Large Deviations for Kac-Like Walks

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.

[1]  Large deviations from a kinetic limit , 1998 .

[2]  Matthias Erbar,et al.  A gradient flow approach to the Boltzmann equation , 2016, Journal of the European Mathematical Society.

[3]  Stefan Adams,et al.  From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage , 2010, 1004.4076.

[4]  G. Basile,et al.  A gradient flow approach to linear Boltzmann equations , 2017, 1707.09204.

[5]  On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations , 1995 .

[6]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[7]  M. A. Peletier,et al.  On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion , 2013, 1312.7591.

[8]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[9]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour XIX, 1989 , 1991 .

[10]  Equipe de Modélisation Stochastique.  On Large Deviations for Particle Systems Associated with Spatially Homogeneous Boltzmann Type Equations , 2004 .

[11]  A. Faggionato,et al.  Large deviations of the empirical flow for continuous time Markov chains , 2012, 1210.2004.

[12]  C. Landim,et al.  Scaling Limits of Interacting Particle Systems , 1998 .

[13]  G. Basile,et al.  Donsker-Varadhan asymptotics for degenerate jump Markov processes , 2013, 1310.5829.

[14]  M. Mariani.  A Gamma-convergence approach to large deviations , 2012, 1204.0640.

[15]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[16]  K. Uchiyama.  A fluctuation problem associated with the Boltzmann equation for a gas of molecules with a cutoff potential , 1983 .

[17]  C. Villani,et al.  Entropy and chaos in the Kac model , 2008, 0808.3192.

[18]  M. Mariani.  A Γ-convergence approach to large deviations , 2018 .

[19]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[20]  K. Uchiyama,et al.  Fluctuations in a Markovian system of pairwise interacting particles , 1988 .