Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

TL;DR

This paper demonstrates that solutions to the homogeneous Boltzmann equation with increasing energy are exponentially unlikely in a microscopic stochastic model, providing precise large deviation estimates and extending Sanov's theorem.

Contribution

It establishes the exponential smallness of energy-increasing solutions in Kac's model by extending large deviation principles to the microcanonical ensemble.

Findings

Probability of energy-increasing solutions is exponentially small.

Extended Sanov's theorem to microcanonical ensemble.

Provided explicit exponential rate of decay.

Abstract

Weak solutions to the homogeneous Boltzmann equation with increasing energy have been constructed by Lu and Wennberg. We consider an underlying microscopic stochastic model with binary collision (Kac's model) and show that these solutions are atypical. More precisely, we prove that the probability of observing these paths is exponentially small in the number of particles and compute the exponential rate. This result is obtained by improving the established large deviation estimates in the canonical setting. Key ingredients are the extension of Sanov's theorem to the microcanonical ensemble and large deviations for the Kac's model in the microcanonical setting.