Scaling invariance for the diffusion coefficient in a billiard system

We investigated the behavior of the diffusion coefficient in a time-dependent oval-shaped billiard, focusing on the connection between this quantity and the system’s transition from unbounded to bounded diffusion caused by inelastic collisions with the boundary. The diffusion coefficient plays a key role in describing the scaling invariance characteristic of this transition. For short times, the low-action regime is characterized by a constant diffusion coefficient, which begins to decay after a crossover iteration, thereby suppressing the unlimited growth of velocity. We demonstrate that this behavior is scaling-invariant concerning the control parameters and can be described by a homogeneous generalized function and its associated scaling laws. This universal function effectively collapses all numerical data onto a single curve, confirming the self-similar nature of the dynamical crossover. The critical exponents governing this scaling were determined both phenomenologically, through extensive numerical simulations, and analytically, by examining the system's equations of motion near the critical point. This analysis confirmed the decay exponent β = -1 for the diffusion coefficient, a value previously identified in related low-dimensional dissipative systems like the dissipative standard map. The consistency between our analytical derivations and numerical results strongly validates the universal framework we propose for describing transport phenomena in open Hamiltonian systems subject to dissipation.