Approach to equilibrium of diffusion in a logarithmic potential

TL;DR

This paper analyzes the late-time behavior of a diffusing particle in a logarithmic potential, revealing a complex scaling structure influenced by initial conditions and showing a transition in scaling exponents.

Contribution

It introduces a novel two-scale scaling solution for diffusion in a logarithmic potential, highlighting the impact of initial conditions on the scaling form and exponent.

Findings

Two distinct scaling forms for diffusive and subdiffusive regimes

Initial conditions determine the scaling function and exponent

Transition from variable to fixed scaling exponent based on initial tail

Abstract

The late-time distribution function P(x,t) of a particle diffusing in a one-dimensional logarithmic potential is calculated for arbitrary initial conditions. We find a scaling solution with three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ t^(1/2)) and a subdiffusive (x ~ t^{\gamma} with a given {\gamma} < 1/2) length scale, respectively, (ii) the overall scaling function is selected by the initial condition, and (iii) depending on the tail of the initial condition, the scaling exponent which characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value.