Central limit theorem for the solution of the Kac equation

TL;DR

This paper establishes a central limit theorem framework for the Kac equation, providing new proofs of convergence to Maxwellian distribution and identifying initial energy finiteness as necessary for weak convergence.

Contribution

It introduces a probabilistic approach to analyze the Kac equation, offering new convergence proofs and clarifying the role of initial energy conditions.

Findings

Convergence to Maxwellian distribution with explicit rate

Finiteness of initial energy is necessary for weak convergence

Probabilistic representation of solutions as sums of random variables

Abstract

We prove that the solution of the Kac analogue of Boltzmann's equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof of the convergence to the Maxwellian distribution is provided, with a rate information both under the sole hypothesis that the initial energy is finite and under the additional condition that the initial distribution has finite moment of order $2+\delta$ for some $\delta$ in $(0,1]$. Moreover, it is proved that finiteness of initial energy is necessary in order that the solution of Kac's equation can converge weakly. While this statement may seem to be intuitively clear, to our knowledge there is no proof of it as yet.