Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations

Abstract

The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function $$f_j(r,t)$$ of a particle moving on a substrate with time delays $$\tau _j$$ . Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability $$P_j$$ is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, $$P_j \propto f_j^{\nu - 1}$$ , the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, $$P_j \propto \tau _j^{-1-\alpha }$$ (with $$0< \alpha < 2$$ ), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.