Retarding Sub- and Accelerating Super-diffusion Governed by Distributed Order Fractional Diffusion Equations

We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion). PACS numbers: 02.50.Ey Stochastic processes; 05.40.-a Fluctuation phenomena, random processes, noise and Brownian motion. Recently, kinetic equations with fractional space and time derivatives have attracted attention as a possible tool for the description of anomalous diffusion and relaxation phenomena, see, e.g., [MK00], [MLP01], [SZ97], [Mai96] and references on earlier studies therein. It was also recognized [HA95], [Com96], [MRGS00], [BMK00] that the fractional kinetic equations may be viewed as ”hydrodynamic” (that is, long-time and long-space) limits of the CTRW (Continuous Time Random Walk) theory [MW65] which was succesfully applied to the description of anomalous diffusion phenomena in many areas, e.g., turbulence [KBS87], disordered medium [BG90], intermittent chaotic systems [ZK93], etc. However, the kinetic equations have two advantages over a random walk approach: firstly, they allow one to explore various boundary conditions (e.g., reflecting and/or absorbing) and, secondly, to study diffusion and/or relaxation phenomena in external fields. Both possibilities are difficult to realize in the framework of CTRW. There are three types of fractional kinetic equations: the first one, describing Markovian processes, contains equations with fractional space or velocity derivative, the second one, describing non-Markovian processes, contains equations with fractional time derivative, and the third class, naturally, contains both fractional space and time derivatives, as well. However, all three types are suitable to describe time evolution of the probability density function (PDF) of a very narrow class of diffusion processes, which are characterized by a unique diffusion exponent showing time-dependence of the characteristic displacement (e.g., of the root mean square) [MK00]. These processes are also called fractal, or self-affine processes, and they are characterized by the exponent H , called the Hurst exponent, which depends on the order of fractional derivative in the kinetic equation. We recall that the stochastic process x(t) is self-affine, or fractal, if its stationary increments possess the following property [ST94]: x(t+ κτ)− x(t) d =κ(x(t+ τ)− x(t)) (1) where κ and H are positive constants. The sign d =implies, that the left and the right hand sides of Eq.(1) have the same PDFs. As a possible generalization of fractional kinetic equations, we propose fractional diffusion equations in which the fractional order derivatives are integrated with respect to the order of differentiation (distributed order fractional diffusion equations). They can serve as a paradigm for the kinetic description of the random processes possessing non-unique diffusion exponent and hence, non-unique Hurst exponent. The processes with time-dependent Hurst exponent are believed to provide useful models for a host of continuous and non-stationary natural signals; they are also constructed explicitly [PV95], [AV99], [AV00]. Ordinary differential equations with distributed order derivatives were proposed in the works by Caputo [Cap69], [Cap95] for generalizing stress-strain relation of unelastic media. In Refs. [BT00], [BT00a] the method of the solution was proposed which is based on generalized Taylor series representation. A basic framework for the numerical solution of distributed order differential equations was introduced in [DF01]. Very recently, Caputo [Cap01] proposed the generalization of the Fick’s law using distributed order time derivative. We write the distributed order time fractional diffusion equation for the PDF f(x, t) as 1 ∫ 0 dβτp(β) ∂f ∂tβ = D ∂f ∂x2 , f(x, 0) = δ(x), (2) where τ and D are positive constants, [τ ] = sec, [D] = cm2/sec, p(β) is a dimensionless nonnegative function of the order of the derivative, and the time fractional derivative of order β is understood in the Caputo sense [GM97]: ∂f ∂tβ = 1 Γ(1− β) t ∫ 0 dτ(t− τ) ∂f ∂t (3) If we set p(β) = δ(β − β0), 0 < β0 ≤ 1, then we arrive at time fractional diffusion equation, whose solution is the PDF of the self-affine random process with the Hurst exponent equal to β0/2. The PDF is expressed through Wright function [Mai97]. The diffusion process is then characterized by the mean square displacement