Second-order asymptotic expansion and thermodynamic interpretation of a fast-slow Hamiltonian system

Matthias Klar, Karsten Matthies, Johannes Zimmer

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Abstract

This article includes a short survey of selected averaging and dimension reduction techniques for deterministic fast–slow systems. This survey includes, among others, classical techniques, such as the WKB approximation or the averaging method, as well as modern techniques, such as the GENERIC formalism. The main part of this article combines ideas of some of these techniques and addresses the problem of deriving a reduced system for the slow degrees of freedom (DOF) of a fast–slow Hamiltonian system. In the first part, we derive an asymptotic expansion of the averaged evolution of the fast–slow system up to second order, using weak convergence techniques and two-scale convergence. In the second part, we determine quantities which can be interpreted as temperature and entropy of the system and expand these quantities up to second order, using results from the first part. The results give new insights into the thermodynamic interpretation of the fast–slow system at different scales.

Original languageEnglish
Article number119
Number of pages32
JournalLetters in Mathematical Physics
Volume112
Issue number6
Early online date23 Nov 2022
DOIs
Publication statusPublished - 31 Dec 2022

Bibliographical note

Funding Information:
MK was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the Project EP/L015684/1. JZ gratefully acknowledges funding by a Royal Society Wolfson Research Merit Award. All authors thank Celia Reina for stimulating discussions and helpful suggestions throughout this project.

Keywords

  • Entropy
  • Fast–slow system
  • Homogenisation in time

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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