Solution of the Fokker-Planck equation with a logarithmic potential

TL;DR

This paper analyzes particle diffusion in a logarithmic potential using the Fokker-Planck equation, revealing that long-term behavior is governed by an infinite covariant density leading to anomalous diffusion and slow decay of average position.

Contribution

It introduces the concept of an infinite covariant density to describe long-time limits, extending understanding beyond the Boltzmann equilibrium in logarithmic potentials.

Findings

Long-time limit characterized by an infinite covariant density.

Mean square displacement exhibits anomalous diffusion.

Average position decays anomalously slowly in asymmetric initial conditions.

Abstract

We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large $|x|$ using the Fokker-Planck equation. An eigenfunction expansion shows that the Boltzmann equilibrium density does not fully describe the long time limit of this problem. Instead this limit is characterized by an infinite covariant density. This non-normalizable density yields the mean square displacement of the particles, which for a certain range of parameters exhibits anomalous diffusion. In a symmetric potential with an asymmetric initial condition, the average position decays anomalously slowly. This problem also has applications outside the thermal context, as in the diffusion of the momenta of atoms in optical molasses.