Asymptotic probability of energy increasing solutions to the homogeneous Boltzmann equation

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 collisions (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.

Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium

“…for an arbitrary α ∈ R, n and λ. For the specific form of the Hamiltonian H = H F + H S (38a) with H S written in the symmetric form (42), one can directly check the above Hamiltonian symmetry boils down to the following property on the diffusive kernel ( 39)…”

“…But interestingly, the diffusive limit can also be obtained from the scattering kinetic regime that we have considered in this paper. Our goal in this subsection is to show how one can derive a path large deviation theory for wave kinetics in the diffusive regime from the large deviation Hamiltonian (42). In a recent paper, a similar weak scattering limit has been considered to derive the path large deviation principle for plasma below the Debye length, related to the Landau equation, from the path large deviation principle for dilute gases, related to the Boltzmann equation [43].…”

“…In order to derive the diffusive limit from the large deviation Hamiltonian, we use Eq. (42). We show in Appendix D that the diffusion kernel Σ transforms into a differential operator such that for any test function f and g:…”

“…Deriving such large deviation principles from deterministic microscopic dynamics is a fundamental endeavor in theoretical and mathematical physics. Recently, the large deviation principles for a number of classical kinetic theories, starting from first principles, have been uncovered: for discrete models that mimic dilute gases and with Boltzmann like behavior [38,39], for dilute gases related to the Boltzmann equation [1,40], for the Kac model [41,42], for plasma at length scales much smaller than the Debye length related to the Landau equation [43], for homogeneous systems with long range interactions related to the Balescu-Guernsey-Lenard equation [44], for weakly interacting waves in a homogeneous setup [2] related to the wave kinetic equation. The large deviation principles describe fluctuations but also uncover gradient structure for the deterministic kinetic equation, see [45] and a simple explanation in [1].…”

“…The large deviation principles describe fluctuations but also uncover gradient structure for the deterministic kinetic equation, see [45] and a simple explanation in [1]. Several mathematical results, usually valid for a fraction of the kinetic time in the spirit of Lanford results for the deterministic equation, have been obtained for the large deviation principles, for instance for the Boltzmann equation [40,46], or for the Kac model with unexpected corrections to the expected large deviation principle [41,42] associated to giant concentrations and solutions of the Boltzmann equation without energy conservation.…”