## Abstract

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 language | English (US) |
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Pages (from-to) | 543-601 |

Number of pages | 59 |

Journal | Journal of Statistical Physics |

Volume | 100 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 2000 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics