Loughborough University
Browse

Unusual properties of adiabatic invariance in a billiard model related to the adiabatic piston problem

Download (551.38 kB)
journal contribution
posted on 2025-05-07, 15:05 authored by Joshua Skinner, Anatoly NeishtadtAnatoly Neishtadt

We consider the motion of two massive particles along a straight line. A lighter particle bounces back and forth between a heavier particle and a stationary wall, with all collisions being ideally elastic. This is one of canonical models in the theory of adiabatic invariants. It is known that if the lighter particle moves much faster than the heavier one, and the kinetic energies of the particles are of the same order, then the product of the speed of the lighter particle and the distance between the heavier particle and the wall is an adiabatic invariant: its value remains approximately constant over a long period. We show that the value of this adiabatic invariant, calculated at the collisions of the lighter particle with the wall, is a constant of motion (i.e., an exact adiabatic invariant). On the other hand, the value of this adiabatic invariant at the collisions between the particles slowly, linearly in time, decays with each collision. The model we consider is a highly simplified version of the classical adiabatic piston problem, where the lighter particle represents a gas particle, and the heavier particle represents the piston.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Theoretical and Applied Mechanics

Publisher

Serbian Society of Mechanics & Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade

Version

  • VoR (Version of Record)

Rights holder

© Serbian Society of Mechanics, Belgrade and Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade

Publisher statement

Theoretical and Applied Mechanics is an Open Access Journal. All articles can be downloaded free of charge and used in accordance with the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Serbia (CC BY NC ND).

Acceptance date

2025-04-07

Publication date

2025-05-05

Copyright date

2025

ISSN

1450-5584

eISSN

2406-0925

Language

  • en

Depositor

Prof Anatoly Neishtadt. Deposit date: 11 April 2025

Usage metrics

    Loughborough Publications

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC