Universal Thermodynamic Extremality

• Universal Thermodynamic Extremality Relation is a framework that fixes thermodynamic data via an extremum functional independent of microscopic details.
• It underpins various formulations including geometrothermodynamics, stochastic variational bounds, free-energy minimization, and black-hole thermodynamics.
• The universality emerges from invariant geometric, variational, or first-law based structures, which offer a common thread through diverse thermodynamic architectures.

“Universal thermodynamic extremality relation” denotes a family of results in which thermodynamic data are fixed by an extremum principle whose formal structure is independent of the particular system once the relevant thermodynamic framework has been specified. In the current literature, the expression does not refer to a single theorem. It instead labels several distinct but structurally analogous statements: the equilibrium manifold as an extremal hypersurface in geometrothermodynamics, variational bounds relating observable statistics to entropy production in stochastic thermodynamics, free-energy minimization and Carnot universality in generalized thermodynamic formalisms, and black-hole identities relating shifts of extremality bounds to entropy corrections (Vazquez et al., 2011, Salazar, 2023, Misra et al., 2015, Goon et al., 2019).

1. Terminological scope and recurring structure

The same expression is used across equilibrium thermodynamics, stochastic thermodynamics, information-theoretic thermodynamics, and gravitational thermodynamics. The common feature is the appearance of an extremal object—volume, free energy, entropy-production functional, transport cost, or extremal mass—whose governing relation is declared universal because it is fixed by geometric, variational, or first-law structure rather than by the microscopic details of a particular model.

Setting	Representative relation	Extremal object
Geometrothermodynamics	$\delta \mathrm{Vol}(\mathcal{E})=0$	Equilibrium-manifold volume
Path-space stochastic thermodynamics	$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$	Observable functional
Thermodynamic tradeoff hierarchy	$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$	Precision at fixed EP
Optimal transport thermodynamics	$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$	Dissipation–mobility product
Black-hole extremality	$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$	Extremal mass shift

This suggests a recurring pattern: system dependence enters through a fundamental relation, path measure, or extremal branch, whereas the extremal functional is determined by the ambient thermodynamic formalism.

2. Geometric extremality in geometrothermodynamics

In geometrothermodynamics (GTD), the starting point is the $(2n+1)$-dimensional thermodynamic phase space $\mathcal{T}$ with coordinates

$Z^A=\{\Phi,E^a,I^a\},$

equipped with the Gibbs 1-form

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$

The contact structure is invariant under Legendre transformations, and GTD requires the phase-space metric $G$ to be Legendre invariant. A central metric family is

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$0

The equilibrium manifold $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$1 is defined by an embedding

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$2

which yields

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$3

The induced metric is

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$4

Thus $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$5 is the Hessian of the thermodynamic potential multiplied by a Legendre-invariant conformal factor (Vazquez et al., 2011).

The extremality statement follows from a Polyakov-like action

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$6

whose variation yields the harmonic-map equations and the constraint that $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$7 is conformal to $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$8. Eliminating $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$9 gives the Nambu–Goto action

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$0

with Euler–Lagrange equations

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$1

equivalently $$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$2. The resulting universal GTD extremality relation is

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$3

Within this framework, every thermodynamic system specified by a smooth fundamental equation $$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$4 defines an extremal hypersurface of the Legendre-invariant phase space.

The formalism also assigns a dynamical interpretation to curves on $$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$5. The thermodynamic length

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$6

leads to the geodesic equation

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$7

and admissible geodesics are interpreted as quasi-static processes. For the ideal gas with

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$8

the induced metric becomes flat,

$$\Omega_\tau^2/\mathrm{Var}[\Theta_\tau^{\Lambda,g}] \le \Sigma_\tau/2$$9

whereas for the van der Waals gas the induced metric is curved, and the curvature scalar is nonzero. GTD therefore ties thermodynamic interaction to curvature while preserving the same underlying extremality principle (Vazquez et al., 2011).

A major delimitation is explicit in the GTD construction itself: if the induced metric becomes degenerate, as can occur at phase-transition points, or if Legendre invariance is abandoned, the extremality relation may fail or require refinement.

3. Variational extremality in nonequilibrium stochastic thermodynamics

A different usage of the term arises in path-space thermodynamics. For forward path measure $W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$0, conjugate measure $W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$1, and entropy production

$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$2

the variational representation of an $W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$3-divergence gives

$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$4

With $W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$5 and the detailed fluctuation theorem, this becomes

$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$6

The thermodynamic variational relation (TVR) is therefore

$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$7

with equality when

$W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$8

or, for differentiable $W_1(p^A,p^B)=\min \sqrt{\Sigma_\tau M_\tau}$9,

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$0

The extremality is exact: the bound is saturated by the observable aligned with the likelihood ratio implied by the fluctuation theorem (Salazar, 2023).

Choosing the $$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$1-divergence yields a universal thermodynamic uncertainty relation valid at arbitrary finite times and without parity restrictions:

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$2

Other choices of $$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$3 generate higher-order inequalities. For $$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$4 one obtains

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$5

and the Hellinger case gives

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$6

These relations universalize extremality from currents to arbitrary observables, moments, and cumulant-generating functions (Salazar, 2023).

A related but distinct hierarchy is built from a pathwise Cramér–Rao inequality. For Langevin and Markov jump dynamics, the extended thermodynamic uncertainty relation (XTUR) reads

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$7

where

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$8

By restricting the observable class, XTUR yields the conventional TUR, the entropic bound, classical speed limits, and the power-efficiency tradeoff. Equality requires

$$\partial_\varepsilon M_{\rm ext}=\lim[-T(\partial_\varepsilon S)_{M,\mathcal Q}]$$9

and the saturation conditions show that state-dependent contributions $(2n+1)$0 are generally necessary; pure current observables do not generically saturate the conventional TUR in nonequilibrium regimes (Kwon et al., 2023).

The scope of these variational extremality statements is explicit. TVR requires a detailed fluctuation theorem and absolute continuity of $(2n+1)$1 with respect to $(2n+1)$2, whereas XTUR assumes Markovian dynamics with local detailed balance and distinguishes total entropy production from pseudo-entropy production in jump processes. Hidden variables, coarse-graining, non-Markovian memory, or singular support can therefore spoil the exact bounds.

4. Extremality as minimum free energy, minimum dissipation, and maximal reversible efficiency

Another line of work identifies universal extremality with minimization principles for free energy or dissipation. In discrete optimal transport for Markov jump processes, the discrete Wasserstein-1 distance satisfies the Benamou–Brenier-type equality

$(2n+1)$3

where $(2n+1)$4 is the irreversible entropy production rate and $(2n+1)$5 is the dynamical state mobility. The immediate consequence is the product extremality inequality

$(2n+1)$6

The same structure extends to Lindblad dynamics through a pseudo-metric $(2n+1)$7,

$(2n+1)$8

and yields improved TURs, thermodynamic speed limits, and finite-time Landauer bounds. In this setting, universality refers to the fact that optimal-transport distance equals the minimum geometric mean of irreversibility and kinetic mobility for all admissible dynamics on the given graph or Hilbert space (Vu et al., 2022).

A free-energy extremality principle appears in Rényi-generalized thermodynamics. With Rényi entropy

$(2n+1)$9

escort internal energy

$\mathcal{T}$0

and the $\mathcal{T}$1-thermal state $\mathcal{T}$2 obtained from the MaxEnt construction, the nonequilibrium free energy retains the form

$\mathcal{T}$3

The extremality statement is

$\mathcal{T}$4

with equality iff $\mathcal{T}$5. This “form invariance” underlies a generalized Clausius inequality and preserves the universality of the Carnot statement. In particular, the Carnot efficiency remains

$\mathcal{T}$6

independently of the Rényi parameter once the thermodynamic quantities are defined consistently (Misra et al., 2015).

The classical Carnot theorem provides the oldest extremality relation of this type. For any reversible heat engine between reservoirs at $\mathcal{T}$7,

$\mathcal{T}$8

and equality is achieved only for reversible cycles. The temperature–entropy plane makes the universality explicit because

$\mathcal{T}$9

A stronger statement established for unconventional equations of state is that Carnot universality does not require ideal-gas asymptotics. Even anomalous cases such as

$Z^A=\{\Phi,E^a,I^a\},$0

still satisfy the Carnot theorem. The underlying geometric reason is the Maxwell/Jacobian identity

$Z^A=\{\Phi,E^a,I^a\},$1

which expresses an area-preserving map between thermodynamic planes and guarantees that the reversible efficiency depends only on reservoir temperatures (Gonzalez-Ayala et al., 2013).

Taken together, these results show that extremality may mean minimum dissipation for a prescribed state transfer, minimum free energy at fixed bath temperature, or maximum efficiency for reversible work extraction. The universality claim is always internal to the chosen thermodynamic architecture.

5. Black-hole extremality: the Goon–Penco relation and its extensions

In black-hole thermodynamics, the dominant formulation is the Goon–Penco relation. For mass $Z^A=\{\Phi,E^a,I^a\},$2, entropy $Z^A=\{\Phi,E^a,I^a\},$3, temperature $Z^A=\{\Phi,E^a,I^a\},$4, additional extensive variables $Z^A=\{\Phi,E^a,I^a\},$5, and perturbation parameter $Z^A=\{\Phi,E^a,I^a\},$6, the exact universal relation is

$Z^A=\{\Phi,E^a,I^a\},$7

A near-extremal perturbative form is

$Z^A=\{\Phi,E^a,I^a\},$8

Under mild stability assumptions, a positive leading entropy correction at fixed extensive data implies a negative leading shift of the extremal mass, which yields Weak Gravity Conjecture-like behavior in charged examples (Goon et al., 2019).

The relation has been generalized repeatedly. In dRGT massive gravity, the extremal derivatives with respect to pressure, charge, and massive-gravity couplings satisfy

$Z^A=\{\Phi,E^a,I^a\},$9

and the whole structure is unified by a triple-product identity (Chen et al., 2020). For rotating BTZ and Kerr–AdS black holes with a constant correction to the cosmological term, the standard Goon–Penco relation is accompanied by a new rotational identity,

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$0

which motivated a broader conjecture involving shifted thermodynamic quantities (Wei et al., 2020).

Several later works extend the same template to more complicated black-hole families. Accelerating AdS black holes satisfy the relation only after imposing a matching between the perturbation parameter and the conical-deficit parameter,

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$1

showing that acceleration introduces a nontrivial normalization constraint (Lee, 2024). Charged AdS black holes surrounded by quintessence admit additional extremality identities involving the quintessence normalization $\Theta=d\Phi-\delta_{ab}I^a dE^b.$2 and, in the nonlinear electrodynamics extension, the magnetic monopole parameter $\Theta=d\Phi-\delta_{ab}I^a dE^b.$3,

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$4

evaluated on the extremal branch (Chen et al., 2020).

Power-law AdS black holes in the Einstein–Power–Maxwell, Einstein–Power–Yang–Mills, and Einstein–Maxwell–Power–Yang–Mills families preserve the same structure; for the cosmological deformation used there,

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$5

in $\Theta=d\Phi-\delta_{ab}I^a dE^b.$6 (Anand, 2024). Einstein–Bel–Robinson gravity pushes the relation in a multi-parameter direction. With $\Theta=d\Phi-\delta_{ab}I^a dE^b.$7 and conjugates $\Theta=d\Phi-\delta_{ab}I^a dE^b.$8, the generalized displacement formula is

$\Theta=d\Phi-\delta_{ab}I^a dE^b.$9

and is proposed as a universal relation among displaced thermodynamic quantities (Zhen et al., 10 Jun 2025).

Higher-dimensional de Sitter black holes further extend the framework to two-horizon systems. There the first laws are written separately at the black-hole and cosmological horizons,

$G$0

and the universal extremality relation is shown to be independent of whether the Nariai point $G$1 or the cold point $G$2 is approached along the black-hole or cosmological branch. Rotating cases add shifted angular-momentum terms of the form $G$3 and lead to a more general conjectured relation involving entropy, angular momentum, charge, and pressure shifts (Zhen et al., 26 Jul 2025).

6. Conditions of validity, failures, and the meaning of “universal”

The adjective “universal” is conditional in every framework. In GTD, the extremal-hypersurface statement requires a contact manifold $G$4, a Legendre-invariant metric $G$5, a smooth equilibrium embedding with $G$6, and a nondegenerate induced metric $G$7; degeneracy at phase transitions or loss of Legendre invariance can obstruct the construction (Vazquez et al., 2011). In the TVR framework, universality requires a detailed fluctuation theorem and absolute continuity of the conjugate measure; hidden variables, coarse-graining, or non-Markovianity can weaken or invalidate the bounds, and empirical estimation of $G$8 is itself delicate (Salazar, 2023). XTUR inherits the constraints of pathwise Cramér–Rao theory, local detailed balance, and Markovianity, while the distinction between pseudo-entropy production and total entropy production limits saturation in jump processes (Kwon et al., 2023). Rényi-based Clausius and monotonicity statements also depend on parameter range: the traditional Rényi DPI is available for $G$9, the sandwiched DPI for $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$00, and commuting cases admit wider validity (Misra et al., 2015).

The black-hole literature contains explicit counterexamples. For black branes in Rastall AdS massive gravity, Einstein–Yang–Mills AdS massive gravity, and anisotropic Hořava–Lifshitz-type gravity, the relation

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$01

is generically violated, with only special limits recovering approximate or exact equality (Sadeghi et al., 2022). A different obstruction appears for Schwarzschild–de Sitter black holes corrected by generalized or extended uncertainty principles. There the tested matching formula

$$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$02

fails at the Nariai limit unless the correction parameters vanish, namely $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$03 for GUP and $$D_f(P\|P')=\sup_{\phi}\{\langle \phi-e^{-\Sigma}f^*(\phi)\rangle\}$$04 for EUP (Zhao et al., 26 Jan 2025).

This suggests that “universal” should not be read as “valid in every thermodynamic model.” It denotes invariance with respect to working substance, observable parity, choice of thermodynamic potential, or microscopic realization within a fixed structural setting. Once the structural hypotheses change—Legendre invariance, detailed fluctuation symmetry, first-law ensemble, horizon structure, correction scheme, or admissible dynamics—the specific extremality relation can change or disappear altogether.