Diffusion in an expanding medium: Fokker-Planck equation, Green's function and first-passage properties

TL;DR

This paper derives the Fokker-Planck equation for diffusion in expanding media, providing analytical Green's functions, studying first-passage properties, and revealing crossover effects influenced by the expansion rate.

Contribution

It introduces a mesoscopic derivation of the Fokker-Planck equation for expanding media and analyzes first-passage properties with new insights into crossover behaviors.

Findings

Green's function explicitly derived for expanding media

Crossover effects at critical expansion rate $oldsymbol{oldsymbol{ ext{ } extstyle rac{1}{2}}}$

Stationary distribution in contracting media

Abstract

We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time $\tau(t)$, which we define via the relation $\dot \tau=1/a^2$, where $a(t)$ is the expansion scale factor. If the medium expansion is driven by a power law [$a(t) \propto t^\gamma$ with $\gamma>0$], we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent $\gamma$ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value $\gamma=1/2$. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long time limit.