Entropy Decay for the Kac Evolution

Abstract

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

Decay of Entropy and Information for multidimensional Kac models

“…The decay rates for the entropy and information for the Kac system coupled to a thermostat or a heat reservoir obtained in the theorems 2.3, 2.6, 3.2 and 3.5 are consistent with the decay rates proven in [4], [1] and [2]. It is interesting to note that the decay rates are quite uniform in the sense that they are independent of the collisions within the Kac system, the probability measure used to sample the scattering angle, as well as the inverse temperature of the thermostat.…”

“…In comparison to [4] and [1] the approach via the information used in this paper and [2] provides a more natural and elegant method with simpler proofs and weaker assumptions. It also allows to treat multidimensional systems as well as the spherical Kac model (see [2] for the latter).…”

“…s T j [h (1) ] = P (1) s MU [h (2) ] = MP (2) s U [h (2) ] = MU P (2) s [h (2) ] = T j P (1) s [h (1) ].…”

The Brascamp-Lieb inequality and its influence on Fourier analysis

“…to prove smoothing estimates for heat kernels of (possibly magnetic) Schrödinger operators ([Bou91, Lie90, LT97] to only mention three), in kinetic theory (e.g. [BGLR18]) and other fields.…”

On a thermostated Kac model with rescaling

“…We mention that the argument behind the model with thermostats was made rigorous in [4] as a mean-field limit of finite-reservoir systems using the L 2 metric and the Fourier-based Gabetta-Toscani-Wennberg metric d 2 . Studying the convergence to equilibrium for the finite-reservoir system is not a simple problem (see [5]).…”