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

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

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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 languageEnglish
Pages (from-to)237-257
JournalMonatshefte fur Mathematik
Volume141
DOIs
Publication statusPublished - 1 Jan 2004

Keywords

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

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    Sparber, C., Carrillo, J. A., Dolbeault, J., & Markowich, P. A. (2004). On the Long-Time Behavior of the Quantum Fokker-Planck Equation. Monatshefte fur Mathematik, 141, 237-257. https://doi.org/10.1007/s00605-003-0043-4