Quasi-Homogeneous Thermodynamic Systems

• Quasi-homogeneous thermodynamic systems are defined by a potential scaling with variable-dependent exponents, extending traditional extensivity.
• They employ generalized Euler and Gibbs-Duhem relations, requiring coupling constants to be treated as thermodynamic variables.
• Applications to black holes and cosmology use geometrothermodynamics to identify phase transitions and universal critical behavior.

A quasi-homogeneous thermodynamic system is one whose fundamental thermodynamic potential satisfies a generalized rescaling law characterized by variable-dependent scaling exponents rather than ordinary linear homogeneity. This framework extends ordinary thermodynamics to encompass systems such as black holes, where traditional extensive and intensive variable assignments break down due to intrinsic non-extensivity and geometric scaling. The quasi-homogeneous structure has deep implications for the formulation of thermodynamic laws, the interpretation of coupling constants as thermodynamic variables, the identification of physical phase transitions, and the adoption of geometric frameworks such as geometrothermodynamics for analyzing equilibrium and criticality (Quevedo, 26 Jun 2025).

1. Mathematical Definition and Core Principles

A function $\Phi(x_1,\ldots,x_n)$ is quasi-homogeneous of degree $k$ with weights $(\alpha_1,\ldots,\alpha_n)$ if under the rescaling $x_i\to\lambda^{\alpha_i}x_i$,

$$\Phi(\lambda^{\alpha_1}x_1,\ldots,\lambda^{\alpha_n}x_n) = \lambda^k \Phi(x_1,\ldots,x_n).$$

The generalized Euler theorem follows: $$\sum_{i=1}^n \alpha_i x_i \frac{\partial\Phi}{\partial x_i} = k\Phi.$$ If all weights $\alpha_i=1$ and $k=1$, one recovers simple homogeneity (extensivity). Quasi-homogeneity allows each variable to scale differently, reflecting the "scaling dimension" of each. This structure is essential for systems in which the entropy or mass–entropy relations are not linearly extensive, such as black holes and certain cosmological horizons (Quevedo, 26 Jun 2025, Quevedo et al., 2018, Romero-Figueroa et al., 8 Jan 2026, Quevedo et al., 2017, Quevedo, 2023).

2. Generalized Thermodynamic Laws and Identities

The first law and Smarr relations in quasi-homogeneous systems must be formulated in terms of the weights. For a fundamental thermodynamic potential $\Phi(E^a)$ with weights $\alpha_a$ and quasi-homogeneity degree $r$: $$\Phi(\lambda^{\alpha_1}E^1,\ldots,\lambda^{\alpha_n}E^n) = \lambda^r \Phi(E^1,\ldots,E^n).$$ Let $I_a = \frac{\partial\Phi}{\partial E^a}$. The generalized Euler relation becomes

$r\,\Phi = \sum_{a=1}^n \alpha_a E^a I_a,$

and the generalized Gibbs–Duhem identity,

$$\sum_a (\alpha_a - r) I_a dE^a + \sum_a \alpha_a E^a dI_a = 0.$$

These relations differ fundamentally from their extensive analogs. In particular, the need to introduce variable scaling weights necessitates that coupling constants—such as the cosmological constant, Born–Infeld parameter, or non-extensivity parameters—must be thermodynamic variables in the extended phase space for consistency (Quevedo, 26 Jun 2025, Quevedo et al., 2018, Quevedo, 2023, Bravetti et al., 2017).

3. Non-Extensive Entropy and Rényi Thermodynamics

The Rényi entropy,

$S_R = \frac{1}{1-q} \ln\left(\sum_i p_i^q\right),$

with $q$ the non-extensivity parameter, serves as a prototype for quasi-homogeneous entropy functions in black hole thermodynamics. In this context, the entropy is expressed as

$$S_R(S_{BH},q) = \frac{1}{q} \ln\left(1 + q S_{BH}\right),$$

which is quasi-homogeneous under simultaneous rescalings of $S_{BH}$ and $q$ with appropriate weights. The Rényi parameter $q$ is an independent thermodynamic variable whose conjugate $\Xi = \partial M/\partial q$ appears in both the generalized first law,

$dM = T\,dS + \Xi\,dq,$

and the Smarr relation,

$$M = \frac{\beta_S}{\beta_M} S\,T + \frac{\beta_q}{\beta_M}q\,\Xi.$$

This extended structure enforces consistency with the underlying quasi-homogeneous scaling, as illustrated explicitly for the Schwarzschild black hole and for more general cases (Quevedo, 26 Jun 2025).

4. Geometrothermodynamics and Curvature Singularities

Geometrothermodynamics (GTD) formalizes thermodynamics as the geometry of an equilibrium manifold, equipped with a Legendre-invariant metric uniquely determined by the quasi-homogeneity structure. For a quasi-homogeneous fundamental potential, the Legendre-invariant metric is

$$g = r\,\Phi\,\frac{\partial^2\Phi}{\partial E^c \partial E^d} dE^c dE^d.$$

Curvature singularities in this metric correspond to phase transitions, as in the Schwarzschild black hole with Rényi entropy, where the condition $e^{qS}=2$ aligns with heat-capacity divergence and stability transitions. For more general black holes and cosmological models, the GTD curvature singularities systematically signal true thermodynamic critical points, and the critical scaling of the curvature near these transitions encodes universal information about the underlying microstructure (Quevedo, 26 Jun 2025, Quevedo et al., 2018, Romero-Figueroa et al., 8 Jan 2026, Quevedo et al., 2017).

5. Applications to Black Holes and Cosmology

Quasi-homogeneous thermodynamics underpins the formulation of black hole thermodynamics, including Kerr–Newman, Kerr–AdS, Born–Infeld, and Gauss–Bonnet black holes, forcing the inclusion of coupling constants such as the cosmological constant and gauge couplings as thermodynamic variables (Quevedo, 2023, Quevedo et al., 2018). In cosmological contexts, the apparent horizon of quantum-corrected FLRW universes exhibits quasi-homogeneous entropy (e.g., arising from the Generalized Uncertainty Principle corrections), with deformation parameters treated as thermodynamic variables. This approach enables the identification of phase transitions, derivation of generalized Maxwell relations, and extraction of universal critical exponents, independent of the dimensionality of equilibrium space (Romero-Figueroa et al., 8 Jan 2026).

6. Phase Structure, Zeroth Law, and Equilibrium

In quasi-homogeneous systems, the usual Maxwell construction and identification of equilibrium points based solely on ordinary intensive variables can be inconsistent with the generalized Gibbs–Duhem identity. The equilibrium condition must be revised by introducing generalized intensive variables,

$$\widetilde{p}_i(X) \equiv [X^i]^{(\beta_i - r)/\beta_i} p_i(X),$$

which are truly homogeneous of degree zero under the scaling. Two quasi-homogeneous systems are in mutual equilibrium if and only if these generalized intensive variables coincide. This modification is essential for consistency in first-order phase coexistence and phase diagrams of black holes, altering or eliminating standard features such as swallowtail structures and van der Waals–type transitions when expressed in terms of the physical quasi-homogeneous variables (Bravetti et al., 2017).

7. Physical Implications and Universality

Quasi-homogeneous thermodynamic systems reveal that the non-extensive and non-ordinary scaling behavior of black holes and cosmological horizons originates in their fundamental geometric and statistical nature. The inclusion of non-extensivity and deformation parameters as bona fide thermodynamic variables leads to a unified, extended thermodynamic framework with generalized first laws, Smarr relations, and accurate identification of phase structure. The universality of the geometric–thermodynamic correspondence, as encoded in the quasi-homogeneous GTD approach, emerges across gravitational models, further cementing the notion that gravitational thermodynamics universally exhibits quasi-homogeneous scaling beyond ordinary extensivity (Quevedo, 26 Jun 2025, Romero-Figueroa et al., 8 Jan 2026).