Fractional Diffusion Processes Research Articles

Simply and multiply scaled diffusion limits for continuous time random walks

First a survey is presented on how space-time fractional diffusion processes can be obtained by well-scaled limiting from continuous time random walks under the sole assumption of asymptotic power laws (with appropriate exponents for the tail behaviour of waiting times and jumps). The spatial operator in the limiting pseudo-differential equation is the inverse of a general Riesz-Feller potential operator. The analysis is carried out via the transforms of Fourier and Laplace. Then mixtures of waiting time distributions, likewise of jump distributions, are considered, and it is shown that correct multiple scaling in the limit yields diffusion equations with distributed order fractional derivatives (fractional operators being replaced by integrals over such ones, with the order of differentiation as variable of integration). It is outlined how in this way super-fast and super-slow diffusion can be modelled.

Tsallis and Rényi entropies in fractional diffusion and entropy production

The entropy production rate for fractional diffusion processes using Shannon entropy was calculated previously, which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. Rényi and Tsallis entropies, which have an additional parameter q generalizing the Shannon case (q=1), are shown here to have similar counter intuitive behaviours. However, the issue can be successfully treated in exactly the same manner as with Shannon entropy for q being not too large (i.e., generalizations near the Shannon case), whereas for larger q the Rényi and Tsallis entropies behave in a different way.