Casimir force between integrable and chaotic pistons

TL;DR

This study numerically analyzes the Casimir force between pistons with different classical dynamics, revealing how spectral oscillations related to integrability or chaos influence the force's correction terms.

Contribution

It provides a detailed numerical comparison of Casimir forces for integrable and chaotic billiard-shaped pistons, highlighting the impact of classical dynamics on quantum vacuum forces.

Findings

The smooth spectral part determines short-distance force based on geometry.

Oscillating spectral terms differ qualitatively between integrable and chaotic systems.

A transition from regular to chaotic geometries causes a sudden change in the force correction.

Abstract

We have computed numerically the Casimir force between two identical pistons inside a very long cylinder, considering different shapes for the pistons. The pistons can be considered as quantum billiards, whose spectrum determines the vacuum force. The smooth part of the spectrum fixes the force at short distances, and depends only on geometric quantities like the area or perimeter of the piston. However, correcting terms to the force, coming from the oscillating part of the spectrum which is related to the classical dynamics of the billiard, are qualitatively different for classically integrable or chaotic systems. We have performed a detailed numerical analysis of the corresponding Casimir force for pistons with regular and chaotic classical dynamics. For a family of stadium billiards, we have found that the correcting part of the Casimir force presents a sudden change in the transition from regular to chaotic geometries.