# Fokker-Planck equations as scaling limits of reversible quantum systems

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.

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Author

Castella, François;
Erdös, László

^{ISTA}^{}; Frommlet, Florian; Markowich, PeterAbstract

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

Acknowledgement

The authors are indebted to H. Spohn for discussions. F.C. and L.E. were partially supported by the Erwin Schrödinger Institute in Vienna (Austria) during their visit, and they thank this institution for its hospitality. This work was supported by the TMR-Network ``Asymptotic Methods in Kinetic Theory'' number ERB FMBX CT97 0157 (F.C., F.F., and P.A.M.) and by NSF Grant DMS-9970323 (L.E.).

Volume

100

Issue

3-4

Page

543 - 601

ISSN

IST-REx-ID

### Cite this

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:1018667323830Castella, F., Erdös, L., Frommlet, F., & Markowich, P. (2000). Fokker-Planck equations as scaling limits of reversible quantum systems.

*Journal of Statistical Physics*. Springer. https://doi.org/10.1023/A:1018667323830Castella, François, László Erdös, Florian Frommlet, and Peter Markowich. “Fokker-Planck Equations as Scaling Limits of Reversible Quantum Systems.”

*Journal of Statistical Physics*. Springer, 2000. 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. Springer, pp. 543–601, 2000.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.