Evolution to the equilibrium in a dissipative and time dependent billiard

We study the convergence towards the equilibrium for a dissipative and stochastic time-dependent oval billiard. The dynamics of the system is described by using a generic four dimensional nonlinear map for the variables: the angular position of the particle, the angle formed by the trajectory of the article with the tangent line at the position of the collision, the absolute velocity of the particle, and the instant of the hit with the boundary. The dynamics of the stationary state as well as the dynamical evolution towards the equilibrium is made by using an ensemble of non interacting particles. Finally, we make a connection with the thermodynamic by using the energy equipartition theorem.