Fokker-Planck equations as scaling limits of reversible quantum systems

We consider a quantum particle moving in a harmonic exterior potential and lineally coupled to a heat bath of quantum oscillators. Caldeira and Legett 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. Tn 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.