Infinite invariant densities for anomalous diffusion in optical lattices and other logarithmic potentials

TL;DR

This paper analyzes the long-term behavior of particles undergoing anomalous diffusion in logarithmic potentials, revealing the existence of infinite invariant densities that govern their dynamics beyond traditional equilibrium states.

Contribution

It introduces the concept of infinite invariant densities for anomalous diffusion in logarithmic potentials, extending understanding of non-normalizable solutions in such systems.

Findings

Infinite invariant densities describe long-time behavior.

Phase diagram for anomalous diffusion is derived.

Application to optical lattices and polyelectrolytes discussed.

Abstract

We solve the Fokker-Planck equation for Brownian motion in a logarithmic potential. When the diffusion constant is below a critical value the solution approaches a non-normalizable scaling state, reminiscent of an infinite invariant density. With this non-normalizable density we obtain the phase diagram of anomalous diffusion for this important process. We briefly discuss the consequence for a range of physical systems including atoms in optical lattices and charges in vicinity of long polyelectrolytes. Our work explains in what sense the infinite invariant density and not Boltzmann's equilibrium describes the long time limit of these systems.