We study the stationary Kac equation in the presence of an external force field and a Gaussian thermostat, and investigate the behavior of a solution to this equation for varying field strength. We prove that for a weak field, the stationary density is continuous, but for a strong field the density has a singularity. Monte Carlo simulations illustrating the same are also presented.

We consider the Kac equation with a thermostatted force field and prove the existence of a global in time solution that converges weakly to a stationary state. As there is no an obvious candidate for the entropy functional, in this case, the convergence result is obtained via Fourier transform techniques.

In a first successful attempt to rigorously derive the Boltzmann equation from a many-particle system, M. Kac introduced a stochastic model, in which particle velocities make jumps due to random interactions between pairs of particles. In this