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Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

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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 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.

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Authors and Affiliations

  1. CNRS et IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Francois Castella

  2. School of Mathematics, Georgia Tech, Atlanta, Georgia, 30332

    László Erdős

  3. Institut für Mathematik, Universität Wien, A-1090, Vienna, Austria

    Florian Frommlet

  4. Institut für Mathematik, Universität Wien, A-1090, Vienna, Austria

    Peter A. Markowich

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  1. Francois Castella
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  2. László Erdős
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  3. Florian Frommlet
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  4. Peter A. Markowich
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Castella, F., Erdős, L., Frommlet, F. et al. Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems. Journal of Statistical Physics 100, 543–601 (2000). https://doi.org/10.1023/A:1018667323830

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