Fractional diffusion, waiting-time distributions, and Cattaneo-type equations

Abstract

We discuss a generalized diffusion equation resulting from an additive two-state process, in combination with an asymptotically fractal (asymptotic power-law) waiting-time distribution. The obtained equation is an extension to previously discussed fractional diffusion equations. Our description leads to a mean squared displacement which describes enhanced, subballistic transport for long times. The short time behavior, however, is of a ballistic nature. This separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting-time distribution. This is also mirrored by the observation that, for small times, our generalized diffusion equation reduces to the standard Cattaneo equation. The asymptotic probability density is of compressed Gaussian type, and thus differs from the L\'evy tail generally found for these kinds of processes.