Fokker-Planck equations as scaling limits of reversible quantum systems Journal Article


Author(s): Castella, François; Erdős, László; Frommlet, Florian; Markowich, Peter A
Article Title: Fokker-Planck 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 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.
Keywords: Coupled harmonic oscillators; Fokker Planck equation; Scaling limit; Wigner distribution
Journal Title: Journal of Statistical Physics
Volume: 100
Issue 3-4
ISSN: 1572-9613
Publisher: Springer  
Date Published: 2000-01-01
Start Page: 543
End Page: 601
DOI: 10.1023/A:1018667323830
Open access: no