Mean-field limit and Vlasov equation for a Gaussian-thermostatted Lorentz gas

We study N particles moving in a dispersing billiard under an electric field and coupled through a Gaussian thermostat. The corresponding mean-field equation is a Vlasov-type equation with specular reflection. We reformulate the equation as a fixed point problem for the time-integrated thermostat coefficient. This yields existence of weak solutions for every absolutely continuous initial datum of finite energy and, for bounded initial densities supported on velocities that are large compared with the field, uniqueness of the weak solution; under the finite horizon assumption, the upper bound on the velocities can be replaced by an exponential tail of the initial density. In these regimes, we prove convergence of the empirical measures of the particle system to the solution of the kinetic equation. In general, convergence holds along subsequences.