Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium

TL;DR

This paper investigates how particles diffuse in a logarithmic potential, revealing two distinct scaling regimes, initial condition dependence, and a phase transition in the approach to equilibrium, with implications for various physical systems.

Contribution

It provides a detailed scaling analysis of diffusion in a logarithmic potential, uncovering dual regimes and a phase transition related to initial conditions, which was not previously understood.

Findings

Two distinct scaling forms for diffusive and subdiffusive regimes.

Scaling exponents depend on initial conditions.

Identification of a phase transition in the scaling behavior.

Abstract

The equation which describes a particle diffusing in a logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. A detailed study of the approach of such systems to equilibrium via a scaling analysis is carried out, revealing three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ \sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii) the scaling exponents and scaling functions corresponding to both regimes are selected by the initial condition; and (iii) this dependence on the initial condition manifests a "phase transition" from a regime in which the scaling solution depends on the initial condition to a regime in which it is independent of it. The selection mechanism which is found has many similarities to the marginal stability mechanism which has been widely studied in the context of fronts propagating into unstable states. The general scaling forms are presented and their practical and theoretical applications are discussed.