Entropy decay for the Kac evolution

TL;DR

This paper proves that solutions to the Kac master equation exhibit exponential entropy decay towards equilibrium, with a rate largely independent of particle number, contrasting previous inverse proportionality results.

Contribution

It establishes explicit exponential decay rates for entropy in the Kac evolution, using hypercontractivity and geometric inequalities, for initial conditions close to equilibrium.

Findings

Entropy decays exponentially with time.

Decay rate is essentially independent of particle number.

Results extend to Kac-Boltzmann equation with uniform cross sections.

Abstract

We consider solutions to the Kac master equation for initial conditions where $N$ particles are in a thermal equilibrium and $M\le N$ 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.