## Kac's process and some probabilistic aspects of the Boltzmann equation

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##### Abstract

We consider a family of stochastic interacting particle systems introduced by Kac as a model for a spatially homogeneous gas undergoing elastic collisions, corresponding to the spatially homogeneous Boltzmann equation. We consider Kac's problem of showing propagation of chaos - that if the velocities of the particles are initially approximately independent, then the same is true at later times - which is equivalent to the convergence of the empirical measures, and which derives the spatially homogeneous Boltzmann equation from the underlying molecular dynamics.

The first two results concern the propagation of chaos for different Kac models. In the first case, we consider the hard spheres kernel, which is appropriate for modelling interactions arising from localised interactions. For this model, we build on previous analyses of the same problem to show that the expected deviation between the Kac process and the Boltzmann equation, measured in expected Wasserstein distance, is of the order

We next consider the case of non-cutoff hard potentials, which arise from modelling a family of long-range interactions. In this case, the collision kernel is doubly-unbounded, both unbounded as the relative velocity increases, and with a non-integrable angular singularity, so that every particle undergoes infinitely many collisions on any nontrivial time-interval. In this context, we introduce a Tanaka-style coupling for a well-chosen distance function on

We further study the dynamical large deviations of the

The final section concerns Smoluchowski and Flory coagulation equations with a particular bilinear form, which arise when studying the interaction structure of the Kac process by forming clusters of particles joined by chains of collisions, corresponding to the cumulant expansion. Exploiting the bilinear structure and a coupling to random graphs, we are able to give a detailed analysis of the coagulation particle system and limiting Flory equation, including showing the emergence of a unique macroscopic cluster at a finite time