Skip to main content

Research Repository

Advanced Search

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

Karlis, A. K.; Diakonos, F. K.; Constantoudis, V.


A. K. Karlis

F. K. Diakonos

V. Constantoudis


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 have 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 standard 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.


Karlis, A. K., Diakonos, F. K., & Constantoudis, V. (2012). A consistent approach for the treatment of Fermi acceleration in time-dependent billiards. Chaos, 22(2), 026120.

Journal Article Type Article
Acceptance Date Mar 7, 2012
Publication Date Jun 25, 2012
Journal Chaos: An Interdisciplinary Journal of Nonlinear Science
Print ISSN 1054-1500
Publisher AIP Publishing
Peer Reviewed Peer Reviewed
Volume 22
Issue 2
Pages 026120
Keywords consistent approach, treatment, Fermi acceleration, time-dependent, billiards
Public URL
Publisher URL


You might also like

Downloadable Citations