We shall revisit the Boltzmann equation for rarefied non-linear particle dynamics, of conservative or dissipative nature, and on the stochastic N-particle model, introduced by M. Kac.
Related to this equation, we consider a a probabilistic dynamics from generalizations to N-particle model which includes multi-particle interactions. From basic symmetries and invariances for a general class of stochastic interactions, we show existence and uniqueness of states and recover the longtime dynamics and decay rates approaching stable laws characterized by self-similar rescaling, with finite or infinity energy initial data. We classify the moments integrability and see that broad tails (Pareto type) attractors are possible.

There is a large class of applications to these models including classical elastic or inelastic Maxwell type interactions with or without a thermostat, and social dynamics such as information percolation models, or wealth distributions models with Pareto tail formation.

This is work in collaboration with A. Bobylev, C. Cercignani and H. Tharkabhushanam.