# Entropy decay for the Kac evolution

Bonetto, Federico; Geisinger, Alissa; Loss, Michael; Ried, Tobias

Abstract:
We consider solutions to the Kac master equation for initial conditions where N particles are in a thermal equilibrium and $M \leq \ N$ particles are out of equilibrium. We show that such solutions have exponential decay in entropy relative to the thermal state. More precisely, the decay is exponential in time with an explicit rate that is essentially independent on the particle number. This is in marked contrast to previous results which show that the entropy production for arbitrary initial conditions is inversely proportional to the particle number. The proof relies on Nelson's hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2017 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-727450 KITopen ID: 1000072745 Verlag KIT, Karlsruhe Umfang 26 S. Serie CRC 1173 ; 2017/20
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