Entropy and chaos in the Kac model

Abstract

We investigate the behavior in N of the N -particle entropy functional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove a number of results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, and obtain a bound showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open, and the upper bounds obtained here show that this problem is somewhat subtle.

Kac’s program in kinetic theory

“…We shall present some results on the construction of chaotic initial data on S N (E), whose proofs are mostly extensions of the precise statements and estimates recently established in [10] on this issue in the setting of Kac on S N −1 ( √ N ). We refer to the work [14] where an extensive study and precise computations of rates are performed.…”

“…Finally one can formulate an even stronger notion of (infinite-dimensional) entropic chaos (see [10] and definition (1.9) of the relative entropy below):…”

“…As a first natural step he raises the question of obtaining such uniform estimates on the L 2 spectral gap of the Markov process. This question has triggered many beautiful works (see the next subsection), however it is easy to convince oneself (see the discussion in [10] for instance) that there is no hope of passing to the limit N → ∞ in this spectral gap estimate, even if the spectral gap is independent of N . The L 2 norm is catastrophic in infinite dimension.…”

“…However, so far this linear strategy has proved to be unsuccessful in relating the asymptotic behavior of the many-particle process and that of the limit equation (cf. the discussion in [10]). …”

“…(2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system.…”