Thermodynamic Variational Principle

• The thermodynamic variational principle is a framework that expresses both equilibrium and nonequilibrium behaviors through constrained variational formulations incorporating irreversible effects.
• It extends classical mechanics by introducing nonholonomic constraints, stochastic forces, and entropy production to model dissipative dynamics accurately.
• This principle underpins modern approaches in stochastic thermodynamics, continuum mechanics, and field theory, providing a systematic method for deriving consistent evolution equations.

A thermodynamic variational principle is a mathematical framework that encodes the evolution of both equilibrium and nonequilibrium thermodynamic systems as solutions to extremal or constrained variational problems. Modern formulations generalize classical variational mechanics to consistently include dissipative, irreversible, stochastic, or open-system effects, thereby unifying mechanical and thermodynamic irreversibility. These principles underpin fundamental and applied developments in stochastic thermodynamics, continuum mechanics, extended irreversible thermodynamics, statistical mechanics, and field-theoretic approaches.

1. Unified Constrained Variational Framework

The archetypal variational formulation for thermodynamic systems extends the classical Hamilton or Lagrange principle by supplementing the action with nonholonomic (typically nonlinear or stochastic) constraints encoding irreversibility and entropy production. For a system with configuration coordinates $q$, entropy variables $S$, and a Lagrangian $L(q,\dot q,S)$, the action

$\delta\int L(q,\dot q,S)dt = 0$

is subject to two levels of constraints:

• Phenomenological constraint: Relates entropy production to friction/dissipation:

$$\partial_S L\,\dot S = \langle F^{\rm fr}(q, \dot q, S), \dot q\rangle$$

• Variational constraint: Links virtual entropy and mechanical variations:

$$\partial_S L\,\delta S = \langle F^{\rm fr}(q, \dot q, S), \delta q\rangle$$

This formalism systematically yields coupled equations for the evolution of all variables, enforces the first and second laws, and can be adapted to incorporate thermal displacements, mass transport, thermal fluxes, stochastic forces, and more complex variables for discrete, continuum, or field-theoretic systems (Carlier, 15 Oct 2024, Gay-Balmaz et al., 2019, Gay-Balmaz, 24 Feb 2025, Pino et al., 2 Oct 2025).

2. Variational Principles for Stochastic and Nonequilibrium Thermodynamics

In stochastic thermodynamics, the variational principle is realized via an extension of the Lagrange–d’Alembert structure:

• Phase space: Includes mechanical, thermal, and possible hidden variables, with conjugate thermodynamic displacements.
• Extended action: Incorporates both deterministic and stochastic virtual-work terms and enforces irreversible dynamics through nonlinear nonholonomic constraints tied to entropy production.
• Euler–Lagrange equations: Directly yield stochastic equations of motion, conservation laws, and total entropy production $\dot S_{\text{tot}} \geq 0$, enforcing the second law at the trajectory level. The mathematical structure guarantees local detailed balance and fluctuation–dissipation relations, including Onsager reciprocity, state dependence, and cross-correlated noise (Pino et al., 2 Oct 2025).

3. Metriplectic, GENERIC, and Bracket Structures

The metriplectic formulation introduces a geometric, bracket-based perspective on irreversible thermodynamics:

• Observables evolve according to coupled Poisson (reversible) and metriplectic 4-bracket (irreversible) dynamics:

$\dot F = \{F, H\} + (F, H; S, H)$

• Poisson bracket encodes Hamiltonian flow, while the metriplectic 4-bracket—constructed to guarantee the monotonic increase of entropy and energy conservation—generates irreversible effects.
• Extensibility: This bracket framework unifies finite-dimensional, discrete, Lie-group-reduced, and continuum theories, and connects directly to the GENERIC formalism of irreversible dynamics (Carlier, 15 Oct 2024, Gay-Balmaz et al., 2019).
• Irreversibility: The entropy $S$ acts as a generator of the metriplectic (dissipative) flow, and its monotonicity is structurally guaranteed.

4. Extended Irreversible Thermodynamics and Higher-Order Fluxes

Extended irreversible thermodynamics (EIT) enhances the variational approach by treating fluxes (e.g., heat, mass, viscous stresses) as independent variables:

• Nonequilibrium Lagrangian: Lagrangian densities depend on additional tensor variables representing fluxes, with corresponding nonholonomic constraints enforcing the appropriate entropy production structure.
• Objective rates: The action principle, respecting covariance or frame-indifference, results in evolution equations employing Truesdell, Lie, or other objective rates, crucial for higher-order flux dynamics (e.g., Cattaneo–Christov heat equations).
• Entropy production law: Emerges as a quadratic or higher-degree function of affinities and fluxes, with constraints or constitutive relations guaranteeing positiveness—thus enforcing the second law.
• Hierarchies: The variational framework naturally accommodates higher-order or tensorial fluxes and the associated entropy flux structures (Gay-Balmaz, 24 Feb 2025).

5. Maximum Entropy and Nonequilibrium Variational Extensions

Equilibrium thermodynamics is traditionally grounded in variational (extremal) principles for thermodynamic potentials (e.g., entropy maximization at fixed energy). In nonequilibrium steady states and phase coexistence:

• Extended entropy functional: The variational principle is enriched by nonequilibrium parameters and their conjugate variables, resulting in a global entropy functional or free energy whose extremum determines the steady state.
• Uniqueness and stationarity: In linear response or small-gradient regimes, the extended variational functional (e.g., for two-phase liquid–gas systems under steady heat conduction) is uniquely determined by the requirement of compatibility with the generalized first law and stationarity conditions, controlling interfacial parameters and phase splits (Nakagawa et al., 2021, Nakagawa et al., 15 May 2025).
• Predictive power: Such extensions yield quantitative predictions, such as shifts in interface temperature and configuration selection governed by "effective gravity" parameters in the presence of gravity and heat flow.

6. Information-Theoretic and Statistical Mechanics Perspectives

Variational principles also appear in statistical and information-theoretic settings:

• Nonlinear pressure and phase transitions: Mean-field and nonlinear thermodynamic formalism associate nonlinear variational functionals (e.g., maximizing $F(\int A\,d\rho) + h(\rho)$ over invariant measures) with equilibrium states, self-consistency equations, and phase transitions (Bru et al., 10 Nov 2025).
• Thermodynamic variational relations: Information-theoretic approaches furnish master variational inequalities (e.g., TVR) relating statistics of observables to entropy production, encompassing and extending classical thermodynamic uncertainty relations (Salazar, 2023).
• Green–function and quasipotential methods: Transport properties in disordered systems and diffusion are rigorously captured as minima of local quadratic forms, unifying computational and theoretical treatments (Trinkle, 2018).

7. Applications and Generalizations

• Field theories: The variational principle is fundamental in nonequilibrium field theory and quantum statistical mechanics, e.g., via analytically continued effective actions whose dissipative contributions are represented by sign-operator terms, achieving manifestly causal, real dissipative equations (Floerchinger, 2016).
• Open systems: In open-system thermodynamics, variational and Hamiltonian formulations encode the principle of maximum entropy generation, with dissipation-driven evolution governed by generalized Lagrangian and Hamiltonian structures (Lucia, 2011).
• Non-equilibrium fluids and beyond: Systematic methodologies via thermodynamic variational principles allow the derivation of the full Navier–Stokes–Fourier system, with consistent treatment of all dissipative couplings, and further extensions for generalized fluids, mixtures, and complex materials (Gay-Balmaz et al., 2019, Gay-Balmaz, 24 Feb 2025).

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The thermodynamic variational principle thus serves as the unifying mathematical and conceptual tool for integrating reversibility, dissipation, stochasticity, and state-dependent interactions into a structurally consistent, extensible framework across the full spectrum of classical and modern nonequilibrium thermodynamics (Carlier, 15 Oct 2024, Gay-Balmaz et al., 2019, Pino et al., 2 Oct 2025, Gay-Balmaz, 24 Feb 2025, Nakagawa et al., 2021).