Relaxation time for nonlinear response of a Brownian particle subject to a step external force:
Analytical solutions for one-dimensional models
Yu. P. Kalmykov
Centre d’Etudes Fondamentales, Universite ´ de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France
and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino,
Moscow Region, 141120, Russian Federation
J. L. De
´
jardin
*
Centre d’Etudes Fondamentales, Universite ´ de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France
W. T. Coffey
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
Received 1 November 1996
The nonlinear response to a step external force of a system with relaxational dynamics governed by a
one-dimensional Fokker-Planck equation is considered. An exact analytical expression for the step response
nonlinear relaxation time is derived in terms of an integral which can be evaluated numerically. Applications
to nonlinear problems concerning the dynamic Kerr effect, dielectric relaxation of liquid dielectrics, and
magnetic relaxation of systems of single domain ferromagnetic particles are given. The results are compared
with solutions previously obtained. S1063-651X9708503-6
PACS numbers: 05.40.+j, 76.20.+q, 77.22.Gm, 78.20.Jq
I. INTRODUCTION
A system initially in an equilibrium stationary state and
suddenly disturbed by an external stimulus e.g., by applying
a step external field will evolve into another equilibrium
stationary state. Presently a satisfactory theory is available
for linear response only where the energy of the system aris-
ing from the external stimulus is much lower than the ther-
mal energy 1,2. Here we need only linear in the external
stimulus deviations of the expectation value of the dynami-
cal variable of interest in the stationary state in order to
evaluate the generalized susceptibility and/or response func-
tions in terms of the appropriate equilibrium stationary cor-
relation function. Linear response theory is widely used for
an interpretation of nonequilibrium phenomena such as di-
electric and magnetic relaxation, conductivity problems, etc.
Here we wish to study relaxation following a steplike
stimulus in systems described by one-dimensional Fokker-
Planck equations for the distribution function W( x , t ) of a
variable x 2,
t
W=L
FP
W. 1.1
We shall therefore first summarize the principal results of
linear response theory 2, Chap. 7 for systems where the
dynamics obey a diffusion equation like Eq. 1.1. Thus let
us consider the Fokker-Planck operator L
FP
of a system sub-
ject to a small perturbing force F ( t ). On account of this, L
FP
may be represented as
L
FP
=
x
D
2
x e
-V x +B x F t
x
e
V x -B x F t
=L
FP
0
x +L
ext
x F t ,
with
L
FP
0
x =
x
D
2
x e
-V x
x
e
V x
, L
FP
0
x W
0
x =0,
1.2
L
ext
x =
x
D
2
x B ' x , B ' x =
d
dx
B ,
where L
FP
0
( x ) is the Fokker-Planck operator in the absence
of the perturbation, W
0
is the equilibrium stationary distri-
bution function, V is called a generalized effective potential
2, D
(2)
( x ) is the diffusion coefficient, and B ( x ) denotes a
dynamical quantity. The step-off and step-on relaxation
functions when, on the one hand, a small constant force F
1
is suddenly switched off and, on the other hand, switched on
at time t =0, respectively, statistical equilibrium having been
achieved prior to the imposition of the stimulus in both in-
stances for a dynamic variable A ( x ) are then
A
off
t - A
0
=F
1
C
AB
t ,
A
on
t - A
0
=F
1
C
AB
0 -C
AB
t t 0 ,
1.3
where the quantity *Author to whom correspondence should be addressed.
PHYSICAL REVIEW E MARCH 1997 VOLUME 55, NUMBER 3
55 1063-651X/97/553/25097/$10.00 2509 © 1997 The American Physical Society