Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions

TL;DR

This paper investigates the long-term behavior of the rate function related to large fluctuations in the homogeneous Boltzmann equation, revealing a phase transition in the asymptotic regime of particles and time.

Contribution

It demonstrates the existence of a phase transition in the rate function for large deviations in the Boltzmann equation and analyzes the probability of subtypical fluctuations.

Findings

Rate function vanishes for subtypical collision counts due to Lu-Wennberg solutions.

Subtypical fluctuations are exponentially unlikely in the number of particles, N.

Established controllability of the homogeneous Boltzmann equation.

Abstract

We analyse the large time behaviour of the rate function that describes the probability of large fluctuations of an underlying microscopic model associated to the homogeneous Boltzmann equation, such as the Kac walk. We consider in particular the asymptotic of the number of collisions, per particle and per unit of time, and show it exhibits a phase transition in the joint limit in which the number of particles N and the time interval [0,T] diverge. More precisely, due to the existence of Lu-Wennberg solutions, the corresponding limiting rate function vanishes for subtypical values of the number of collisions. We also analyse the second order large deviations showing that the probability of subtypical fluctuations is exponentially small in N, independently on T. As a key point, we establish the controllability of the homogeneous Boltzmann equation.