Explaining a changeover from normal to super diffusion in time-dependent billiards

TL;DR

This paper provides an analytical explanation for the transition from normal to super diffusion in time-dependent billiards, highlighting how energy growth regimes depend on initial velocities and collision dynamics.

Contribution

It introduces a two-dimensional mapping approach to analytically describe the changeover from normal to super diffusion in billiard systems.

Findings

Low initial velocities lead to normal diffusion with velocity growth exponent ~1/2.

Super diffusion with velocity growth exponent ~1 occurs at later stages.

Initial velocity symmetry causes a plateau in average velocity for larger initial velocities.

Abstract

The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ~1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ~1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity.