Fokker-Planck equations as scaling limits of reversible quantum systems

F. Castella, L. Erdös, F. Frommlet, P. Markowich, Journal of Statistical Physics 100 (2000) 543–601.

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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.
Publishing Year
Date Published
2000-01-01
Journal Title
Journal of Statistical Physics
Volume
100
Issue
3-4
Page
543 - 601
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Castella F, Erdös L, Frommlet F, Markowich P. Fokker-Planck equations as scaling limits of reversible quantum systems. Journal of Statistical Physics. 2000;100(3-4):543-601. doi:10.1023/A:1018667323830
Castella, F., Erdös, L., Frommlet, F., & Markowich, P. (2000). Fokker-Planck equations as scaling limits of reversible quantum systems. Journal of Statistical Physics, 100(3–4), 543–601. https://doi.org/10.1023/A:1018667323830
Castella, François, László Erdös, Florian Frommlet, and Peter Markowich. “Fokker-Planck Equations as Scaling Limits of Reversible Quantum Systems.” Journal of Statistical Physics 100, no. 3–4 (2000): 543–601. https://doi.org/10.1023/A:1018667323830.
F. Castella, L. Erdös, F. Frommlet, and P. Markowich, “Fokker-Planck equations as scaling limits of reversible quantum systems,” Journal of Statistical Physics, vol. 100, no. 3–4, pp. 543–601, 2000.
Castella F, Erdös L, Frommlet F, Markowich P. 2000. Fokker-Planck equations as scaling limits of reversible quantum systems. Journal of Statistical Physics. 100(3–4), 543–601.
Castella, François, et al. “Fokker-Planck Equations as Scaling Limits of Reversible Quantum Systems.” Journal of Statistical Physics, vol. 100, no. 3–4, Springer, 2000, pp. 543–601, doi:10.1023/A:1018667323830.

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