Hamiltonian thermodynamics on symplectic manifolds

TL;DR

This paper presents a symplectic Hamiltonian framework for thermodynamics, modeling thermodynamic processes as Hamiltonian dynamics on symplectic manifolds, with applications to ideal gases and irreversible processes.

Contribution

It introduces a novel symplectic Hamiltonian approach to thermodynamics, including explicit examples and extensions to irreversible processes and port-Hamiltonian systems.

Findings

Describes thermodynamic transformations as symplectic Hamiltonian dynamics.

Provides explicit examples with ideal gases.

Extends the framework to irreversible processes and heat transfer.

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.