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Thermodynamic reduction of contact dynamics

Hyun-Seok Do, Yong-Geun Oh

TL;DR

This work extends the thermodynamic reduction framework of Lim–Oh to contact dynamics, treating the pair $(\lambda,\mathcal{F})$ as two information systems and employing relative information entropy as a generating function for non-equilibrium thermodynamics on contact manifolds. It introduces big and small contact kinetic theory phase spaces (b-CKTPS and s-CKTPS), builds a moment-map/Marsden–Weinstein reduction to obtain $\mathcal{F}$-reduced CKTPS, and shows how the reduced entropy generates a Legendrian equilibrium in the jet space. The key technical contributions include regularity and transversality criteria for observable maps, a two-stage reduction producing a family of reduced entropies $\mathcal{S}^{\mathrm{red}}_{\mathcal{F}}$, and a generating-function description of contact thermodynamic equilibrium distributions. The results culminate in a Gibbs-type description of equilibrium measures under holonomic constraints and provide a robust geometric framework for a thermodynamic formalism in dissipative contact dynamics, with implications for a future do-oh:formalism. These developments offer a principled route to encode dissipative behavior and thermodynamic structure directly on the contact phase space.

Abstract

A universal algorithm to derive a macroscopic dynamics from the microscopic dynamical system via the averaging process and symplecto-contact reduction was introduced by Jin-wook Lim and the second-named author in [LO23]. They apply the algorithm to derive non-equilibrium thermodynamics from the statistical mechanics utilizing the relative information entropy as a generating function of the associated thermodynamic equilibrium. In the present paper, we apply this algorithm to the contact Hamiltonian dynamical systems. We describe a procedure of obtaining a discrete set of dynamical invariants of the given contact Hamiltonian system, or more generally of a contact multi-Hamiltonian system in a canonical way by deriving a (finite-dimensional non-equilibrium) thermodynamic system. We call this reduction the thermodynamic reduction of contact dynamics.

Thermodynamic reduction of contact dynamics

TL;DR

This work extends the thermodynamic reduction framework of Lim–Oh to contact dynamics, treating the pair as two information systems and employing relative information entropy as a generating function for non-equilibrium thermodynamics on contact manifolds. It introduces big and small contact kinetic theory phase spaces (b-CKTPS and s-CKTPS), builds a moment-map/Marsden–Weinstein reduction to obtain -reduced CKTPS, and shows how the reduced entropy generates a Legendrian equilibrium in the jet space. The key technical contributions include regularity and transversality criteria for observable maps, a two-stage reduction producing a family of reduced entropies , and a generating-function description of contact thermodynamic equilibrium distributions. The results culminate in a Gibbs-type description of equilibrium measures under holonomic constraints and provide a robust geometric framework for a thermodynamic formalism in dissipative contact dynamics, with implications for a future do-oh:formalism. These developments offer a principled route to encode dissipative behavior and thermodynamic structure directly on the contact phase space.

Abstract

A universal algorithm to derive a macroscopic dynamics from the microscopic dynamical system via the averaging process and symplecto-contact reduction was introduced by Jin-wook Lim and the second-named author in [LO23]. They apply the algorithm to derive non-equilibrium thermodynamics from the statistical mechanics utilizing the relative information entropy as a generating function of the associated thermodynamic equilibrium. In the present paper, we apply this algorithm to the contact Hamiltonian dynamical systems. We describe a procedure of obtaining a discrete set of dynamical invariants of the given contact Hamiltonian system, or more generally of a contact multi-Hamiltonian system in a canonical way by deriving a (finite-dimensional non-equilibrium) thermodynamic system. We call this reduction the thermodynamic reduction of contact dynamics.
Paper Structure (24 sections, 24 theorems, 223 equations)

This paper contains 24 sections, 24 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Contact dynamics is not measure-preserving
  3. Contact dynamics and its thermodynamics reduction
  4. Where are we heading for?
  5. Contact kinetic theory phase space: big phase space
  6. Contact kinetic theory phase space: small phase space
  7. Moment maps and symplectic reductions
  8. Multi-Hamiltonian system and relative information entropy
  9. Observables and regularity of observation map
  10. Some contact Hamiltonian calculus
  11. A regularity criterion of the observation on b-CKTPS
  12. A regularity criterion of the observation on s-CKTPS
  13. Reduction of CKTPS
  14. A generating function of contact thermodynamic equilibrium
  15. Transversality of the reduced entropy function as a generating function
  16. ...and 9 more sections

Key Result

Lemma 2.1

Assume that $M$ is oriented. For any pair of orientation-preserving diffeomorphism and positive contact form, we have

Theorems & Definitions (63)

  • Definition 1.1: Contact form and its mass
  • Definition 1.2: Contact pair
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Remark 4.1
  • Definition 4.2
  • ...and 53 more