Discussing a transition from bounded to unbounded energy in a time-dependent billiard

TL;DR

This paper investigates a phase transition in a time-dependent oval billiard system, where energy growth shifts from bounded to unbounded, exhibiting properties akin to continuous phase transitions with critical phenomena.

Contribution

It introduces a detailed analysis of the transition from bounded to unbounded energy growth in a time-dependent billiard, highlighting critical behavior and scale invariance.

Findings

Identifies a phase transition with scale invariance.

Shows critical exponents follow scaling laws.

Demonstrates unbounded energy growth is limited by inelastic collisions.

Abstract

We revisit a time-dependent, oval-shaped billiard to investigate a phase transition from bounded to unbounded energy growth. In the static case, the phase space exhibits a mixed structure. The chaotic sea in the static scenario leads to average energy growth for a time-dependent boundary. However, inelastic collisions between the particle and the boundary limit this unbounded energy increase. 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 an elementary excitation that promotes particle diffusion and lacks topological defects that provide modifications to the probability distribution function.