A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

TL;DR

This paper introduces a self-consistent method for analyzing Fermi acceleration in time-dependent billiards, overcoming limitations of traditional diffusion-based models by using transition probabilities and the Chapman-Kolmogorov equation.

Contribution

It proposes a novel, general approach that does not rely on ensemble averages or assumptions of process continuity for Fermi acceleration analysis.

Findings

Provides a consistent methodology applicable to any time-dependent billiard.

Successfully applied to the Fermi-Ulam model as a case study.

Addresses transient evolution of the velocity distribution, not just long-term behavior.

Abstract

The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions $n$ is determined by the Fokker-Planck equation (FPE). In the literature the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions has been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a stadanrd simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.