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Entropy decay for the Kac evolution. (English) Zbl 1405.35126
Summary: 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.
##### MSC:
 35Q20 Boltzmann equations 35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
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