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A Contact Topological Glossary for Non-Equilibrium Thermodynamics

Michael Entov, Leonid Polterovich, Lenya Ryzhik

TL;DR

This work develops a geometric framework for non-equilibrium thermodynamics by encoding thermodynamic equilibria as Legendrian submanifolds in a jet-space (extended and reduced) phase space and interpreting non-equilibrium transformations as non-negative paths or Reeb chords. It establishes a direct link between slow, quasi-static evolutions (non-negative Legendrian isotopies) and ultrafast processes (Reeb chords), with a partial order on Legendrians encoding thermodynamic constraints. The authors provide general theorems connecting slow global connections to the existence of ultrafast, energy-non-increasing jumps and illustrate the theory with explicit ultrafast processes in ideal-gas/Stirling-like settings and Curie-Weiss magnet models, deriving chord conditions and energy implications. The results offer a topological toolkit for analyzing non-equilibrium thermodynamics and suggest experimental interpretations for ultrafast transitions, including Stirling-engine segments and magnetic demagnetization phenomena.

Abstract

We discuss the occurrence of some notions and results from contact topology in the non-equilibrium thermodynamics. This includes the Reeb chords and the partial order on the space of Legendrian submanifolds.

A Contact Topological Glossary for Non-Equilibrium Thermodynamics

TL;DR

This work develops a geometric framework for non-equilibrium thermodynamics by encoding thermodynamic equilibria as Legendrian submanifolds in a jet-space (extended and reduced) phase space and interpreting non-equilibrium transformations as non-negative paths or Reeb chords. It establishes a direct link between slow, quasi-static evolutions (non-negative Legendrian isotopies) and ultrafast processes (Reeb chords), with a partial order on Legendrians encoding thermodynamic constraints. The authors provide general theorems connecting slow global connections to the existence of ultrafast, energy-non-increasing jumps and illustrate the theory with explicit ultrafast processes in ideal-gas/Stirling-like settings and Curie-Weiss magnet models, deriving chord conditions and energy implications. The results offer a topological toolkit for analyzing non-equilibrium thermodynamics and suggest experimental interpretations for ultrafast transitions, including Stirling-engine segments and magnetic demagnetization phenomena.

Abstract

We discuss the occurrence of some notions and results from contact topology in the non-equilibrium thermodynamics. This includes the Reeb chords and the partial order on the space of Legendrian submanifolds.
Paper Structure (14 sections, 4 theorems, 83 equations, 5 figures, 1 table)

This paper contains 14 sections, 4 theorems, 83 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Thermodynamic equilibria as the Legendrian sub-manifolds
  3. Microscopic and macroscopic thermodynamic states
  4. A geometric interpretation of the equilibria and the extended thermodynamic phase space
  5. The reduced thermodynamic phase space
  6. Positivity of the thermodynamic paths
  7. The thermodynamic paths in the extended thermodynamic phase space
  8. The thermodynamic paths in the reduced thermodynamic phase space
  9. Classification of thermodynamic processes
  10. Slow global processes and partial order on Legendrians
  11. Ultrafast processes as Reeb chords
  12. An isochoric temperature and pressure jump
  13. A temperature and magnetic field jump with constant magnetization
  14. An ultrafast (de)magnetization for the Curie-Weiss model

Key Result

Theorem 5.1

For any (open or closed) smooth connected manifold $X$, the class $[0_X]$ of the zero section $0_X$ in $J^1 X= T^*X \times {\mathbb{R}}$ is orderable. If two distinct equilibrium Legendrians $\Lambda_0,\Lambda_1 \in [0_X]$ are related by the partial order, $\Lambda_0 \preceq \Lambda_1$, there is no

Figures (5)

  • Figure 1: Ultrafast process for $T_0=1,T_1=5,c=2$
  • Figure 2: The Stirling engine cycle, with $T_0=T_C<T_H=T_1$.
  • Figure 3: Reeb chord for $T_0=1,T_1=5,c=2$
  • Figure 4: $b=1,\beta_0 = 0.5, \beta_1 = 0.3, c= 1$
  • Figure 5: Interpolation and gluing in $(Z,Q)$-plane

Theorems & Definitions (20)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Example 3.3
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 5.1
  • ...and 10 more