Temporal Fokker-Planck Equations

TL;DR

This paper generalizes the temporal Fokker-Planck equation to include nonlinear and fractional time delay effects, modeling complex diffusive processes with temporal dispersion and power-law distributed delays.

Contribution

It introduces two new generalizations of the temporal Fokker-Planck equation, one nonlinear with concentration-dependent delays and one fractional with power-law delays.

Findings

Derived a nonlinear temporal Fokker-Planck equation with concentration dependence.

Formulated a fractional propagation-dispersion equation for power-law delays.

Established the temporal analog of fractional spatial diffusion equations.

Abstract

The temporal Fokker-Plank equation [{\it J. Stat. Phys.}, {\bf 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. %\cite{boon-grosfils-lutsko}. We present two generalizations of the temporal Fokker-Plank 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 master equation. 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-Plank 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).