Transition from bounded to unbounded energy in a time-dependent billiard

We revisit a time-dependent, oval-shaped billiard to investigate a phase transition from bounded to unbounded energy growth. The introduction of inelastic collisions between the particle and the boundary limit the unbounded energy increase after the crossover iteration observed for the conservative dynamics. The central phenomenology uses a probability distribution that satisfies the system's boundary conditions to solve the diffusion equation. From this, we express the observables and, using a set of scaling hypotheses and a generalized homogeneous function, obtain a relation between the critical exponents, leading to a scaling law validated through numerical simulations and analytical methods. This transition displays properties similar to continuous phase transitions in statistical mechanics, including scale invariance, interrelated critical exponents governed by scaling laws, and an order parameter (susceptibility) approaching zero (infinity) at the transition. Furthermore, the system exhibits a variation in velocity between collisions mediated by control parameters, known as elementary excitations, which facilitate particle diffusion. The system also does not present topological defects that could alter the probability distribution function.