Irreversible Thermodynamic Potential

• Irreversible thermodynamic potentials are scalar functions or functionals that characterize nonequilibrium states by tracking dissipation and entropy production.
• They operationally govern force–flux relationships and appear in diverse formulations from classical linear theories to stochastic and quantum frameworks.
• Their formulations extend equilibrium potentials like Helmholtz and Gibbs energies, providing bounds on work reversal and optimal finite-time driving.

Searching arXiv for recent and foundational papers on irreversible thermodynamic potentials across classical, stochastic, quantum, and finite-time formulations. I’ll inspect a focused set of papers that directly treat irreversible thermodynamic potentials or closely related Lyapunov, free-energy, action, and entropy-production functionals. An irreversible thermodynamic potential is a scalar function or functional that characterizes directed nonequilibrium evolution by being monotonic along the dynamics, by generating thermodynamic forces through gradients or variational derivatives, or by quantifying dissipated work, entropy production, or recoverability. In the literature it is not a single universally fixed object. In linear irreversible thermodynamics it is embodied in entropy and in Onsager–Machlup functionals; in stochastic nonequilibrium steady-state theory it appears as a Lyapunov functional such as a relative-entropy-like quantity; in quantum thermodynamics it is often realized as a free-energy difference or a relative-entropy functional; and in finite-time theories it can take an action-like form that bounds the product of dissipation and duration (Cordoba et al., 2013, Tomé et al., 27 Jan 2025, Alhambra et al., 2015, Hanel, 2016, Hanel et al., 2020).

1. Conceptual scope and defining features

In equilibrium thermodynamics, familiar potentials such as Helmholtz and Gibbs free energies are state functions whose extrema characterize equilibrium under specified constraints. Out of equilibrium, several papers extend this logic by introducing monotonic quantities that organize relaxation toward equilibrium or toward a nonequilibrium steady state. In this sense, an irreversible thermodynamic potential is the nonequilibrium analogue of an equilibrium potential: it is a function or functional that tracks distance from the stationary state, determines the direction of spontaneous evolution, or encodes dissipation through its time derivative (Tomé et al., 27 Jan 2025).

Two recurrent features appear across otherwise different frameworks. First, the potential is tied to entropy production. A monotonic increase or decrease is typically equivalent to nonnegative entropy production, excess entropy production, or free-energy dissipation. Second, the potential is often operational rather than purely state-theoretic: it may govern path probabilities, appear in a variational principle for optimal finite-time driving, or bound the work cost of reversing a process (Cordoba et al., 2013, Alhambra et al., 2015, Rolandi, 2024).

This breadth produces several technically distinct realizations. In stochastic thermodynamics the relevant object can be a relative-entropy functional on probability space. In continuum theories it can be a free-energy density whose gradients generate chemical potentials, stresses, and affinities. In open quantum systems it can be a non-equilibrium free energy or dissipated work functional. In finite-time thermodynamics it can be an action-like quantity such as $\tau\,\Delta W_{\mathrm{irrev}}$ or a thermodynamic length controlling minimal dissipation (Sahu et al., 2017, Santillan et al., 2010, Rolandi, 2024).

2. Linear irreversible thermodynamics and Onsager–Machlup structure

A canonical formulation appears in classical irreversible thermodynamics near a stable equilibrium. For a single extensive variable $y$, entropy is expanded as

$$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$

so the thermodynamic force is

$Y=\frac{dS}{dy}\approx -s\,y,$

and the flux is $\dot y=dy/dT$, with $T$ thermodynamic time. In the linear regime the Onsager relation is

$\dot y=L\,Y,\qquad L>0,$

or equivalently $Y=R\,\dot y$ with $R=L^{-1}$. The entropy production rate is then

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$

which fixes the thermodynamic arrow of time (Cordoba et al., 2013).

Onsager–Machlup theory elevates this local force–flux structure into a path-space description. The conditional probability for a thermodynamic trajectory is weighted by

$y$0

with thermodynamic Lagrangian

$y$1

Here $y$2 has dimensions of entropy per unit time, is quadratic in flux and force, and suppresses strongly dissipative fluctuations through the exponential weight. In this formulation the irreversible thermodynamic potential is no longer just $y$3; it is the combined entropy-based Lagrangian structure that governs relaxation and fluctuations (Cordoba et al., 2013).

The associated thermodynamic momentum is

$y$4

and the Hamiltonian is

$y$5

It generates thermodynamic time evolution through

$y$6

Because this evolution acts on Banach spaces of probability densities rather than on a Hilbert space with a scalar product, it is nonunitary and encodes irreversibility directly. In this setting, entropy, $y$7, and $y$8 jointly realize the irreversible thermodynamic potential of the linear theory (Cordoba et al., 2013).

3. Lyapunov functionals, excess entropy production, and nonequilibrium steady states

In stochastic thermodynamics, irreversible thermodynamic potentials are usually formulated as Lyapunov functionals on probability space. For a continuous-time Markov process with probabilities $y$9 and transition rates $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$0,

$$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$1

the Gibbs–Shannon entropy is

$$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$2

and the entropy production rate is the Schnakenberg expression

$$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$3

Tomé and de Oliveira show that $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$4 is upward convex in the probability vector and, because the macroscopic fluxes $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$5 are linear functionals of $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$6, also convex in the fluxes. This convexity yields the Glansdorff–Prigogine inequality and the nonnegativity of the excess entropy production $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$7 (Tomé et al., 27 Jan 2025).

For a stationary nonequilibrium state with stationary distribution $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$8, stationary fluxes $$S(y)=S_0-\frac{1}{2}s\,y^2+\dots,\qquad s=\left.\frac{d^2S}{dy^2}\right|_{y=0}>0,$$9, and stationary entropy production $Y=\frac{dS}{dy}\approx -s\,y,$0, the excess entropy production is defined macroscopically by

$Y=\frac{dS}{dy}\approx -s\,y,$1

and microscopically by

$Y=\frac{dS}{dy}\approx -s\,y,$2

The crucial Lyapunov functional is

$Y=\frac{dS}{dy}\approx -s\,y,$3

which satisfies

$Y=\frac{dS}{dy}\approx -s\,y,$4

and reaches its extremal value $Y=\frac{dS}{dy}\approx -s\,y,$5 at the stationary state. Here $Y=\frac{dS}{dy}\approx -s\,y,$6 measures a Kullback–Leibler-type distance to the nonequilibrium steady state and is the clearest stochastic realization of an irreversible thermodynamic potential (Tomé et al., 27 Jan 2025).

In the special isothermal case without temperature gradients, this microscopic Lyapunov functional becomes a macroscopic thermodynamic potential,

$Y=\frac{dS}{dy}\approx -s\,y,$7

with

$Y=\frac{dS}{dy}\approx -s\,y,$8

The stationary state then extremizes $Y=\frac{dS}{dy}\approx -s\,y,$9, and $\dot y=dy/dT$0 differs from $\dot y=dy/dT$1 only by a constant. This gives a direct bridge between nonequilibrium stochastic dynamics and a potential built from macroscopic variables $\dot y=dy/dT$2, $\dot y=dy/dT$3, and $\dot y=dy/dT$4 (Tomé et al., 27 Jan 2025).

A related statistical-mechanical construction appears in multiscale Markov systems. There the non-equilibrium free energy

$\dot y=dy/dT$5

acts as a Lyapunov functional even without detailed balance, and under time-scale separation it reduces to the coarse-grained free energy

$\dot y=dy/dT$6

In this setting the free energy remains invariant under coarse-graining over fast variables, whereas the internal energy and entropy each acquire identical fast entropic contributions (Santillan et al., 2010).

For reaction networks in the limit where some reactions become irreversible, Gorban and Yablonsky derive an extended principle of detailed balance. The reversible sector obeys classical detailed balance, while the irreversible sector admits a linear Lyapunov functional

$\dot y=dy/dT$7

with

$\dot y=dy/dT$8

Here $\dot y=dy/dT$9 is a genuinely irreversible potential for the limit kinetics, even though the classical free energy diverges when some equilibrium concentrations tend to zero (Gorban et al., 2012).

4. Quantum, semiclassical, and information-theoretic formulations

Quantum thermodynamics reformulates irreversibility through free energy, relative entropy, and recoverability. In the resource-theoretic setting of thermal operations or Gibbs-preserving maps, the irreversible content of a transition $T$0 is quantified by the free-energy drop

$T$1

with

$T$2

Using the Gibbs state $T$3, this becomes

$T$4

This quantity is nonnegative, bounds deterministic work extraction, and also bounds how well the process can be reversed without work investment (Alhambra et al., 2015).

The same work makes the operational meaning explicit by introducing a recovery channel $T$5, which for thermal operations is itself a thermal operation: $T$6 The main inequality is

$T$7

Thus the irreversible potential is simultaneously a free-energy loss, a lower bound on reversal error, and—when the standard free energy is sufficient—a measure of work gain in the forward process (Alhambra et al., 2015).

In fully microscopic finite-time quantum thermodynamics, the same role is played by entropy production

$T$8

with non-equilibrium free energy

$T$9

The thesis literature further converts this dissipative potential into a geometric object: $\dot y=L\,Y,\qquad L>0,$0 where $\dot y=L\,Y,\qquad L>0,$1 is a thermodynamic metric on the control manifold. The corresponding thermodynamic length $\dot y=L\,Y,\qquad L>0,$2 satisfies

$\dot y=L\,Y,\qquad L>0,$3

so geodesic distance becomes an irreversible potential for optimal finite-time protocols (Rolandi, 2024).

A different quantum interpretation appears in the thermodynamics–quantum duality of the linear Onsager regime. Under the identifications

$\dot y=L\,Y,\qquad L>0,$4

the thermodynamic path integral and the thermodynamic evolution operator map into the Feynman path integral and the Schrödinger propagator. This establishes a semiclassical correspondence in which the irreversible thermodynamic Lagrangian and Hamiltonian play the role of the classical action and Hamiltonian of quantum mechanics (Cordoba et al., 2013).

5. Finite-time dissipation, action principles, and speed limits

A distinct class of irreversible thermodynamic potentials is action-like rather than free-energy-like. In gas-kinetic finite-time thermodynamics, Hanel derives a thermodynamic action principle for an ideal gas in a cylinder with a moving piston. For any thermodynamic cycle there exists a constant action $\dot y=L\,Y,\qquad L>0,$5 such that

$\dot y=L\,Y,\qquad L>0,$6

The quantity $\dot y=L\,Y,\qquad L>0,$7 depends on molecular mass, geometry, and temperature, but not on particle number $\dot y=L\,Y,\qquad L>0,$8 or cycle period $\dot y=L\,Y,\qquad L>0,$9. It therefore acts as a cycle-specific irreversible potential measured in action units (Hanel, 2016).

For an isothermal compression–expansion cycle,

$Y=R\,\dot y$0

while for an adiabatic cycle

$Y=R\,\dot y$1

These relations show explicitly how finite-time irreversibility scales with microscopic parameters and with cycle geometry. The same framework yields a finite-time correction to Landauer erasure,

$Y=R\,\dot y$2

so the irreversible potential is the sum of the quasi-static Landauer term and an action-controlled finite-time term (Hanel, 2016).

Hanel and Jizba sharpen this into a time–irreversible-work uncertainty principle for ideal-gas heat engines,

$Y=R\,\dot y$3

and for Carnot-like cycles

$Y=R\,\dot y$4

The lower-bound constant has the dimension of an action and becomes comparable to Planck’s constant when the characteristic length scale is of the order of the Bohr radius (Hanel et al., 2020).

In finite-time quantum thermodynamics the action-like picture reappears geometrically. For Landauer erasure beyond weak coupling, the heat cost satisfies

$Y=R\,\dot y$5

with $Y=R\,\dot y$6 in the resonant-level model studied there. This makes the Planckian time a lower speed scale for optimal erasure and converts the irreversible potential into an explicit finite-time correction to Landauer’s bound (Rolandi, 2024).

These formulations suggest a broad dichotomy. In some theories the irreversible thermodynamic potential is a state function or relative-entropy functional; in others it is a process functional proportional to dissipation integrated over time or to thermodynamic length. A plausible implication is that nonequilibrium thermodynamics has at least two complementary “potential” structures: one organizing stationary-state relaxation and another constraining finite-time driving (Hanel, 2016, Rolandi, 2024).

6. Representative domain-specific realizations

The concept acquires different concrete meanings in different subfields.

Domain	Potential-like quantity	Thermodynamic role
Curved lipid membranes	Helmholtz free-energy density $Y=R\,\dot y$7	Generates stresses, chemical potentials, and affinities (Sahu et al., 2017)
Thermoelectric conversion	Voltage $Y=R\,\dot y$8 with Guy–Stodola correction	Reversible work minus lost work $Y=R\,\dot y$9 (Markvart, 7 Jul 2026)
Open-system cosmology	Creation pressure $R=L^{-1}$0, $R=L^{-1}$1, or $R=L^{-1}$2	Encodes irreversible matter creation and entropic dark energy (Gohar, 8 Jul 2025, Harko et al., 2012)
Photon-driven biospheres	Photon affinities $R=L^{-1}$3	Drive pigment proliferation and entropy production (Michaelian, 2013)
Non-isothermal Brownian motion	$R=L^{-1}$4, $R=L^{-1}$5, and $R=L^{-1}$6	Capture cumulative irreversibility even when rates vanish (Taye, 8 Sep 2025)

In curved membrane thermodynamics, the relevant potential is the Helmholtz free-energy functional. For a single-component membrane one writes $R=L^{-1}$7, or equivalently $R=L^{-1}$8, and for multicomponent systems $R=L^{-1}$9 also depends on composition fields. Its derivatives generate in-plane stresses, bending moments, chemical potentials, and reaction affinities. Irreversibility then enters through entropy production

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$0

with forces derived from variational derivatives of $\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$1 and fluxes related to them by linear irreversible laws (Sahu et al., 2017).

In thermoelectric energy conversion, the paper on Thomson irreversibility treats the open-circuit thermoelectric generator as a heat engine and identifies the electrochemical potential difference as the relevant macroscopic potential. The produced work per carrier is

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$2

while irreversibility introduces the reduction

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$3

Here voltage is an irreversible thermodynamic potential in the Guy–Stodola sense: a reversible maximum reduced by entropy generation proportional to the Thomson coefficient (Markvart, 7 Jul 2026).

In open-system cosmology, irreversible matter creation enters through the creation pressure

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$4

or, in the adiabatic-creation notation,

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$5

Recent horizon-thermodynamic models additionally construct an effective entropic dark-energy density

$\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$6

derived from a generalized horizon entropy $\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$7 and the Hawking temperature $\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$8. In this context the irreversible potential is distributed among $\frac{dS}{dT}=Y\dot y=L\,Y^2\ge 0,$9, $y$00, and $y$01, all of which encode nonequilibrium driving of the cosmic expansion (Gohar, 8 Jul 2025, Harko et al., 2012).

Michaelian’s nonlinear thermodynamic account of pigment proliferation shifts the notion from chemical free energies to radiative affinities. The relevant potentials are the photon affinities

$y$02

which drive conversion of solar photons into terrestrial and cloud-top spectra and the production of pigments. The central claim is that dissipative structures are sustained by ongoing dissipation of such generalized potentials rather than by minimization of equilibrium free energy (Michaelian, 2013).

Underdamped Brownian motion with spatially varying temperature provides a different cautionary example. There the instantaneous entropy production and entropy extraction rates can decay to zero at long times even for a free particle in a temperature gradient, yet the cumulative quantities remain finite. Exact identities lead to

$y$03

for a particle crossing from hot to cold in several temperature profiles, and the free energy change is correspondingly

$y$04

This shows that cumulative heat and entropy functionals, rather than instantaneous rates alone, may be the correct irreversible potentials in non-isothermal underdamped systems (Taye, 8 Sep 2025).

Across these realizations, the common thread is structural rather than notational. An irreversible thermodynamic potential is whatever quantity simultaneously encodes directionality, dissipation, and admissible asymptotic behavior for the nonequilibrium dynamics under study. In some theories that quantity is a free energy, in others a relative entropy, a Lagrangian, a creation pressure, an affinity, a cumulative heat functional, or an action-like bound. The term therefore denotes a family of mathematically distinct but functionally analogous objects rather than a single canonical scalar.