The precise time-dependent solution of the Fokker-Planck equation with anomalous diffusion

TL;DR

This paper derives an exact time-dependent solution to the Fokker-Planck equation for anomalous diffusion in complex media, showing it approaches a stationary power-law distribution consistent with nonextensive statistics.

Contribution

It provides the first precise analytical time-dependent solution for the Fokker-Planck equation with anomalous diffusion, validated by numerical tests.

Findings

Solution approaches a stationary power-law distribution at long times

Analytical solution matches numerical simulations accurately

Supports nonextensive statistical mechanics framework

Abstract

We study the time behavior of the Fokker-Planck equation in Zwanzig rule (the backward-Ito rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation-dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker-Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.