F. Castella, L. Erdos, F. Frommlet, P. A. Markowich
Fokker-Planck equations as scaling limits of reversible quantum systems.
(161K, Latex)
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 (Physica A, 121, 587-616 (1983)) have 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 unnatural under their 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