Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 145, Issue 6, Pages 1524-1545Publisher
SPRINGER
DOI: 10.1007/s10955-011-0363-z
Keywords
Anomalous diffusion; Fokker-Planck equation; Logarithmic potential; Ergodicity breaking
Categories
Funding
- Israel Science Foundation
- DFG [LU1382/1-1]
- cluster of excellence Nanosystems Initiative Munich
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We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large |x| using the Fokker-Planck equation. An eigenfunction expansion shows that the Boltzmann equilibrium density does not fully describe the long-time limit of this problem. Instead this limit is characterized by an infinite covariant density. This non-normalizable density yields the mean square displacement of the particles, which for a certain range of parameters exhibits anomalous diffusion. In a symmetric potential with an asymmetric initial condition, the average position decays anomalously slowly. This problem also has applications outside the thermal context, as in the diffusion of the momenta of atoms in optical molasses.
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