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

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

doi: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

^*Corresponding author for this work

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

19 Scopus citations

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 (US)
Pages (from-to)	237-257
Number of pages	21
Journal	Monatshefte fur Mathematik
Volume	141
Issue number	3
DOIs	https://doi.org/10.1007/s00605-003-0043-4
State	Published - Mar 2004
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

General Mathematics

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10.1007/s00605-003-0043-4

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title = "On the Long-Time Behavior of the Quantum Fokker-Planck Equation",

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",

author = "C. Sparber and Carrillo, {J. A.} and J. Dolbeault and Markowich, {P. A.}",

year = "2004",

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AU - Sparber, C.

AU - Carrillo, J. A.

AU - Dolbeault, J.

AU - Markowich, P. A.

PY - 2004/3

Y1 - 2004/3

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

UR - http://www.scopus.com/inward/record.url?scp=1842536803&partnerID=8YFLogxK

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DO - 10.1007/s00605-003-0043-4

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EP - 257

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 3

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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