Communications in Mathematical Physics

, Volume 363, Issue 3, pp 847–875 | Cite as

Entropy Decay for the Kac Evolution

  • Federico Bonetto
  • Alissa Geisinger
  • Michael Loss
  • Tobias Ried


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.


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A.G. and T.R. would like to thank Georgia Tech for its hospitality. The work of M.L. and A.G. was supported in part by NSF grant DMS-1600560 and the Humboldt Foundation. A.G. and T.R. gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through GRK 1838 (A.G.) and CRC 1173 (T.R.). F.B. gratefully acknowledges financial support from the Simons Foundation award number 359963. T.R. thanks the Karlsruhe House of Young Scientists (KHYS) for a Research Travel Grant supporting the stay at Georgia Tech.


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© The Author(s) 2018

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Universität Tübingen, Fachbereich MathematikTübingenGermany
  3. 3.Institute for AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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