On the Long-Time Behavior of the Quantum Fokker-Planck Equation

C. Sparber; J. A. Carrillo; J. Dolbeault; P. A. Markowich

doi:https://doi.org/10.1007/s00605-003-0043-4

On the Long-Time Behavior of the Quantum Fokker-Planck Equation

C. Sparber, J. A. Carrillo, J. Dolbeault, P. A. Markowich

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

16 Citations (Scopus)

Abstract

We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.

Original language	English
Pages (from-to)	237-257
Journal	Monatshefte fur Mathematik
Volume	141
DOIs	https://doi.org/10.1007/s00605-003-0043-4
Publication status	Published - 1 Jan 2004

Keywords

Fokker-Planck operator
Lindblad condition
Logarithmic Sobolev inequality
Long-time asymptotic
Open quantum system
Wigner transform

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abstract = "We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.",

keywords = "Fokker-Planck operator, Lindblad condition, Logarithmic Sobolev inequality, Long-time asymptotic, Open quantum system, Wigner transform",

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AU - Sparber, C.

AU - Carrillo, J. A.

AU - Dolbeault, J.

AU - Markowich, P. A.

PY - 2004/1/1

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N2 - We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.

AB - We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.

KW - Fokker-Planck operator

KW - Lindblad condition

KW - Logarithmic Sobolev inequality

KW - Long-time asymptotic

KW - Open quantum system

KW - Wigner transform

U2 - https://doi.org/10.1007/s00605-003-0043-4

DO - https://doi.org/10.1007/s00605-003-0043-4

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EP - 257

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