Continuous time random walks and space-time fractional differential equations

Abstract

The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of this model is governed by a fractional diffusion equation. If the step length of the random walk follows a power law, we get a spacefractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives.

On relation between generalized diffusion equations and subordination schemes

“…(3) 1/ ML (u) for the case Eq. ( 4), (5) where M... (u) = ∞ 0 M ... (t)e −ut dt. We note that for special cases of integral kernels corresponding to fractional or distributed-order derivatives Eqs.…”

“…The situation changed when the generalized Fokker-Planck equations with power-law memory kernels gained popularity in different fields. This kind of GFPEs, called fractional Fokker-Planck equations (FFPEs) describe the continuous (long time-space) limit of continuous time random walks (CTRW) with power-law waiting times [3][4][5], which gives a physical foundation and explains broad applicability of FFPE. Such equations proved useful for description of anomalous transport processes in different media.…”

The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

“…Quasi-diffusion imaging [1] is a novel diffusion magnetic resonance imaging (dMRI) technique based on a special case of the continuous time random walk (CTRW) diffusion model [2][3][4][5]. The diffusion dynamics are represented by an effective normal diffusion where the mean squared-displacement of the diffusing particles is proportional to time,…”