Symmetry Breaking in Time-Dependent Billiards

We investigate symmetry breaking in a time-dependent billiard that undergoes a continuous phase transition when dissipation is introduced.We provide the identification of symmetry breaking explicitly linked to the boundary velocity limits in time-dependent billiards, resolving the open question of fully characterizing this phase transition.The system presents unlimited velocity, and thus energy growth for the conservative dynamics.When inelastic collisions are introduced between the particle and the boundary, the velocity reaches a plateau after the crossover iteration.The system presents the expected behavior for this type of transition, including scale invariance, critical exponents related by scaling laws, and an order parameter approaching zero in the crossover iteration.We analyze the velocity spectrum and its averages for dissipative and conservative dynamics.The transition point in velocity behavior caused by the physical limit of the boundary velocity and by the introduced dissipation coincides with the crossover interaction obtained from the root-mean squared velocity, or , curves.Additionally, we examine the velocity distributions, which lose their symmetry once the particle's velocity approaches the lower limit imposed by the boundary's motion and the system's control parameters.This distribution is also characterized analytically by an expression P(V, n), which attains a stationary state, with a well-defined upper bound, only in the dissipative case.