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We solve the Fokker-Planck equation for Brownian motion in a logarithmicpotential. When the diffusion constant is below a critical value the solutionapproaches a non-normalizable scaling state, reminiscent of an infiniteinvariant density. With this non-normalizable density we obtain the phasediagram of anomalous diffusion for this important process. We briefly discussthe consequence for a range of physical systems including atoms in opticallattices and charges in vicinity of long polyelectrolytes. Our work explains inwhat sense the infinite invariant density and not Boltzmann's equilibriumdescribes the long time limit of these systems.

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