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Hamiltonian thermodynamics on symplectic manifolds

Aritra Ghosh, E. Harikumar

Abstract

We describe a symplectic approach towards thermodynamics in which thermodynamic transformations are described by (symplectic) Hamiltonian dynamics. Upon identifying the spaces of equilibrium states with Lagrangian submanifolds of a symplectic manifold, we present a Hamiltonian description of thermodynamic processes where the space of equilibrium states of a system in a certain ensemble is contained in the level set on which the Hamiltonian assumes a constant value. In particular, we work out two explicit examples involving the ideal gas and then describe a Hamiltonian approach towards constructing maps between related thermodynamic systems, e.g., the ideal (non-interacting) gas and interacting gases. Finally, we extend the theory of symplectic Hamiltonian dynamics to describe (a) the free expansion of the ideal gas which involves irreversible generation of entropy, and (b) a symplectic port-Hamiltonian framework for the ideal gas which is exemplified through two problems, namely, the problem of isothermal expansion against a piston and that of heat transfer between a heat bath and the gas via a thermal conductor.

Hamiltonian thermodynamics on symplectic manifolds

Abstract

We describe a symplectic approach towards thermodynamics in which thermodynamic transformations are described by (symplectic) Hamiltonian dynamics. Upon identifying the spaces of equilibrium states with Lagrangian submanifolds of a symplectic manifold, we present a Hamiltonian description of thermodynamic processes where the space of equilibrium states of a system in a certain ensemble is contained in the level set on which the Hamiltonian assumes a constant value. In particular, we work out two explicit examples involving the ideal gas and then describe a Hamiltonian approach towards constructing maps between related thermodynamic systems, e.g., the ideal (non-interacting) gas and interacting gases. Finally, we extend the theory of symplectic Hamiltonian dynamics to describe (a) the free expansion of the ideal gas which involves irreversible generation of entropy, and (b) a symplectic port-Hamiltonian framework for the ideal gas which is exemplified through two problems, namely, the problem of isothermal expansion against a piston and that of heat transfer between a heat bath and the gas via a thermal conductor.
Paper Structure (24 sections, 3 theorems, 72 equations)

This paper contains 24 sections, 3 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Motivation and results
  3. A brief review of symplectic geometry
  4. Symplectic manifolds
  5. Hamiltonian dynamics
  6. Lagrangian submanifolds
  7. Symplectic geometry of thermodynamics
  8. Change of representation
  9. Legendre transforms
  10. Hamiltonian description of thermodynamic processes
  11. Relationship with the contact-geometric framework
  12. Thermodynamic processes of the ideal gas
  13. Isochoric process of the ideal gas
  14. Isothermal-isochoric process of the ideal gas
  15. Infinitesimal transformations between systems
  16. ...and 9 more sections

Key Result

Proposition 3.1

Thermodynamic equilibrium states are described (locally) by points on a Lagrangian submanifold of a symplectic manifold with the thermodynamic potential as the generator of this submanifold.

Theorems & Definitions (6)

  • Proposition 3.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Theorem 3.2
  • Definition 7.1