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 thismodel is governedbya fractional diffusion equation. If the step lengthof the randomwalk followsapower law,weget 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.