Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient
Kwok Sau Fa and E. K. Lenzi
Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900, Maringá-PR, Brazil
Received 20 September 2004; published 20 January 2005
We investigate the solutions and the first passage time for anomalous diffusion processes governed by the
usual diffusion equation. We consider a space- and time-dependent diffusion coefficient and the presence of
absorbing boundaries. We obtain analytical results for the probability distribution and the first passage time
distribution for finite and semi-infinite intervals. In addition, we compare our results for the first passage time
distribution with the one obtained by the usual diffusion equation with constant diffusion coefficient.
DOI: 10.1103/PhysRevE.71.012101 PACS numbers: 05.40.-a, 05.60.-k, 66.10.Cb
I. INTRODUCTION
Anomalous diffusion is a ubiquitous phenomenon in na-
ture and it appears in several contexts related to physics,
chemistry, and biology. The processes associated with
anomalous diffusion are investigated, in general, by using the
Langevin equation or differential equations for the probabil-
ity density x , t. Nowadays, there are several approaches at
our disposal to describe these processes. For instance, the
well-known cases are the Langevin equation and the corre-
sponding Fokker-Planck equation, and the master equation.
The other ones we could mention are the generalized Lange-
vin equations 1, the generalized Fokker-Planck equation
with memory effect 2, generalized thermostatistics 3,
generalized master equations 4, continuous time random
walk models 5, and fractional equations 6. In connection
to these approaches, the investigation of a stochastic process,
such as anomalous diffusion, is also associated with the
mean first passage time MFPT. The MFPT is defined as the
time T when a process, starting from a given point, reaches a
predetermined level for the first time. Examples of the MFPT
are the escape time from a random potential, intervals be-
tween neural spikes 7, and fatigue failure 8. In this con-
text, the knowledge of the first passage time FPT distribu-
tion Ft is also essential. However, in only a few cases one
has explicit analytical expressions for the FPT distribution,
as was pointed out in Ref. 9. In this direction, our focus on
this work is to analyze the MFPT, the FPT distribution, and
the solutions related to the following diffusion equation:
t
x, t =
x
Dt, x
x
x, t
, 1
where the diffusion coefficient is given by Dt , x
= Dt|x|
-
. Note that Eq. 1 has as particular cases several
situations present in the literature and it brings further as-
pects to explore, for example, physical systems whose dy-
namic aspects are governed by fractal-like structure and non-
Markovian processes. The above equation has been applied
to investigate turbulence 10,11, fast electrons in a hot
plasma in the presence of a dc electric field 12, and diffu-
sion on fractals 13.
The plan of this work is to investigate Eq. 1. In Sec. II,
we present the solutions of Eq. 1 with natural boundary
conditions and diffusion coefficient given by Dt , x
= Dt|x|
-
. We analyze the mean squared displacement of
these processes with different forms for Dt. In Sec. III, we
investigate Eq. 1 subjected to the boundary condition
0, t = L , t = 0 and the initial condition x ,0 = ¯ x. We
also analyze this result by extending it to a semi-infinite
interval, i.e., L → . Furthermore, we analyze the FPT distri-
bution and the MFPT. Next in Sec. IV, we present our con-
clusion by giving a discussion about our results.
II. DIFFUSION EQUATION WITH NATURAL
BOUNDARY CONDITIONS
The diffusion equation 1 with variable diffusion coeffi-
cient in space and/or in time has been considered by several
authors. Richardson considered, by empirical argument, the
spatial diffusion coefficient given by Dt , x|x|
-
with
=-4/3 in order to study turbulent diffusivity 10, whereas
Batchelor suggested Dt , x t
2
for the same problem 14.
In a later step, Okubo 15 and Hentschel and Procaccia 16
OHP suggested mixed algebraic forms given by Dt , x
t
a
|x|
-
, with the initial condition 0, x = x. The solution
of Eq. 1 for Dt , r = KDtr
-
, in n dimensions with spheri-
cal symmetry, is given by
r, t
¯
=
2+
n„n/2+ …
1
K2+
2
t
¯
n/2+
e
-r
2+
/k2+
2
t
¯
,
2
where t
¯
= Dtdt. The corresponding mean squared displace-
ment is r
2
t
¯
2/2+
. Note that the non-Gaussian solution of
Eq. 2 is due to the spatial diffusion coefficient. In particu-
lar, for the Batchelor model =0, the probability distribu-
tion has the Gaussian form. For Dt = t
we recover the OHP
solution which yields r , t t
-31+a/2+
e
-C
2
r
2+
/t
1+a
. In par-
ticular, for 2 -3 =4 one has r
2
t
3
which leads to the
same behavior of the Richardson and Batchelor models. One
can see that the above three models are linear in the logarith-
mic scale. To deviate from the linear behavior one can con-
sider, for example, Dt d1+ at
b
c
/ 1+ gt
q
h
/ dt. For this
last case, r
2
is shown in Fig. 1. For t small r
2
is domi-
nated by the initial distance, and for large time the rate of
r
2
is less than that of the intermediate time. These behav-
iors seem to be verified in turbulent processes 11.
PHYSICAL REVIEW E 71, 012101 2005
1539-3755/2005/711/0121014/$23.00 ©2005 The American Physical Society 012101-1