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

Carrillo, José A.; Dolbeault, Jean; Markowich, Peter; Sparber, Christof

Carrillo, José A.; Dolbeault, Jean; Markowich, Peter; Sparber, Christof (2004), On the Long-Time Behavior of the Quantum Fokker-Planck Equation, Monatshefte für Mathematik, 141, 3, p. 237-257. http://dx.doi.org/10.1007/s00605-003-0043-4

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Type

Article accepté pour publication ou publié

Date

2004

Journal name

Monatshefte für Mathematik

Volume

141

Number

Publisher

Springer

Pages

237-257

CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Markowich, Peter

Sparber, Christof

Abstract (EN)

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.

Subjects / Keywords

Open quantum system; Lindblad condition; Wigner transform; Fokker-Planck operator; long-time asymptotic; logarithmic Sobolev inequality