Journal of Statistical Physics

, Volume 145, Issue 6, pp 1524–1545 | Cite as

Solution of the Fokker-Planck Equation with a Logarithmic Potential

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


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.


Anomalous diffusion Fokker-Planck equation Logarithmic potential Ergodicity breaking 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • A. Dechant
    • 1
  • E. Lutz
    • 1
  • E. Barkai
    • 2
  • D. A. Kessler
    • 2
  1. 1.Department of PhysicsUniversity of AugsburgAugsburgGermany
  2. 2.Department of PhysicsBar-Ilan UniversityRamat-GanIsrael

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