Phase Space Interpretation of Exponential Fermi Acceleration

TL;DR

This paper explains the physical mechanism behind exponential Fermi acceleration in a rectangular billiard with an oscillating bar, modeling it as a correlated sequence of motions and a velocity-space random walk, and predicts growth rates.

Contribution

It introduces a new physical interpretation and a random walk model for exponential Fermi acceleration, providing quantitative predictions and identifying conditions for this phenomenon.

Findings

The acceleration results from correlated motions along invariant curves.

A velocity-space random walk model accurately predicts growth rates.

The analysis identifies key conditions for exponential Fermi acceleration.

Abstract

Recently, the occurrence of exponential Fermi acceleration has been reported in a rectangular billiard with an oscillating bar inside [K. Shah, D. Turaev, and V. Rom-Kedar, Phys. Rev. E {\bf 81}, 056205 (2010)]. In the present work, we analyze the underlying physical mechanism and show that the phenomenon can be understood as a sequence of highly correlated motions, consisting of alternating phases of free propagation and motion along the invariant spanning curves of the well-known one-dimensional Fermi-Ulam model. The key mechanism for the occurrence of exponential Fermi acceleration can be captured in a random walk model in velocity space with step width proportional to the velocity itself. The model reproduces the occurrence of exponential Fermi acceleration and provides a good ab initio prediction of the value of the growth rate including its full parameter-dependency. Our analysis clearly points out the requirements for exponential Fermi acceleration, thereby opening the perspective of finding other systems exhibiting this unusual behaviour.