CTRW Pathways to the Fractional Diffusion Equation

The foundations of the fractional diffusion equation are investigated based on coupled and decoupled continuous time random walks (CTRW). For this aim we find an exact solution of the decoupled CTRW, in terms of an infinite sum of stable probability densities. This exact solution is then used to understand the meaning and domain of validity of the fractional diffusion equation. An interesting behavior is discussed for coupled memories (i.e., L\'evy walks). The moments of the random walk exhibit strong anomalous diffusion, indicating (in a naive way) the breakdown of simple scaling behavior and hence of the fractional approximation. Still the Green function $P(x,t)$ is described well by the fractional diffusion equation, in the long time limit.