Author(s):

Castella, François; Erdős, László; Frommlet, Florian; Markowich, Peter A

Article Title: 
FokkerPlanck equations as scaling limits of reversible quantum systems

Affiliation 

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 largetemperature 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 largetime 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.

Keywords: 
Coupled harmonic oscillators; Fokker Planck equation; Scaling limit; Wigner distribution

Journal Title:

Journal of Statistical Physics

Volume: 
100

Issue 
34

ISSN:

15729613

Publisher:

Springer

Date Published:

20000101

Start Page: 
543

End Page:

601

DOI: 
10.1023/A:1018667323830

Open access: 
no 