Universal energy diffusion in a quivering billiard

TL;DR

This paper introduces the quivering billiard model, demonstrating that small boundary oscillations lead to universal stochastic energy diffusion and Fermi acceleration, resolving discrepancies in existing models.

Contribution

The paper presents a simple, analyzable model of time-dependent billiards that captures universal energy diffusion and stochastic behavior in the quivering limit.

Findings

Particle ensembles evolve to a universal energy distribution

Energy diffusion occurs regardless of billiard shape or dimension

The model resolves discrepancies in the Fermi-Ulam model

Abstract

We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.