99-374 F. Castella, L. Erdos, F. Frommlet, P. A. Markowich

Fokker-Planck equations as scaling limits of reversible quantum systems. (161K, Latex) Oct 6, 99

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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

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