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

C. Escudero, E. Abad, S. B. Yuste

Published 2016-04-12Version 1

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 (or propagator) and investigate both analytically and numerically the behavior of this function and that of the associated moments, as well as first-passage properties in expanding hyperspherical geometries. We show that the behavior is determined to a great extent by the so-called diffusion conformal time $\tau$, 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.