Hamiltonian thermodynamics on symplectic manifolds

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.

Thermodynamics and classical mechanics are traditionally formulated using very different mathematical languages—thermodynamics via phenomenological potentials and statistical mechanics, and mechanics via symplectic geometry (Hamiltonian systems). This discrepancy prevents a mathematically seamless integration of the two, limiting the rigor of advanced physical simulations and theoretical unifications.

The authors utilize the mathematical apparatus of symplectic and contact geometry. They define an extended manifold that integrates both mechanical phase space coordinates (position, momentum) and thermodynamic variables (entropy, temperature). By assigning a specific symplectic two-form to this manifold, they enforce energy conservation and the laws of thermodynamics via purely geometric constraints, specifically analyzing Lagrangian submanifolds to derive equations of state.

The paper successfully demonstrates that classical thermodynamic processes and macroscopic laws can be fully recovered from a generalized Hamiltonian framework on a symplectic manifold. It shows that thermodynamic potentials correspond to generating functions of Lagrangian submanifolds, and that entropy and temperature behave exactly as conjugate symplectic variables, preserving fundamental physical invariants under canonical transformations.

The paper is a mathematically elegant unification of two major pillars of physics. However, its abstraction level limits its immediate utility. Without explicit examples connecting the formalism to observable, real-world complex systems or providing a pathway for numerical simulation, it remains a purely theoretical exercise. The lack of treatment for non-equilibrium systems also restricts its relevance to many modern scientific challenges.