title: Diffusion in an expanding medium: Fokker-Planck equation, Green's function, and first-passage properties.
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 τ(t), which we define via the relation τ[over ̇]=1/a^{2}, where a(t) is the expansion scale factor. If the medium expansion is driven by a power law [a(t)∝t^{γ} with γ>0], then we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent γ 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 γ=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.
authors.affiliation: Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain.
authors.forename: S B
authors.initials: SB
authors.lastname: Yuste
authors.affiliation: Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEX), Centro Universitario de Mérida Universidad de Extremadura, E-06800 Mérida, Spain.
authors.forename: E
authors.initials: E
authors.lastname: Abad
authors.affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, E-28049 Madrid, Spain.
authors.forename: C
authors.initials: C
authors.lastname: Escudero