A stochastic particle system modeling the Carleman equation

Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient σ > 0 but different velocities (drift), 1 for one species and -1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ≤ var-epsilon (var-epsilon being a positive parameter), they both may simultaneously flip their velocity according to a poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when var-epsilon goes to zero and the total number of particles increases like var-epsilon -1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, σ depends on var-epsilon and suitably vanishes when var-epsilon goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation