Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

💡 Research Summary

The paper presents a geometric framework for constructing structure‑preserving numerical integrators for adiabatically closed simple thermodynamic systems. Building on the variational formulation of nonequilibrium thermodynamics introduced by Gay‑Balmaz and Yoshimura, the authors adopt partially cosymplectic structures—a generalisation of cosymplectic geometry where the two‑form ω is closed but the one‑form η need not be—to encode both the symplectic part of the mechanical variables and the irreversible thermodynamic processes.

In the continuous setting, the phase space is M = T*Q × ℝ, with coordinates (qᵢ, pᵢ, S). The Hamiltonian H(q,p,S) determines the temperature through ∂H/∂S, assumed non‑zero, and the friction forces are represented by a semibasic one‑form F_fr. The pair (ω, η) with ω = dqᵢ∧dpᵢ and η = −∂H/∂S dS − F_fr constitutes a partially cosymplectic structure. The Reeb vector field R satisfies ι_R ω = 0, ι_R η = 1, and the evolution vector field E_{H,F_ext} is defined by the relation ♭(ω,η)(E_{H,F_ext}) = dH + η − F_ext. In coordinates this yields the familiar set of first‑order ODEs for (q, p, S) that combine Hamiltonian dynamics with irreversible entropy production.

Switching to the Lagrangian picture, a regular Lagrangian L(q, v, S) on TQ × ℝ is Legendre‑transformed to the Hamiltonian side. Pull‑back of ω and η via the Legendre map gives ω_L and η_L, which again form a partially cosymplectic pair on the tangent bundle. The resulting thermodynamic Euler–Lagrange equations read
d/dt(∂L/∂vᵢ) = ∂L/∂qᵢ + F_{fr,i} + F_{ext,i}, ∂L/∂S · Ṡ = ṙᵢ F_{fr,i},
where the second equation couples entropy change to the work done by friction.

The core contribution is an discrete variational principle for such systems. The time interval is partitioned into N steps, and a discrete Lagrangian L_d(q_k,q_{k+1},S_k,S_{k+1}) together with a discrete one‑form η_d are introduced. By extremising the discrete action sum, the authors obtain discrete thermodynamic Euler–Lagrange equations that mimic the continuous equations while preserving the underlying partially cosymplectic structure. The associated discrete flow Φ_d is shown to be symplectic on the mechanical subspace and to respect the η‑distribution, guaranteeing that the discrete dynamics respects the same geometric constraints as the continuous model.

A discrete version of Noether’s theorem is proved: any discrete symmetry of the discrete Lagrangian and η_d yields a conserved quantity along the discrete flow. For time‑translation invariance this leads to a discrete analogue of the total energy‑entropy balance, confirming that the integrator does not artificially create or destroy thermodynamic potentials.

Four numerical experiments illustrate the method’s performance:

1. Damped harmonic oscillator – the integrator reproduces the expected exponential decay of mechanical energy while correctly accounting for entropy increase due to friction.
2. Ideal gas in a piston – the scheme conserves the combined mechanical‑thermodynamic energy and reproduces the adiabatic relation between pressure, volume, and entropy.
3. Van der Waals gas in a piston – despite the nonlinear equation of state and strong coupling between volume and temperature, the discrete flow respects the thermodynamic constraints and remains stable over long integration times.
4. Expansion against external pressure – the discrete Noether theorem correctly captures the work done on the environment and the associated entropy production.

In each case the geometric integrator outperforms standard non‑structure‑preserving methods, showing superior long‑term stability and accurate preservation of the thermodynamic balance laws.

The authors compare their approach with the earlier variational integrators of Gay‑Balmaz and Yoshimura. While the latter rely on a GENERIC framework and often introduce two entropy variables, the present method stays within the partially cosymplectic formalism, allowing the direct use of classical geometric‑mechanics tools such as reduction, momentum maps, and cotangent bundle techniques. The paper also outlines future directions: extending the framework to systems with multiple entropy variables, incorporating matter exchange, developing coisotropic reduction and Hamilton–Jacobi theory for thermodynamic systems, and applying the method to more complex, possibly field‑theoretic, models.

Overall, the work provides a rigorous, geometrically grounded recipe for constructing variational integrators that respect both the symplectic structure of the mechanical variables and the irreversible thermodynamic processes encoded in the η‑form. It bridges the gap between abstract geometric thermodynamics and practical numerical simulation, offering a valuable tool for researchers needing accurate, long‑time integration of nonequilibrium thermodynamic systems.