Solution of the Fokker-Planck Equation with a Logarithmic Potential

A. Dechant, E. Lutz, E. Barkai, D. A. Kessler

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Abstract

We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large {pipe}x{pipe} 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.

Original languageEnglish
Pages (from-to)1524-1545
Number of pages22
JournalJournal of Statistical Physics
Volume145
Issue number6
StatePublished - Dec 2011

Bibliographical note

Funding Information:
Acknowledgements This work was supported by the Israel Science Foundation, the Emmy Noether Program of the DFG (contract No LU1382/1-1) and the cluster of excellence Nanosystems Initiative Munich.

Funding

Acknowledgements This work was supported by the Israel Science Foundation, the Emmy Noether Program of the DFG (contract No LU1382/1-1) and the cluster of excellence Nanosystems Initiative Munich.

FundersFunder number
Deutsche ForschungsgemeinschaftLU1382/1-1
Israel Science Foundation

    Keywords

    • Anomalous diffusion
    • Ergodicity breaking
    • Fokker-Planck equation
    • Logarithmic potential

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