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.

💡 Research Summary

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The paper presents a comprehensive symplectic‑geometric formulation of equilibrium thermodynamics and thermodynamic processes. The authors begin by identifying the space of equilibrium states of a thermodynamic system with a Lagrangian submanifold E of a 2n‑dimensional symplectic manifold (M, ω), where ω = dqᵢ∧dpᵢ in Darboux coordinates. The fundamental thermodynamic potential Φ(qᵢ) generates this submanifold through the relation dΦ = pᵢ dqᵢ, which is precisely the condition that ω restricts to zero on E. In this way, equilibrium states are encoded geometrically as points on a Lagrangian submanifold, and the potential Φ plays the role of a generating function.

Having established the geometric setting, the authors introduce Hamiltonian dynamics on (M, ω). A Hamiltonian function H∈C∞(M) defines a vector field X_H via ι_{X_H}ω = dH. If H is chosen so that it takes a constant value on the Lagrangian submanifold (i.e., H|_E = c), the flow generated by X_H stays on E while preserving H. This construction provides a symplectic analogue of the contact‑Hamiltonian description of reversible thermodynamic transformations. The authors illustrate the idea with the ideal gas: the internal energy E(S,V,N) serves as the fundamental equation, and appropriate Hamiltonians reproduce isothermal, isobaric, and isoentropic processes as restricted Hamiltonian flows.

The paper then treats Legendre transforms, which in thermodynamics correspond to changes of ensemble (e.g., from the microcanonical internal energy to the canonical Helmholtz free energy). In the symplectic picture, a (partial) Legendre transform replaces a subset of the configuration variables qᵢ by their conjugate momenta pᵢ, thereby mapping one Lagrangian submanifold to another. When the Hessian of Φ with respect to the transformed variables is non‑singular, the Legendre transform is a local diffeomorphism, guaranteeing a one‑to‑one correspondence between the two equilibrium manifolds. The authors give explicit examples for the ideal gas and for a simple interacting gas, showing how the same physical system can be described in different ensembles by different generating functions.

To address irreversible phenomena, the authors relax the condition that H be constant on E. They construct a Hamiltonian that includes a non‑conservative term, allowing the flow to leave the original Lagrangian submanifold and generate entropy. Applying this to the free expansion of an ideal gas, they recover the well‑known entropy increase ΔS > 0 while still working within the symplectic framework. This demonstrates that the symplectic approach can accommodate simple phenomenological irreversibility, extending the contact‑geometric treatments.

Finally, the paper introduces a port‑Hamiltonian extension of the formalism. By augmenting the symplectic system with input‑output ports (u, y) that represent mechanical work (pressure–volume) and heat exchange (temperature–entropy), the dynamics can be written in the standard port‑Hamiltonian form \