Aspects of diffusion in the stadium billiard

TL;DR

This paper conducts a detailed numerical analysis of diffusion in the stadium billiard, confirming normal diffusion, inhomogeneous diffusion characteristics, and validating an empirical model that describes the diffusion process effectively.

Contribution

It introduces an empirical model for local and global diffusion in the stadium billiard, confirming its accuracy and applicability across various parameters and initial conditions.

Findings

Diffusion is normal for all tested epsilon values up to 0.3.

The diffusion constant varies parabolically with momentum, indicating inhomogeneous diffusion.

The empirical model accurately describes diffusion including boundary effects.

Abstract

We perform a detailed numerical study of diffusion in the $\varepsilon$ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of $\varepsilon$ with the following conclusions: (i) the diffusion is normal for all values of $\varepsilon \leq(0.3)$ and all initial conditions, (ii) the diffusion constant is a parabolic function of the momentum (i.e., we have inhomogeneous diffusion), (iii) the model describes the diffusion very well including the boundary effects, (iv) the approach to the asymptotic equilibrium steady state is exponential, (v) the so-called random model (Robnik et al., 1997) is confirmed to apply very well, (vi) the diffusion constant extracted from the distribution function in momentum space and the one derived from the second moment agree very well. The classical transport time, an important parameter in quantum chaos, is thus determined.