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

2010/05/26 by David A. Kessler, Kessler, David A., Eli Barkai +1
Biochemistry, Genetics and Molecular Biology · Materials Science · #Diffusion and Search Dynamics #FOS: Physical sciences #Material Dynamics and Properties #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech)

paper · pdf · doi:10.48550/arxiv.1005.4737

openalex publication_date 2010/05/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

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