DIFFUSION IN HETEROGENEOUS MEDIA
Rytis Kazakevičius, Julius Ruseckas
Institute of Theoretical Physics and Astronomy, Vilnius University, Lithuania
[email protected]
There are many systems and processes where the time dependence of the central second moment is not linear as in
the classical Brownian motion. Such family of processes is called anomalous diffusion [1]. Recently [2] it was suggested
that the anomalous diffusion can be a result of heterogeneous diffusion process, where the diffusion coefficient depends
on the position. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on
the behavior of the diffusion coefficient. Spatially dependent diffusion can occur in heterogeneous systems. For example,
heterogeneous medium with steep gradients of the diffusivity can be created in thermophoresis experiments using a local
variation of the temperature [3]. Here we consider heterogeneous diffusion processes with the power-law dependence
of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous
diffusion. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the
position [4].
ν 2η−1
dt + σ xη dWt .
(1)
x
dx = σ 2 η −
2
Here η is the power-law exponent of multiplicative noise, σ is the amplitude of noise and Wt is a standard Wiener process
(Brownian motion). This stochastic differential equation is interpreted in Itô sense. Here ν is a new parameter describing
the additional drift term. The meaning of the parameter ν is as follows: when the reflective boundaries at small positive
x = xmin and large x = xmax are present, the steady-state PDF is a power-law function of position with the power-law
exponent ν, P0 (x) ∼ x−ν .
2
1
1
(b) 10
(a) 10
(c) 10
100
100
⟨x⟩
⟨x⟩
⟨x⟩
101
100
10-1
10-1
10-1
100
101
10-2
10-1
102
10-1
10-2
100
101
10-3
10-1
102
100
t
t
1
(d) 10
101
102
101
102
t
0
(f) 10
(e)
10-1
⟨(x - ⟨x⟩)2⟩
⟨(x - ⟨x⟩)2⟩
⟨(x - ⟨x⟩)2⟩
102
100
100
10-4
10-2
10-2
10-1
100
101
t
102
10-1
10-2
100
101
t
102
10-1
100
t
Fig. 1. Dependence of the mean (a,b,c) and variance (d,e,f) on time for various values of the
parameters η and ν when the position of the diffusing particle changes according to Eq. (1).
Dashed black lines show numerical result, dotted lines show the power-law dependence on
time ∼ t 1/[2(1−η)] for (a,b,c) and ∼ t 1/(1−η) for (d,e,f). The parameters are σ = 1 and η = − 21 ,
ν = −1 for (a,d); η = 12 , ν = 0 for (b,c); η = 32 , ν = 5 for (c,f). The initial position is x0 = 1.
We found that the power-law exponent in the dependence of the mean square displacement on time does not depend
on the external force; this force changes only the anomalous diffusion coefficient (see Fig. 1 (d) and (e)). Anomalous
diffusion occur only for specific parameters values if ν < 3 and η < 1 (or ν < 1 and η < 1 ). As we can see in Fig. 1 (c)
and (f), in other cases anomalous diffusion do not occur due localization of particles. Also, we obtain analytic expressions
for the transition probability in two cases: when the power-law exponent in the external force is equal to 2η − 1, where
2η is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force
has a linear dependence on the position.
[1] J. P. Bouchaud and A. Georges,Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , Phys. Rep.
195, 127-293 (1990).
[2] A. G. Cherstvy and R. Metzler, Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes, Phys. Rev. E 90, 012134 (2014).
[3] C. B. Mast, S. Schink, U. Gerland and D. Braun, Escalation of polymerization in a thermal gradient, Proc. Natl. Acad. Sci. USA 110, 8030-8035
(2013).
[4] J. Ruseckas and B. Kaulakys, Scaling properties of signals as origin of 1/f noise, J. Stat. Mech. P06005 (2014).