Thermodynamic Lagrangian Methods

• Thermodynamic Lagrangian is a variational construct that encodes the evolution of thermodynamic states by reformulating energy, entropy, and transport in a mechanics-like language.
• It unifies equilibrium formulations—using quadratic metrics and geometric submanifolds—with nonequilibrium descriptions that incorporate doubled densities and nonholonomic constraints.
• Various formulations yield actionable transport laws and entropy production statements, bridging traditional thermodynamics with geometric and variational methodologies.

Searching arXiv for the cited work and closely related formulations of thermodynamic/Lagrangian thermodynamics. [arXiv search] Query: "thermodynamic Lagrangian thermodynamics action principle irreversible thermodynamics" A thermodynamic Lagrangian is a variational object used to encode thermodynamic evolution in a form analogous to analytical mechanics, but the literature does not attach a single universally fixed meaning to the term. In one line of work, equilibrium thermodynamic processes are treated as trajectories on a configuration space endowed with a metric, with a quadratic Lagrangian equal to the invariant distance between equilibrium states and determined from a complete set of equations of state (Vaz, 2011). In other lines of work, the relevant object is a doubled Lagrangian density for irreversible transport (Glavatskiy, 2015), a constrained Lagrangian principle with thermodynamic displacements and nonlinear nonholonomic entropy-production constraints (Gay-Balmaz et al., 2015), an augmented action for homogeneous nonequilibrium systems (Podio-Guidugli et al., 2022), or a Lagrangian 1-form whose closure expresses path independence in equilibrium thermodynamics (Yoo-Kong, 2023). Geometric reformulations connect these constructions to Lagrangian submanifolds of symplectic thermodynamic spaces and to metriplectic dynamics generated jointly by Hamiltonian and entropy (Ghosh et al., 2024, Carlier, 2024).

1. Principal meanings of the term

The expression “thermodynamic Lagrangian” refers to several structurally related, but non-identical, objects. Taken together, the literature indicates that the term is best understood as a family resemblance concept rather than a single canonical definition.

Formulation	Characteristic object	Thermodynamic role
Equilibrium state-space mechanics	Quadratic Lagrangian on thermodynamic configuration space	Distance between equilibrium states
Irreversible transport theory	Doubled Lagrangian density $L(\varphi,\varphi^*)$	Variational derivation of transport laws and entropy production
Nonequilibrium variational thermodynamics	$L(q,\dot q,S)$ with constraints	Coupled mechanical and thermodynamic evolution
Equilibrium 1-form approach	$L=\mathcal{L}_V dV+\mathcal{L}_S dS$	Closure/path independence of state functions
Metriplectic reformulation	$\dot F=\{F,H\}+(F,H;S,H)$	Reversible plus irreversible generators

In Cenalo Vaz’s equilibrium construction, thermodynamic processes are the evolution of a dynamical system on a configuration space, described by a quadratic Lagrangian and a metric, and the Lagrangian is the invariant distance between equilibrium thermodynamic states (Vaz, 2011). In the irreversible theory of linear transport, the Lagrangian density is explicitly

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$

defined on a doubled space containing a normal and a mirror-image system (Glavatskiy, 2015). In the constrained framework for nonequilibrium thermodynamics, the Lagrangian is an ordinary mechanical or continuum Lagrangian enlarged by thermodynamic variables, while irreversibility is carried by nonlinear nonholonomic constraints built from entropy production (Gay-Balmaz et al., 2015). In the equilibrium 1-form formulation, the genuine variational object is not a scalar $L(q,\dot q,t)$ but the 1-form

$L=\mathcal{L}_V dV+\mathcal{L}_S dS,$

whose line integral gives the change of a thermodynamic potential (Yoo-Kong, 2023).

A recurrent theme is that the Lagrangian structure is thermodynamic not because it always replaces the first and second laws, but because it reorganizes them into a variational statement. This suggests that the central issue is less the name of the object than the way entropy, temperature, free energy, and irreversible production enter the action principle.

2. Equilibrium formulations

The equilibrium case admits several distinct Lagrangian-style descriptions. Vaz’s formulation is explicitly mechanical in style: equilibrium thermodynamics is reformulated so that thermodynamic processes are trajectories on a configuration space, and the metric on that space is obtained from a complete set of equations of state (Vaz, 2011). The resulting quadratic Lagrangian gives an invariant notion of distance between equilibrium states. The abstract motivation is that if spacetime has microstructure, then Riemannian geometry itself may be an effective macroscopic description, and thermodynamics may provide a natural setting in which geometric structure emerges (Vaz, 2011).

A different equilibrium construction is developed through a Lagrangian 1-form. Starting from

$dU = TdS - PdV,$

the line integral of

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$

along a reversible path $\Gamma$ is identified with the change of internal energy,

$L(q,\dot q,S)$0

The central variational statement is not an equation of motion but a closure relation,

$L(q,\dot q,S)$1

interpreted as the integrability condition ensuring path independence of the state function (Yoo-Kong, 2023). In this setting, the least-action principle acts on deformations of the path in the $L(q,\dot q,S)$2 plane, and the main output is exactness rather than second-order dynamics (Yoo-Kong, 2023).

Equilibrium thermodynamics has also been recast in symplectic language. In the Hamiltonian symplectic formulation, spaces of equilibrium states are modeled as Lagrangian submanifolds of a symplectic manifold, generated by thermodynamic potentials, and thermodynamic transformations are described by Hamiltonian dynamics on thermodynamic spaces (Ghosh et al., 2024). That framework is geometric rather than action-based in the mechanics sense. It does not introduce an explicit variational thermodynamic Lagrangian, but it clarifies how equilibrium thermodynamic potentials can function as generating functions of Lagrangian submanifolds (Ghosh et al., 2024).

These approaches share a common equilibrium restriction. Reversible paths, exact differentials, and state-function structure are primary. Irreversible entropy production does not drive the formalism; instead, it is excluded or postponed.

3. Irreversible and nonequilibrium variational thermodynamics

The most direct attempts to define a thermodynamic Lagrangian for irreversible processes occur in nonequilibrium thermodynamics. In the doubled formulation of linear irreversible transport, the normal system $L(q,\dot q,S)$3 is paired with a mirror-image system $L(q,\dot q,S)$4, and the action

$L(q,\dot q,S)$5

is extremized with the Lagrangian density

$L(q,\dot q,S)$6

The resulting Euler–Lagrange equations yield the irreversible evolution equations

$L(q,\dot q,S)$7

and, with the balance law, the constitutive relations

$L(q,\dot q,S)$8

Entropy production then becomes

$L(q,\dot q,S)$9

for the two branches (Glavatskiy, 2015). The time-symmetric Lagrangian and the irreversible single-branch evolution are therefore separated conceptually.

A broader nonequilibrium framework is provided by the Lagrange–d’Alembert-type formalism with thermodynamic displacements. For a simple closed system one starts with

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$0

and imposes the variational condition

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$1

subject to the phenomenological constraint

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$2

and the variational constraint

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$3

The resulting equations are

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$4

(Gay-Balmaz et al., 2015). In this formalism, irreversibility is not represented by a separate dissipative potential alone; it is encoded in nonlinear nonholonomic constraints built from entropy production.

For homogeneous nonequilibrium systems, Podio-Guidugli and Virga formulate an augmented stationarity principle with a slow sector $L=\mathcal{L}_V dV+\mathcal{L}_S dS$5, a fast thermal variable $L=\mathcal{L}_V dV+\mathcal{L}_S dS$6, Lagrangian

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$7

and dissipation potential

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$8

Their core variational statement is

$L=\mathcal{L}_V dV+\mathcal{L}_S dS$9

leading to

$\dot F=\{F,H\}+(F,H;S,H)$0

The framework is explicitly intended to bridge analytical mechanics and nonequilibrium thermodynamics for homogeneous systems with finitely many time-dependent variables (Podio-Guidugli et al., 2022).

4. Geometric, Hamiltonian, and large-deviation structures

Several formulations reinterpret thermodynamic Lagrangians in geometric or Hamiltonian terms rather than as direct analogues of $\dot F=\{F,H\}+(F,H;S,H)$1 mechanics. In macroscopic fluctuation theory, the object playing the Lagrangian role is the dynamic large-deviation functional

$\dot F=\{F,H\}+(F,H;S,H)$2

which measures the cost of a density trajectory in a stationary nonequilibrium state (Bertini et al., 2010). Its associated Hamiltonian generates canonical equations on density–momentum space, the unstable manifold is a Lagrangian manifold, and the quasi-potential

$\dot F=\{F,H\}+(F,H;S,H)$3

acts as the natural nonequilibrium free energy (Bertini et al., 2010). Singularities arising from multiple optimal fluctuation paths are interpreted as Lagrangian phase transitions (Bertini et al., 2010).

A more explicitly algebraic synthesis is given by the metriplectic reformulation of constrained thermodynamic variational principles. There the dynamics is written as

$\dot F=\{F,H\}+(F,H;S,H)$4

with a Poisson bracket for reversible motion and a dissipative 4-bracket generated by Hamiltonian $\dot F=\{F,H\}+(F,H;S,H)$5 and entropy $\dot F=\{F,H\}+(F,H;S,H)$6 (Carlier, 2024). For simple systems, after Legendre transform from $\dot F=\{F,H\}+(F,H;S,H)$7 to $\dot F=\{F,H\}+(F,H;S,H)$8, the construction proves equivalence between the constrained thermodynamic variational equations and metriplectic evolution (Carlier, 2024). This framework shows that entropy can be a Casimir of the Poisson structure and simultaneously a generator of the dissipative sector.

The Hamiltonian symplectic approach to equilibrium thermodynamics reaches a related geometric endpoint from the opposite direction. There equilibrium spaces are Lagrangian submanifolds, and thermodynamic transformations become Hamiltonian trajectories on a symplectic manifold (Ghosh et al., 2024). The shared point is that “Lagrangian” may refer either to a variational integrand or to a geometric submanifold. The literature treats both usages as relevant, but not interchangeable.

5. Constitutive realizations and representative applications

The thermodynamic Lagrangian idea has been specialized to several constitutive settings. For viscoelasticity, a genuine Lagrangian material with an internal variable $\dot F=\{F,H\}+(F,H;S,H)$9 reproduces the Oldroyd B fluid in Eulerian form. The free energy is

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$0

the polymer stress is

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$1

and the Lagrangian flow rule yields

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$2

The internal dissipation is

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$3

and the second law holds conditionally with admissible initial internal variables

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$4

(Dret et al., 2023). By contrast, analogous Lagrangian constructions for the Zaremba–Jaumann and Oldroyd A fluids do not satisfy the second law conditionally in that formulation (Dret et al., 2023).

In stochastic nonequilibrium environments, the relevant thermodynamic Lagrangian can be an Onsager–Machlup-type path Lagrangian derived from the Fokker–Planck equation. The resulting single-particle effective Lagrangian,

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$5

encodes friction and dissipation through the environment-dependent factor $$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$6, with dissipative force

$$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$7

(Jurisch, 2020). In this usage, “thermodynamic” refers to the encoding of thermal fluctuations, dissipation, and system–environment energy exchange in a variational path-integral formalism.

Optimal control of slowly driven nonequilibrium steady states gives another concrete form: $$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$8 Here the full total entropy production is approximated by a local functional of the control protocol, and minimizing the action yields the low-dissipation protocol between steady states (Kamp et al., 17 Jun 2025). This extends equilibrium thermodynamic geometry by retaining a nonzero housekeeping entropy production term $$L = \ell(\varphi^{eq}) \nabla \varphi \cdot \nabla \varphi^* + \frac{1}{2}\left(\varphi \dot{\rho}^* - \varphi^* \dot{\rho}\right),$$9 (Kamp et al., 17 Jun 2025).

6. Terminological ambiguities and broader usage

A persistent difficulty is that “thermodynamic Lagrangian” is sometimes used literally, sometimes analogically, and sometimes only heuristically. An explicit caution comes from the reformulation of classical mechanics by analogy with thermodynamic potentials. There the sign-redefined object $L(q,\dot q,t)$0 is put in one-to-one correspondence with Helmholtz free energy, but the paper does not derive a thermodynamic action principle; it constructs a formal Legendre-transform dictionary between mechanical and thermodynamic potentials (Teruel, 2013). In that setting, calling $L(q,\dot q,t)$1 a thermodynamic Lagrangian would be structurally motivated but physically misleading (Teruel, 2013).

A comparable broadening occurs in more recent extensions outside traditional thermodynamics. In the transformer-attention model, the “thermodynamic Lagrangian” or “intelligence Lagrangian” is the action density for attention dynamics on an information manifold, with softmax identified as a stationary thermodynamic equilibrium minimizing Helmholtz free energy (Kim, 9 Feb 2026). The paper is strongest where it supplies explicit formulas for the Lagrangian, equations of motion, and free-energy minimization, and more heuristic in its discussion of scaling, grokking, and Goldstone physics (Kim, 9 Feb 2026). This suggests an expanding use of thermodynamic Lagrangian language in information geometry and machine learning.

Other domains use “Lagrangian” in yet different senses. Black-hole thermodynamics often traces thermodynamic behavior back to an underlying field-theory Lagrangian rather than defining a standalone thermodynamic Lagrangian (Diaz-Alonso et al., 2012, Hendi et al., 2015). Financial model-risk theory uses a path-integral interaction Lagrangian to encode exponential tilting of path measures in a thermodynamic analogy, but not a thermodynamic variational principle over entropy and state variables (Feng, 2019). Finite-temperature effective field theory may speak of a thermodynamic potential extracted from an effective chiral Lagrangian rather than of a separate thermodynamic Lagrangian object (Huang et al., 2011).

The most defensible general conclusion is therefore restricted. A thermodynamic Lagrangian is not a single universally standardized construction. It is a class of variational, geometric, or action-based devices by which thermodynamic state change, entropy production, or nonequilibrium fluctuation cost is represented in a Lagrangian idiom. The strongest formulations are those in which the action principle yields transport equations and entropy-production statements directly (Glavatskiy, 2015, Gay-Balmaz et al., 2015, Podio-Guidugli et al., 2022). The weakest are formal analogies in which the Lagrangian language organizes thermodynamic structure without producing a genuinely new thermodynamic dynamics (Teruel, 2013).