Inertial effects in the fractional translational diffusion of a Brownian particle
in a double-well potential
Yuri P. Kalmykov
Lab. Mathématiques et Physique des Systèmes, Université de Perpignan, 52, Avenue de Paul Alduy, 66860 Perpignan Cedex, France
William T. Coffey
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
Sergey V. Titov
Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino,
Moscow Region, 141190, Russian Federation
Received 31 October 2006; published 1 March 2007
The anomalous translational diffusion including inertial effects of nonlinear Brownian oscillators in a double
well potential Vx = ax
2
/2+ bx
4
/ 4 is considered. An exact solution of the fractional Klein-Kramers Fokker-
Planck equation is obtained allowing one to calculate via matrix continued fractions the positional autocor-
relation function and dynamic susceptibility describing the position response to a small external field. The
result is a generalization of the solution for the normal Brownian motion in a double well potential to fractional
dynamics giving rise to anomalous diffusion.
DOI: 10.1103/PhysRevE.75.031101 PACS numbers: 05.40.-a, 05.45.Df
I. INTRODUCTION
The Brownian motion in a field of force is of fundamental
importance in problems involving relaxation and resonance
phenomena in stochastic systems 1,2. An example is the
translational diffusion of noninteracting Brownian particles
due to Einstein 3 with a host of applications in physics
chemistry, biology, etc. Einstein’s theory relies on the diffu-
sion limit of a discrete time random walk. Here the random
walker or particle makes a jump of a fixed mean square
length in a fixed time and the inertia is ignored so that the
velocity distribution instantaneously attains its equilibrium
value. Thus the only random variable is the jump direction
leading automatically via the central limit theorem in the
limit of a large sequence of jumps to the Wiener process
describing the normal Brownian motion. The Einstein theory
of normal diffusion has been generalized to fractional diffu-
sion see Refs. 4 –6 for a review in order to describe
anomalous relaxation and diffusion processes in disordered
complex systems such as amorphous polymers, glass form-
ing liquids, etc.. These exhibit temporal nonlocal behavior
arising from energetic disorder causing obstacles or traps si-
multaneously slowing down the motion of the walker and
introducing memory effects. Thus in one dimension the dy-
namics of the particle are described by a fractional diffusion
equation for the distribution function f x , t in configuration
space incorporating both a waiting time probability density
function governing the random time intervals between single
microscopic jumps of the particles and a jump length prob-
ability distribution. The fractional diffusion equation stems
from the integral equation for a continuous time random
walk CTRW introduced by Montroll and Weiss 7,8. In the
most general case of the CTRW, the random walker may
jump an arbitrary length in arbitrary time. However, the jump
length and jump time random variables are not statistically
independent 7–9. In other words a given jump length is
penalized by a time cost, and vice versa.
A simple case of the CTRW arises by assuming that the
jump length and jump time random variables are decoupled.
Such walks possessing a discrete hierarchy of time scales,
without the same probability of occurrence, are known as
fractal time random walks 5. They lead in the limit of a
large sequence of jump times and the non inertial limit to the
following fractional Fokker-Planck equation in configuration
space for a review see Refs. 5,7
f x, t
t
=
0
D
t
1-
K
x
x
f x, t +
f x, t
kT
x
Vx, t
. 1
Here x specifies the position of the walker at time t,
- x , kT is the thermal energy, K
=
/ kT is a gener-
alized diffusion coefficient,
is a generalized viscous drag
coefficient arising from the heat bath and Vx , t denotes the
external potential. The operator
0
D
t
1-
t 0
D
t
-
in Eq. 1 is
given by the convolution the Riemann-Liouville fractional
integral definition6
0
D
t
-
f x, t =
1
0
t
f x, t'dt'
t - t'
1-
, 2
where z is the gamma function. The physical meaning of
the parameter is the order of the fractional derivative in the
fractional differential equation describing the continuum
limit of a random walk with a chaotic set of waiting times
fractal time random walk. Values of in the range
0 1 correspond to subdiffusion phenomena =1 cor-
responds to normal diffusion.
Since inertial effects are ignored the fractional Fokker-
Planck equation in configuration space Eq. 1 only describes
the long time low frequency behavior of the ensemble of
particles. In order to give a physically meaningful descrip-
tion of the short time high frequency behavior, inertial ef-
fects must be taken into account just as in normal diffusion
PHYSICAL REVIEW E 75, 031101 2007
1539-3755/2007/753/0311018 ©2007 The American Physical Society 031101-1