Large time analysis of the rate function associated to the Boltzmann equation: dynamical phase transitions
Giada Basile, Dario Benedetto, Lorenzo Bertini, Daniel Heydecker
TL;DR
The paper investigates the large-time fluctuations of the total number of collisions per particle in the Kac walk linked to the homogeneous Boltzmann equation, revealing a dynamical phase transition driven by Lu-Wennberg solutions. Using a detailed large-deviation framework with static and dynamical rate function components, it shows first-order costs i_e(q) vanish on the subcritical interval [0,\bar{q}_e], while second-order costs j_e arise from Gamma-convergence under Maxwellian initialization. Reversibility identities and an entropy-chain rule are established, along with a controllability result ensuring finite-cost connections between distributions, enabling rigorous Gamma-convergence arguments. The work also demonstrates that time ordering matters for second-order asymptotics and provides explicit bounds and variational characterizations for both first- and second-order rate functions, offering a precise quantification of dynamical phase transitions in kinetic systems.
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.