Scaling invariance for the diffusion coefficient in a billiard system

TL;DR

This paper studies how inelastic collisions in a time-dependent billiard suppress unbounded diffusion, revealing a scaling-invariant behavior of the diffusion coefficient characterized by specific critical exponents.

Contribution

It introduces a scaling-invariant framework for understanding diffusion suppression in billiards with inelastic collisions, including analytical derivation of critical exponents.

Findings

Diffusion coefficient remains constant at short times in the low-action regime.

A crossover causes the diffusion coefficient to decay, suppressing unlimited velocity growth.

The decay exponent beta is analytically determined as -1.

Abstract

We investigated the unbounded diffusion observed in a time-dependent oval-shaped billiard and its suppression owing to inelastic collisions with the boundary. The main focus is on the behavior of the diffusion coefficient, which plays a key role in describing the scaling invariance characteristic of this transition. For short times, the low-action regime is characterized by a constant diffusion coefficient, which begins to decay after a crossover iteration, thereby suppressing the unlimited growth of velocity. We demonstrate that this behavior is scaling-invariant concerning the control parameters and can be described by a homogeneous generalized function and its associated scaling laws. The critical exponents are determined both phenomenologically and analytically, including the decay exponent beta = -1, previously identified in the diffusion coefficient of the dissipative standard map.