p(0))
approximates the solution of (2.13) with this particular ().
It is interesting to note that due to pathwise convergence, the result of
the Theorem holds for all initial distributions for the bath that are absolutely
continuous with respect to the microcanonical one. Any initial distribution
on and { p ,} ^ , will induce a probability distribution on the noise
w(0- This, in turn, induces a probab ility distribu tion on the limiting process
26
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
39/161
(t) = lini/v-xxj w(0- Th e limiting noise will generally no t be Gau ssian, and
will not satisfy a fluctuation-dissipation relation.
As an example, take P (u ) to be the one sided Lorentz distribution
P M = 2, ^ > 0 (2.19)7T OL U)
and p(u>)= 0 otherwise, where a > 0. In a previous pape r [5] we stud ied this
model both formally and numerically in the case when V(x0) is a double-well
potential an d th e dynamics is metastab le. In Section 2.3.3 we prove th a t th e
effective equation for x 0(t) is
m 0X 0 + V ' ( X 0) + 7 j \ ~ a't- T' X 0( r)dr = ~ ^ t ) , (2.20)
and (t) is an Ornstein-Uhlenbeck (OU) process at equilibrium with zero
drift and an exponentially decaying covariance with rate a.
2 .3 .3 Pr oof o f the theorem
Although the appropriate invariant measure for the Hamiltonian dynamics
(2 .1 2 ) is the microcano nical one, it is technically simpler to use the cano ni
cal invariant measure. Under the canonical ensemble, initial conditions are
indepen dent G aussian random variables. The ir distribution is given by the
27
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
40/161
Gibbs measure
Xi = N ( x 0,N /pyi ) ,
( 2 .21 )
Pi = N(0,rrii /f3) , i = l . . . N ,
where N( fi, 1 / a 2) denotes the G aussian distribution with mean n and vari
ance a 2. Most auth ors studying such models consider only the canonical
ensemble [29, 32, 38, 40, 41, 42, 57, 58, 74]. We will first prove the Theo
rem for the case of canonical initial conditions and then prove that the same
result holds for the microcanonical one.
Canonical initial conditions
We begin with a few prelimina ry calculations. Using either variation of pa
rameters or the Laplace transform, (2.12) can be solved for x ^ .. . x N. Sub
stituting into the equation for x 0(t) and integrating by parts, the equation
for Xq (t) can be written as
(2 .22)
where
N
(2.23)
and /v(t) is given by
(2.24)
28
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
41/161
Note th a t th e bath in itia l co nd itions appear only in ^ . For th is reason we
will refer to /v(t) as a ran dom noise term. Changing variables into dimen-
sionless, centered coordinates
IQ'Y
hi = V77 ~~
9i = ~ P*(0)>V rr i i
(2.25)
the noise term r(t) can be written as
I N= J ^ 2 (hi cosujit + gi sin LUit). (2.26)
1=1
The T heorem is proved in three steps. The first considers convergence
of the memory kernel i?/v(f). For fixed time t, the random variable cosu t is
bo unded. Hence, th e law of large nu mbers implies th a t Rn(I) converges to
its average for almost all u>
N
lim f?Ar(t) = lim -7- cosujt =7 ECi)cosu t R( t), (2.27)N-+O O N OO N /
1= 1
In Lemma 1 we prove that Rn(I) converges to R(t) strongly. The second
step considers the noise. In Lem ma 2 we prove th a t jv() converges strongly
to a limiting Gaussian process, (), with zero mean and covariance R( t) .
The final step considers the position and momentum of the distinguished
pa rticle. Stron g convergence of x ff(t) to x 0(t) and of (t) to p0(t) is proved
in Theorem 1. All steps consider a finite time interval 0 < t < T, T
-
7/22/2019 Stochastic Dynamics
42/161
Initial conditions of the b ath have a Gibbs distribution. Hence, hi and i, and the definition of R(t),
(2.27), yields (2.28). Th e strong convergence is also an imm ediate conse
quence of Birkoffs ergodic theorem in L 2 [73].
A similar calculation shows that for large N,
sup E J i ? ( t ) - i ? ( t ) |4 < I ^ . (2.30)0 < t < T (V
This is used in the proof of Theorem 1.
30
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
43/161
L e m m a 2 : For all T < oo, there exists a constant C independent of T , N
and p(u>) such that,
C T 2
sup E j ( f ) - 6 v ( * ) | 2 ^ (231)0 < t < T A
where (f) is a Gaussian process with zero mean and covariance R( t) . The
rate of convergence, I, is determined by the tail of the distribution ofp(u>) as
detailed before.
Proof: For fixed N and t, jv() is a linear combination of independent Gaus
sian rando m variables and is therefore Gaussian itself. Hence, () is also
Gaussian for fixed t. As a result, all marginals of the form ( (f i) ,. .. ,(&))
are Gaussian vectors, which implies th a t () is a Gaussian process. It is
interesting to note that (2.31) implies that jv() converges in distribution
to (). The ra te of this weaker convergence is always N ~ 1/2 and does not
depend on the tail of p(u>).
In order to prove strong convergence, we must identify how the random
initial conditions hi and gi blend into the limiting process. In other words,
both th e sequence of processes i( t) , 2^) , and the limiting process (t)
have to be defined on the same pro bability space. To achieve this we write
31
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
44/161
the limiting process as
m = \ / 7 / p l/ 2(ui)[costutdhu+ sin u>tdgj\ , (2.32)Jo
where and gw are independent copies of the standard Brownian motion
(BM) on [0, oo). To see this is indeed (), we use the Ito iso metry to show
that the average of this process is zero and the covariance is
We now wish to define the initial conditions, hi and gx,on the same probabil
ity space of hMand gw. The difficulty is in finding a scaling tha t relates th e
random variables hi and ^ to parts of h^ and g^. We begin with a few defini
tions. Let a be a permutation of {1,.. . , N } that rearranges the frequencies
in increasing order, i.e., ui^i) ). For short
hand we denote p x = ipf(u>a(i)). We take the maps to b e increasing in the
sense that
-
7/22/2019 Stochastic Dynamics
45/161
Th e rand om variables are defined similarly using gw instead of h^. The
motivation for this definition is that for any family of such maps, hi and gl
are independen t random v ariables with normal distribution. This is easily
verified using the Ito isometries. We now look for the scaling maps th a t
would yield strong convergence, and show that for all 0 < t < T,
for some I > 0. The constant C depends only on the density function p(u>).
A necessary condition is to have convergence at t 0. Th is simple case will
provide us with th e requ ired scaling. Substi tu ting in the representation s for
at and ,
where x a {u) denotes the indicator function of the set A evaluated at u>.
Using the Ito isometries yields
(2.35)
Eo.0,16v(0) - mi2=27 - 27 L V Pi-Pi-1- (2-37)
It is clear that we want to find a map ip such that
(2.38)
33
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
46/161
Squaring this expression and using Jense ns inequality yields an up per bound:
^(^Ev^wn) < 1. (2.39)
Equality holds if all the terms Pj P,_i are equal, i.e., Pj = i / N . Hence,
strong convergence is obtained if
( d ' K w ) = P - 1 ( - 0 = P - \ ( 2 . 40)
rpr l K( i) = V N / p l,2{ u) dh u, (2.41)
Jp- \
where P 1denotes the inverse of the distribution function Pw{u < z). There
fore, initial conditions take the simple representation
rp,r
and similarly for gt .
For t > 0, we have, substituting in the representations for f, N, hi an d
9i
Eo.c m ) - ZN{t)I2 = 27 (1 - S N) (2.42)
with
x
5 iv = E 0iW / p 1 2(x) cos(xt)dhx + / p 1 2(x) sin (xt)dgxL v R Jr
N ( rpi X P E 1V ] / p 1/2(x)dhx cos(uaii)t) + / p 1/2(x )dg x sm(u
-
7/22/2019 Stochastic Dynamics
47/161
Note th a t averaging with respec t to in itia l co nd itions implies averaging over
the two BM hx and gx, while averaging with respect to u implies averaging
over mu ltiple samples of Clearly, we need to show th a t Sn > 1 .
Using the Ito isometries yields
N f P r 1SV = y ~ ' E u, / p( x) cos( (x - u a(i))t)dx. (2.44)
i = i J p r - \
Bounding the cosine by 1 yields an up per bo und , Sn < 1. To get a lower
boun d, we show th a t for most i, (x oJa{i)) is small, and the cosine is almost
one. The reason why (x u>a(i)) is not small for all i is due to the tail of
the density p(uj). We therefore break the sum into two parts: up to kN and
above. For i = 1 . . . k^ , we use co sx > 1 x 2/2. Let u>k be such that
Pw[u> < u>k] kN/ N . For i = (kn + 1) N , we bou nd the cosine by 1.
Hence,
fcjv pP~l N rW 1I , p{x ) [ ^ ~ i.x - ^ a ( i ) f T2/ ^ ] d x - / p{x)dx
1 = 1 J p i - l i = k N + l ^ p i - \
(2.45)
p - 1
Noting th a t f p p(x)dx 1 /N yields
\SN - 1| < ~ P^ E^ X ~ u (i))2(lx+ 2 ( l (2-46)
Th e main difficulty is in evaluating the exp ecta tion E w(x tu^j))2. Noting
35
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
48/161
that for x G [P ^ ),Pi *]
K , ( x ~ ^ ) 2 < 2 ( Pr ' - ^ - l ) 2 + 2EW( P - 1 - coa { i ) ) \ (2.47)
we proceed in several steps:
L e m m a 2 .1 : For all T < oo, there exists a constant C such that for all
i < kf j, and except fo r a possible fini te number of indices,
We begin with the second term of (2.47), which has a form similar to the
setu p of the Kolmogorov-Smirnov statistics. Bre iman [10] gives a proof of
the following Theorem:
Let U \ , . . . ,Un denote n independent samples from a random variable, un i
forml y dis tributed on [0,1]. Let P CT(i ), . . . , Ua(n) denote the same set of sam
ples arranged in increasing order. Then, the random variable
L > n = Vnm ax \Uaii) - i / n \ ,i < n
has a limiting distribution with finite variance.
This implies that,
(2.48)
Proof:
m ax E ( U. (2.49)
36
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
49/161
Noting tha t P(u>lT ')) is uniformally distributed [0,1], i / n = P(Pi : )
P (v) > 0 for all u>< u>k and for k 1 , . . . , we have
e u p - 1 - u a{i))2 0j)(w ) ) ]2
Since p{u ) is strictly positive in its support, the minimum is bounded and
E ( / r UVW)2 < (2.51)
If there are sp ectral gaps then (2.51) fails whenever two subsequent samples
u>cr(i) and u>tr(i+i) are in different conectivity classes of the s upp ort ofp(u>). In
the latter case, which may happen in a finite set of indices, we have
(jEu)( i^ _1 - u a{i))2 < + largest gap. (2.52)
Substituting (2.51) and (2.52) into (2.46) yields,
+T ?
-
7/22/2019 Stochastic Dynamics
50/161
for some I2> 0 that depend on l\.
Proof:
We prove Lemma 2.2 for different types of distributions p(u), and show
that the rate of convergence of t) to (t) depends on the tail of p(uj).
Suppose that the tail of the density function p(oj) decreases polynomially,
i.e.,
C u~ q < p{uj) < D u~ q, (2.55)
for some q > 1 and C, D > 0. We use C and D to denote generic constants
whose values may vary between expressions. Den ote l - k N/ N = N ~ l, I > 0.
Since p{u>) is strictly positive, then for large enough N we have,
Cu k q )COk] = / p(u})du> < D qdu = ----- 7^l~q.
J^k Jwk Q~ k
(2.57)
Using the opposite bound yields
C N h/(q-i) < U k
-
7/22/2019 Stochastic Dynamics
51/161
Hence, for i < k^,
. r 'N\j = ( ' p(u>)dw > C(P- ' - P ~\ y i > C ( p r ' - p r_ \) N - W i -'), (2.59)v Jpr~\
which implies that
|P f1 - < N hqn* - l)- 1. (2.60)
Similar evaluations can be done with different tails. For instance, if p(u>)
decreases like e~ax, e~ax or has finite sup port th an th e rate is the sup over
all qs, I2= h 1 .
Combining the above bounds we find that with a w~ g tail, the optimal
rate is obtained with
With an exponential tail or better,
l = i . (2.62)
This concludes the proof of Lemma 2.
Figure 2.1 compares the random noise at() for N = 2000 obtained by
drawing random initial conditions hi and gi, with (i) obtained by sampling
39
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
52/161
two BMs hw and th a t satisfy the con straints (2.41). The sample is pre
pared in th e following way: we first draw th e frequencies, {uJi}i=i,..N, and
then initial conditions hi and from the canonical distribution. For the fre
quencies we use the one sided exponential distribution p(u>) = e~w for u > 0 ,
and p(u) = 0 otherwise. We can then comp ute /v(t). The exponen tial distri
bu tion is convenient since th e an ti-de riva tiv e of e~u^2cosuit, which appears
in (2.41), is easily calcula ted. We the n app rox imate two BMs hu and that
satisfy the constraints (2.41) by a linear interpolation between the points
P _1(i/./V), i = 0 . . . N . This partition becomes finer as N >oo. Hence, the
approx imation converges to BM by the invariance principle. The limiting
noise (t) is then obtained through its representation given by (2.32). Since
the approximation is piece-wise linear, the integration w.r.t. dh^ and dg^
can be preformed analytically.
Theorem 1 : For all T < oo, there exists a constant C(T), independent of
N such that,
sup E0,w0< t < T
|xK(t ) - X ( t ) I2+ 1 ^ ( 0 - F ( ( )|2] < ^ 2 , (2.63)
where the rate of convergence, I, is the same as in Lemma 2.
This implies that Xq (t), the solution of the full equations of motion (2.12),
40
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
53/161
2.5
0.5
- 0.5
- 1.520
t
Figure 2.1: Com parison of the rando m noise, jv(f) (solid line), and the
limiting one (t) (dashed line). The frequencies are drawn from a one sided
exponential distribution and N = 2000. The ba th initial conditions are
canonical.
converges strongly to X ( t ) , the solution of the limiting stochastic equation
(2.13).
Proof: Compare the solutions of the limiting equation for (X ,P):
X = P / m 0
t (2 -64 )
P = - f - V ( X ) - ^ f R ( t - T ) P ( T ) d T + - ; ( ( t ) ,m0 mlQJo v/3m0
41
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
54/161
(2.65)
with the one obtained for finite N:
%o = P o / m o
Po = - V ' ( x Z ) - f R N(t - r )p g ( r )d r + N(t).m 0 rag J0 V/5m0
For shorthand, we drop the subscript zero from mo,XQ and Pq for the rest
of the section. We wish to show th at
sup E 0,a, [X (t) - ^ { t ) ] 2+ [P(t) ~ pN(t)]2 oo, one has to
consider the convergence of R ^ ( t ) , iv(t) and (x q (t ) ,p(t )) . However, since
jRiv(t) does not depend on initial cond itions, th e only difference between the
microcanonical ensemble and the canonical one considered in Section 2.3.3 is
the convergence of the noise, #() Once this is established, convergence of
Xq (t) follows by the exac t same arg um ent d etailed in the proof of Theorem 1,
(2.79)
The normalization Zis obtained by taking /( h i) = 1. S etting /( h i) = e lthl
yields the characteristic function of the marginal h\
(2.80)
46
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
59/161
Section 2.3.3. In this Section we use the s up er sc rip tsm and c to distinguish
between micro- and canonical initia l co nd itions or noise.
In the case of Gaussian initial conditions, %(t) is trivially a Gaussian
process for any N. Hence, the sequence %(t) converges to a Gaussian pro
cess, c(f). Th e situa tion is not as simple with th e microcanonica l ensemble.
We recall the definition of jv, (2.26):
1 NG W = - j = ^ 2 y / T i ( h ^ c o s u } i t + g ^ s i n u i t) . (2.81)
' i = l
For fixed t, it is easy to see th at the characte ristic function of converge,
as IV > oo to the sam e charac teristic function in th e can onical case, c().
This is because contributions of order iV- 1 do not sum up to 0(1) due to
the N ~ 1/2 prefactor in (t). Hence, () converge in distr ibut ion to the
same limit c(f). The rat e is prop ortiona l to N ~ ^ 2. Th e same holds for
any linear com bination of marginals a t different times. This implies tha t the
whole process $() converges in distrib utio n to (), which is Gaussian. For
this reason, from now on we can drop the ensemble label from the limiting
noise ().
In order to prove strong convergence, we need to map the probability
space of the initial conditions hand gto the one on which () was defined:
the two BM hu and gu. The idea is to draw independent initial conditions
47
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
60/161
according to the Gibbs distribution and th en scale them to the desired energy.
The dependence between different variables comes from the scaling factor.
Define
K ^ h y r ; g? = r f / r , ( 2 .8 2 )
where
= v / 2H / N =
\ 1 = 1Then, with hc{ and g\ given by (2.41), h' f and ghave the same distribution
as the microcanonical initial conditions hand g. We can therefore drop
the tilde notation. Substituting into (2.81) yields
( 0 = i& W - (2.84)
An elementary calculation shows that, in the limit N > oo, the average of
r te nds to 1 while the variance is 0 (N ~ 1). Hence, $() convergence to (t)
strongly. The dependence of the ra te on N is the same as in Sections 2.3.3.
To sum up, we have proved the following Theorem:
Theorem 2: For all T < oo, there exists a constant C(T), independent of
48
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
61/161
N such that,
sup E 0)U0 < t < T
X, \
7 t = 7 If the product 'y(N,u>)pn{uj) converge in Li(ui) to a limiting
function p(u>), then the cosine transform ofp(uj) is bounded and continuous.
We denote R(t) = J p(u>) cosuitdu). Note tha t p(u/) may not be normalized.
Assuming that 7 (N,lo)pn(u>) converge also in L 2 (w), and that 7 ( N , u ) < C N
for some constant C > 0, it is easily shown that R E (t) converges to R(t) in
L 2 (u>). The proof is the same as in Lemma 1, however, the convergence rate
may be smaller. Once strong convergence of the covariance function is estab
lished, stron g convergence of the noise w(t ) and the trajectory ( x ^(t),p$ (t))
follows. For instance , S tu ar t and W arren [58] suggest the following example
Pn {u ) = t ^ X [ o , j v ] ( w )
N (2 .86 )
7(NM=7r a z + ur
where a, 7 > 0 and 0 < a
-
7/22/2019 Stochastic Dynamics
63/161
(2.19).
If we remove the requirement for L\ (u>) convergence, then the model ad
mits a m uch larger variety of limiting processes. For instance, if y (N , cu)p^r(u)
is not uniformly bounded, then the covariance function R(t) is not necessar
ily bounded and continuous. Taking p(u>) = N a / n / ( N 2a -f- u 2) and 7j = N a,
yields, for any 0 < a < 1/2, i?jv(t) = N ae~N
-
7/22/2019 Stochastic Dynamics
64/161
fortunately, it is often these long time scales, or even the asy mp totic behavior
of the pa rticle th a t are interesting. An example of this is a case in which
the external potential V(x0) is metastable [5]. However, it seems reasonable
that the process (x q,Pq) should converge to the limiting one weakly or in
distribution uniformly for all times T > 0. This is because the invariant
measure of the finite system should converge to the invariant measure of the
limiting one. This problem is beyond the scope of the present m anuscript
and will be trea ted in a later publication. A proof for a specific case similar
to (2.86) was given in [42].
52
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
65/161
2.4 Testing Transition State Theory on Kac-
Zwanzig Model
2.4.1 Introduction
Deterministic dynamical systems, in particular, Hamiltonian systems, often
display very complicated chaotic behavior when the number of degrees of
freedom in the systems is large. Amid the complexity of individual traje c
tories, it is sometimes the case that most trajectories remain confined for
very long periods of time in well separated regions of phase-space and only
switch from one region to ano ther occasionally. The confinement is due to
the presence of dynam ical bottlenecks between these regions. The system is
then said to display metastability, and the regions in which the trajectories
remain confined are referred to as me tastable sets. Example of systems dis
playing m eta stabil ity abound in natu re , with exam ples ar ising from physics
(phase transitions), chemistry (chemical reactions, conformation changes of
bio-molecules), biology (switches in po pula tion dynamics), and many others.
In these systems, it is worth verifying whether the dynamics can be approx
imated by a Markov chain over the state-space of the metastable sets with
approp riate rate constants. The m ain question of interest then becomes the
53
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
66/161
determ ination of these rate con stants. This question is nontrivial because it
amounts to understanding the pathways by which the transitions between
the m etastab le sets occur. These pathways are usually non-trivial and com
plicated.
One of the earliest attempts to determine transition pathways and rate
con stants is tran sition s tate theory (TST ) [22, 35, 75]. Under the sole as
sumption that the dynamics of the system is ergodic with respect to some
known equilibrium distribution, TST gives the exact average frequency at
which trajectories cross a given hypersurface or hyperplane which separates
two m etasta ble sets of interest. This average frequency can be used as a
first approximation for the frequency of transition between the metastable
sets. Unfortunately, it was recognized early on tha t this approxim ation can
be qu ite po or , for not every crossing of th e dividing surface corresponds to
a transition between the meta stable sets. Indeed, the trajectorie s can cross
the dividing surface many times in the course of one transition. As a result,
the TST prediction for the frequency of transition always overestimate the
actua l frequency, sometimes grossly so. One way to minimize this problem
is to use the freedom in the choice of dividing surface. The best prediction
for the frequency from TST is the one corresponding to the dividing surface
54
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
67/161
with minimum crossing frequency. This idea is at th e core of the so-called
variational transition state theory (VTST) [35, 51, 65, 70], which aims at
identifying the dividing surface with minimum crossing rate which is the
lift-up in phase space of a surface defined in configuration space.
Unfortunately, VTST (just like TST) is an uncontrolled approximation,
for it only provides an upper bound on the transition frequency between
the m etastab le sets. In general, one does not know how sharp this b oun d is.
Other assumptions beyond TST and VTST are usually difficult to assess too.
Are successive transitions between the metastable sets well approximated by
Poisson events (i.e. statistically indepen dent and with exponentially dis
tributed waiting times) as required for the approximation of the dynamics
by a Markov chain to hold? How does th is pro perty dep en ds on th e definition
of the metastable sets? Etc.
In this section we study a benchmark problem, which is simple enough
so that many of the assumptions and approximations underlying TST and
VT ST can be examined. Bu t the m odel is also complex enough to display a
wide variety of phenomena common to many dynamical systems exhibiting
me tastability. The problem we consider is a variant of a model originally pro
posed by Ford, Kac, and Mazur [24, 25] and Zwanzig [76]. It was revisited
55
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
68/161
in the context of transition rates in [13, 30, 52, 53, 54, 55] and more re
cently, from a more analytical point of view in [11, 29, 32, 36, 40, 41, 42, 58].
The Kac-Zwanzig model is a Hamiltonian system describing the evolution
of a distinguished particle moving in a double-well external potential and
coupled to a bath of N free particles via linear springs. The dynamics in
this model is deterministic. However with app rop riate choice of the pa ram e
ters, the evolution of the distinguished particle can be captured by a closed
stochastic differential equation in the limit of infinite bath, N >oo. W ith
the double-well external potential, the model displays metastability over two
sets (bistability). Tran sition rate con stants between these sets can be com
puted exac tly from th e effective stocha stic dynamics in th e lim it N oo.
The values for these exact transition rate constants can then be compared
to the predictions of TST and V TST. This comparison is the m ain objective
of this paper. In particular , we will show tha t the application of TS T w ith a
naive (but natural) choice of dividing surface based only on the position of
the distinguished particle leads to a wrong prediction for the transition rate
constants. This is because the naive dividing surface is crossed many times
in the course of each transition between the two me tastable sets. However,
we will also show that if one optimizes over the dividing surface following
56
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
69/161
VT ST, these many spurious crossings can be eliminated completely. Hence,
the correct transition rate constants for the model can be computed within
VTST. The optimal dividing surface which allows one to do so is then a
surface whose normal spans all the configurational degrees of freedom in the
system and n ot only the one associated with the distinguished particle. We
shall try to explain why this is the case and when a similar success of VTST
can be expected in other more realistic systems.
In [52], Poliak et al. app rox imate a generalized Langevin equation , which
has the same form of the limiting equation in our case, by the Hamiltonian
dynamics of the Kac-Zwanzig model, and then use TST for obtaining the
escape rates. In [54, 55], they ob tain th e same rate s from the limiting equa
tion by a generalization of Kram ers meth od [28]. Although our results are
similar, th e po int of view is different. Here, the Kac-Zwanzig model is used
as a platform for analyzing the predictions of TST and VTST and testing
the und erlying assumptions of these theories. Our results are also all de
rived from basic principles, and rely on the only (uncontrolled) assumption
of ergodicity.
The rem ainder of this section is organ ized as follows. In Section 2.4.2
we present the model and derive some of its basic properties. Section 2.4.4
57
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
70/161
details numerical experiments, discusses some of the properties of the sys
tem conjectured in Section 2.4.2 and presents results for the transition rate.
In Section 2.4.5 we derive the effective stochastic differential equation that
describes the dynamics of the distinguished particle in the limit TV > oo.
We then calculate trans ition ra tes for the limiting dynamics. This is done by
spectral decomposition of the backward operator of the stochastic equations.
Eigenvalues and eigenfunctions are obtaine d using matched asym ptotics. In
Section 2.4.6 we develop TST and VTST. We find predictions for the tran
sition rates from these theories. Finally, in Section 2.4.8 we summarize our
findings and discuss possible generalizations.
2.4 .2 Th e m odel
In order to simplify the discussion, for the rest of this section we will restrict
our analysis to the following particular choice of probabilty density for the
frequencies
p(u) = * 1 y r ~ 2 if w > o
tt 0,2 + ^ 2 (2.87)
0 otherwise
where o,* > 0 is a parameter playing the role of a characteristic frequency.
Unless sta ted otherwise, we will take o,* = 1. Notice th a t all the moments
58
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
71/161
of (2.87) are infinite, i.e. u>* ^ E ul = oo, where E denotes expectation with
respect to (2.87). Hence, using the no tation of the previous sections
R(t) = qEfcoswf] = 7 e- ^. (2.88)
In section 2.4.5, we proved that if we choose initial conditions (x(0), p(0))
so that
E = N/P, (2.89)
for some P > 0 playing the role of an inverse temperature, then in the
limit N >oo, the evolution of the distinguished particle is described by the
stochastic equations
f t iX o + V ' { x o ) + 7 j e~(t~T)x0(T)dr= (2-90)
where (t) is a G aussian process with zero mean a nd covariance function
7 e- ^. Hence, the noise is an Ornstein-Uhlenbeck process at equilibrium
which solves the stochastic differential equation
d= - d t + x/27 dWt , C(O) = A7(0,1 ), (2.91)
where Wt isa stan da rd Brow nian motion. Using (2.90) and (2.91), the lim-
59
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
72/161
iting equation (2.90) can be written as [29, 58]:
xq = Po
* Po = s / i s - V \ x q) (2.92)
s = - s - ^ p Q+y/W~mt
Hence, with this particu lar choice of model parameters th e limiting dynamics
satisfy a stochastic differential equation.
In Section 2.4.5 we will find the following form of (2.92) useful:
Ldt
( \Xq
Po
\ S /
= -K X 7H (x 0, po, s) + x / ^ W t.
Here H ( x q , P q , s ) = V(x0)+ \ p l+ | s 2,
/ \0
(2.93)
(2.94)
V 1 /
60
produced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
73/161
and K= K s+ K A, with
/0 0 0
K s = aaT = 0 0 0
V0 0 1
/(2.95)
0 - 1 0
K A = 1 0 -V 7
2.4.3 M etastabi l ity in Kac-Zwanzig
In order to study metastability, the distinguished particle is placed in a
double-well poten tial. Unless state d o ther wise, for the rest of this Section
we take
When the inverse temperature (3 and the size of the bath N are large
enough and satisfy 1
-
7/22/2019 Stochastic Dynamics
74/161
instance be taken as
S - ( N , P , 5 ) = {(x, p) :H(x , p) = N/(3,x0 < 0, and H 0{x0,po) < 5}(2.97)
S+(N,0 ,6) = {(x,p) :H (x, p) = N/(3,x0 > 0, and H 0{ x0,Po) oo. (2.98)
Here (Xe is the microcanonical distribution on H(x , p) = E,
(2.99)
62
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
75/161
element (Lebesgue measure) on H(x , p) = E, and 2 ( E ) is a normalization
constan t. The ergodic assumption can not be proved rigorously for the po
tential in (2.96), but it will be corroborated by the numerical experiments
presen ted in section 2.4.41.
The key property implying metastability over the sets in (2.97) is that
for every 5 E (0,1], and similarly for the integral over S -( N , (3,6). Note that
the o rder in which the limits are taken m atters. The validity of (2.100) can
be checked by direct ca lculation. Indeed, performing first th e in tegrat ion
over x \ , . . . , xn and p i , . . . , p we have
where Z 0(N/ /3) = f Ho
-
7/22/2019 Stochastic Dynamics
76/161
stant. In the limit as Noo, this is
lim [ dtiE=N/0(X;P) = Z q 1 [ e~/3Hodx0dp0 (2 .1 0 2 )N ^ J S + ( N , 0 , 8 ) J H 0< 6
where Zq = f R2 e~^Hodx0dp0. The B oltzmann-Gibbs probability density func
tion ZQ1e~/3H is in fact the marginal density for the position and momentum
of the distinguished particle in the limit of infinite bath, N oo. For every
5G (0 , 1], the minimum at (x0,po) = (1 , 0 ) is the only minimum that belongs
to the domain where H 0 < 8. Therefore, (2.100) follows from (2 .1 0 2 ) by
simple evaluation of this integral by Laplace method. It is im po rtan t to note
that S are cylindrical sets in R27V+ 2 and n ot small neighborhoods around of
the energy minima. Due to the high dimensionality of the model, the mass
of the equilibrium measure is not concentrated in a small volume of phase
space since the Laplace approximation does not hold when N > (3.
Equation (2.100) implies that, when 1 -C j3 N , any generic trajectory
solution of (2.2) spends most of its time in either S+(N, (3,8) or S- (N , (3,8).
However, under the ergodicity assumption, this trajectory must switch be
tween S+(N,(3,8) and S-(N,(3,8) infinitely often. W hat are the rate con
stants of these transitions ? How do they depend on 81 Are they statistically
independent, with transitio n events Poisson distributed? In othe r words, can
the dynamics in (2.2) be reduced to a Markov process over S+(N,(3,8) and
64
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
77/161
S-(N, /3 ,5) for some suitable choice of 6? These are the questions which we
shall investigate in the remainder of this paper, first via a series of numer
ical experiments with (2.2) (section 2.4.4), then using the limiting equation
in (2.92) (section 2.4.5), and finally within TST and VTST (section 2.4.6).
2.4 .4 N um erical experim ents
In this section, we perform a series of numerical experiments with (2.2) to
investigate when the dynamics can be app roxim ated by a Markov process over
the two m etastab le sets in (2.97). The questions we are especially interested
in are:
1 . W hat are the rate of the transition? How do they depend on the
para m ete r 7 (interaction strength with the bath) in the model? How
do they depend on the choice of 5 in (2.97)?
2. Are successive transitions to a good approximation statistically inde
pendent? Are th e transit io n tim es in th e sets (2.97) Poisson dis tr ibuted
with intensity equal to the rate of transition? How do these properties
depend on 5?
65
roduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
7/22/2019 Stochastic Dynamics
78/161
The second question is especially important since it determines when the
dynamics in (2.2) can be approximated by a Markov process, and how the
metastable sets have to be chosen in this case to get the correct transition
rate constants to use in the chain.
For these experiments, we will take N = 1000 and /? = 7. We will
also consider two different values of 7: 7 = 1 and 7 = 10. In (2.2), the
equations of motion of the bath are integrated explicitly, while the equations
of motion describing the distinguished particle are integrated numerically
using the Verlet algo rithm [71]. Each time step is made reversible by a
Tro tter expansion of the time evolution ope rator [27, 67]. In all the results
reported below we use the double-well potential (2.96), but the integration
scheme was also checked using the harmonic potential, V(xq) ^Xq, for
which (2.2) can be integra ted analytically. Initial conditions are chosen once
from the microcanonical invariant distribution on the energy shell E = N/ /?.
The integration is performed up to time T = 2 106. T he param eters N , T
and the step size are chosen so that further increase (in Nand T) or decrease
(in step size) does not change the average rates considerably. Transition ra tes
are obtained by counting the number of times the trajectory switches between
S + and S - . Table 1 deta ils resu lts obtained for
-
7/22/2019 Stochastic Dynamics
79/161
7 5 = 1 (5 ==0 .1
1 4.1 1 0 4 3.7 1 0 ~ 4
1 0 3.5 1 0 ~ 4 1 .1 1 0 ~ 4
Table 2.1: Com parison between tran sition ra tes with two different candida te
metastable sets.
5 = 0.1 is arb itrary . Any choice of 0.05 <
-
7/22/2019 Stochastic Dynamics
80/161
between successive transi tio ns, we ex pect th a t it will have an expo nen tia l
distribution,
P[tS> s ] = e ~ k s , (2.103)
for some rate constant k,which is also the average transitio n rate. Figure 2.2
depicts the distribution of the waiting times between transitions, P [ts > s],
on a semi-log plot for the case 7 = 10. For 5 = 1 , the graph is not linear
near the origin. However, for 5 = 0.1, the linear fit is very good, indicating
th a t transitio ns events have
top related