Large Deviations of Kac's Conservative Particle System and Energy Non-Conserving Solutions to the Boltzmann Equation: A Counterexample to the Predicted Rate Function

Abstract

We consider the dynamic large deviation behaviour of Kac's collisional process for a range of initial conditions including equilibrium. We prove an upper bound with a rate function of the type which has previously been found for kinetic large deviation problems, and a matching lower bound restricted to a class of sufficiently good paths. However, we are able to show by an explicit counterexample that the predicted rate function does not extend to a global lower bound: even though the particle system almost surely conserves energy, large deviation behaviour includes solutions to the Boltzmann equation which do not conserve energy, as found by Lu and Wennberg, and these occur strictly more rarely than predicted by the proposed rate function. At the level of the particle system, this occurs because a macroscopic proportion of energy can concentrate in o(N ) particles with probability e −O(N ) . CONTENTS

Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

“…In [2] a large deviation upper bound is achieved for a homogeneous model which conserves momentum but not energy, while the matching lower bound is obtained for a restricted class of paths. A similar result, in the case of energy and momentum conservation, has been proven in [7]. In this case the upper and lower bound match for a subset of paths for which energy is conserved.…”

“…In particular we consider a Kac walk with the hard sphere cross section, and prove the large deviation upper bound with such rate function. The matching large deviation lower bound is achieved for both Lu and Wennberg solutions and the same restricted class of path as in [2,7].…”

“…Another example of large deviation asymptotic for non-conserving energy path has been constructed in [1], for a Kac-like microscopic dynamics with discrete energies. More precisely, as proven in [7], the upper bound rate function derived in [10] vanishes on Lu and Wennberg solutions, while their asymptotic probability is e −cN , which implies the upper bound rate function in [10] is not optimal.…”

“…For energy preserving microscopic dynamics with unbounded velocities, a main obstacle to a complete proof of large deviations is the occurrence of macroscopic paths with finite rate function that violate the conservation of the energy. In particular, as discussed in [7], a class of such paths is given by the solutions to the homogeneous Boltzmann equations constructed by Lu and Wennberg in [12], for which the energy is increasing. Another example of large deviation asymptotic for non-conserving energy path has been constructed in [1], for a Kac-like microscopic dynamics with discrete energies.…”

Dynamical large deviations for homogeneous systems with long range interactions and the Balescu--Guernsey--Lenard equation

“…They basically prove the validity of the large deviation Hamiltonian for times of order one collision time, in the spirit of Lanford's proof for the Boltzmann equation. Large deviation principles have also been obtained for Kac's models [25,3]. As far as we know, no mathematical results exists for the large deviations of plasma or systems with mean-field interactions.…”