We explore Fermi acceleration in a stochastic oval billiard which shows unlimited to limited diffusion in energy when passing from the free to the dissipative case. We provide evidence for a transition from limited to unlimited energy growth taking place while detuning the corresponding restitution coefficient responsible for the degree of dissipation. A corresponding order parameter is suggested, and its susceptibility is shown to diverge at the critical point. We show that this order parameter is also be applicable to the periodically driven oval billiard and discuss the elementary excitation of the controlled diffusion process.

• Received 11 November 2022
• Revised 28 July 2023
• Accepted 26 September 2023

DOI:https://doi.org/10.1103/PhysRevE.108.054206