Abstract
We introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.
Highlights
The statistics of rarefied gas is described, at the kinetic level, by the Boltzmann equation
Regarding the large deviation asymptotics, we point out that while the law of large numbers depends on the validity of the Stosszahlansatz with probability converging to one as N → ∞, the large deviation principle requires that it holds with probability super-exponentially close to one for N large
In [18] a large deviation result has been derived for a stochastic model in the setting of one dimensional spatially dependent Boltzmann equation with discrete velocities
Summary
The statistics of rarefied gas is described, at the kinetic level, by the Boltzmann equation. It has become paradigmatic since it encodes most of the conceptual and technical issues in the description of the statistical properties for out of equilibrium systems. In the spatially homogeneous case the Boltzmann equation reads. ∂t ft (v) dv = r (v , v∗; dv, dv∗) ft (v ) ft (v∗) dv dv∗ − ft (v) dv r (v, v∗; dv , dv∗) ft (v∗) dv∗,
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