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On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation
Author(s) -
Desvillettes Laurent,
Villani Cédric
Publication year - 2001
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/1097-0312(200101)54:1<1::aid-cpa1>3.0.co;2-q
Subject(s) - fokker–planck equation , mathematics , degeneracy (biology) , degenerate energy levels , logarithm , planck , entropy (arrow of time) , statistical physics , operator (biology) , mathematical analysis , variable (mathematics) , kinetic energy , classical mechanics , physics , partial differential equation , quantum mechanics , bioinformatics , biochemistry , chemistry , repressor , gene , transcription factor , biology
We study the long‐time behavior of kinetic equations in which transport and spatial confinement (in an exterior potential or in a box) are associated with a (degenerate) collision operator acting only in the velocity variable. We expose a general method, based on logarithmic Sobolev inequalities and the entropy, to overcome the well‐known problem, due to the degeneracy in the position variable, of the existence of infinitely many local equilibria. This method requires that the solution be somewhat smooth. In this paper, we apply it to the linear Fokker‐Planck equation and prove decay to equilibrium faster than O ( t −1/ϵ ) for all ϵ > 0. © 2001 John Wiley & Sons, Inc.