Infinite covariant density for diffusion in logarithmic potentials and optical lattices

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 an infinite covariant density. With this non-normalizable solution we obtain the phase diagram of anomalous diffusion for this process. We briefly discuss the physical consequences for atoms in optical lattices and charges in the vicinity of long polyelectrolytes. Our work explains in what sense the infinite covariant density and not Boltzmann's equilibrium describes the long time limit of these systems.