Fokker-Planck equations as scaling limits of reversible quantum systems

Francois Castella*, László Erdős, Florian Frommlet, Peter A. Markowich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit: however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.

Original languageEnglish (US)
Pages (from-to)543-601
Number of pages59
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - Aug 2000
Externally publishedYes


  • Coupled harmonic oscillators
  • Fokker Planck equation
  • Scaling limit
  • Wigner distribution

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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