Ballistic dynamics of a convex smooth-wall billiard with finite escape rate along the boundary

TL;DR

This paper develops a supersymmetric non-linear ballistic sigma-model to analyze the response functions of a convex smooth-wall billiard with boundary-dependent escape rates, highlighting the role of whispering gallery modes in surface diffusion.

Contribution

It introduces a non-perturbative supersymmetric functional approach for open billiards with boundary coupling, emphasizing the surface diffusion modes from whispering gallery states.

Findings

Derivation of a non-linear ballistic sigma-model for open billiards.

Identification of whispering gallery modes as surface diffusion channels.

Analysis of response functions in the presence of boundary escape rates.

Abstract

We focus on the problem of an impurity-free billiard with a random position-dependent boundary coupling to the environment. The response functions of such an open system can be obtained non-perturbatively from a supersymmetric generating functional. The derivation of this functional is based on averaging over the escape rates and results in a non-linear ballistic $\sigma $-model, characterized by system-specific parameters. Particular emphasis is placed on the {}``whispering gallery modes'' as the origin of surface diffusion modes in the limit of large dimensionless conductance.