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

Ralf Metzler, Theo F. Nonnenmacher
1998 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We discuss a generalized diffusion equation resulting from an additive two-state process, in combination with an asymptotically fractal ͑asymptotic power-law͒ 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
more » ... rom 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 Lévy tail generally found for these kinds of processes. ͓S1063-651X͑98͒01606-7͔ PACS number͑s͒: 05.40.ϩj, 05.60.ϩw
doi:10.1103/physreve.57.6409 fatcat:6t6b2jsnnjbx5nmhj5ms6jeeou