we focus on extensions of pair interactions and the stochastic N-particle model, introduced by M. Kac, to N-particle model which includes multi-particle interactions to study the evolution of probability density of multi-particle interactions.
Under the assumption that temporal evolution of the system is invariant under scaling transformations of the phase space and contractive properties, we obtain a full description of existence, uniqueness and long time behavior from its spectral properties. This model can also be seen as an extention of the Boltzmann dynamics of Maxwell type for conservative or dissipative interactions and the formation of power tails for long time behavior.
We also focus on a couple of new examples of multi-agent dynamics and information sharing and percolation effect.
This is work in collaboration with A. Bobylev, C. Cercignani and also with H. Tharkabhushanam.