Basile, Benedetto, Bertini and Heydecker's large-time analysis of the large-deviation rate function for the number of collisions per particle per unit time in the Kac walk, whose hydrodynamic limit is the homogeneous Boltzmann equation. In the joint limit of diverging particle number N and time horizon T, the collision count exhibits a dynamical phase transition: the limiting rate function vanishes on subtypical values because of time-independent-cost Lu-Wennberg (energy-increasing) solutions, so subtypical fluctuations are exponentially small in N but T-independent while supertypical ones cost order NT. The work also gives a second-order rate function (explicit via relative entropy of Maxwellians for uniform initial data) and proves an entropy chain rule for non-energy-conserving paths and the controllability of the homogeneous Boltzmann equation.
This paper analyses the large-time behaviour of the large-deviation rate function governing atypical fluctuations of a microscopic model (the Kac walk) whose hydrodynamic limit is the spatially homog…