Kac Equation

Entropy decay for the Kac evolution

The Kac model is an attempt to "derive" the homogeneous Boltzmann equation from a probabilistic point of view rather than based on deterministic mechanical principles. There is still no satisfactory derivation of the Boltzmann equation as a certain limit of a microscopic system of classical particles: the existing rigorous proofs only work up to a fraction of the first average collision time. In Kac's approach, the notion of chaoticity and propagation of chaos, can be used to show that the Kac walk, a probabilistic sketch of what happens during collisions, approaches the Boltzmann equation if one lets the number of particles tend to infinity. An attractive feature of the Kac walk is that it gives a precise meaning of what it means to approach an equilibrium state, that is, a stationary solution of the respective equation that is invariant under the collision processes. Mischler and Mouhot were able show relaxation towards equilibrium in relative entropy as well as in Wasserstein distance with a rate that is independent of the particle number. As expected, they achieve this not for any initial condition, but rather for a natural class of chaotic states. The rate of relaxation is, however, polynomial in time. So far, there is no mathematical evidence that the entropy in the Kac model in general decays exponentially with a rate that is independent of the number of particles, and physical intuition suggests that for highly "improbable" states this cannot be expected. In [1] we took a different approach, one which is based on the idea of coupling a system of particles to a reservoir. 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. Based on Nelson's hypercontractive estimate and the geometric form of the Brascamp-Lieb inequalities, 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. Similar results hold for the Kac-Boltzmann equation with uniform scattering cross sections. 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.

Related publications

1. Entropy decay for the Kac evolution. Communications in Mathematical Physics 363 (2018), 847–875.
   (with F. Bonetto, A. Geisinger, M. Loss)