Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2010-05-26
Physics
Condensed Matter
Statistical Mechanics
4 pages, 2 figures
Scientific paper
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.
Barkai Eli
Kessler David A.
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