Geometrothermodynamics: Unified Thermodynamic Geometry

• Geometrothermodynamics is a geometric formalism that unifies thermodynamics using contact and Riemannian geometries to describe equilibrium states and phase transitions.
• It employs Legendre invariance to maintain metric consistency under potential transformations, directly linking curvature with thermodynamic interactions.
• Applications range from ideal gases and van der Waals fluids to black hole thermodynamics and cosmology, revealing critical phenomena through curvature singularities.

Geometrothermodynamics (GTD) is a covariant geometric formalism developed to describe the thermodynamic properties of macroscopic systems in terms of differential geometry. Its core aim is to provide an invariant mathematical framework that unifies classical thermodynamics, the structure of equilibrium states, and phase transitions in a manifestly Legendre-invariant and coordinate-independent manner. Built on the fusion of contact geometry and Riemannian geometry, GTD enables a geometric interpretation of thermodynamic interactions and critical phenomena, establishing direct connections between geometric invariants—such as curvature—and thermodynamic behavior across ordinary systems, critical fluids, black holes, and cosmological models (Quevedo et al., 2011).

1. Phase Space Structure and Legendre Invariance

GTD formulates thermodynamics on a $(2n+1)$-dimensional differential manifold $\mathcal T$, the thermodynamic phase manifold, with local coordinates

$Z^A = \{\Phi,\,E^a,\,I_a\}, \quad a=1,\dots,n.$

Here, $\Phi$ is a chosen thermodynamic potential, $E^a$ are extensive variables (e.g. energy, volume, charge), and $I_a$ their conjugate intensive variables (e.g. temperature, pressure, potential). The contact structure is defined via the 1-form

$\Theta = d\Phi - I_a\,dE^a.$

The contact manifold condition, $\Theta \wedge (d\Theta)^n \ne 0$, is always satisfied for non-degenerate thermodynamics. Legendre transformations,

$$\{\Phi, E^a, I_a\} \rightarrow \{\tilde{\Phi}, \tilde{E}^a, \tilde{I}_a\},$$

implement changes of thermodynamic potential and exchange extensive and intensive variables such that the contact form becomes invariant: $$\Theta \to \tilde\Theta = d\tilde\Phi - \tilde{I}_a\,d\tilde{E}^a = \Theta$$. A Riemannian metric $G$ on $\mathcal T$ is Legendre-invariant if the functional form of $G$ is preserved under any (total or partial) Legendre transformation. Canonical classes of Legendre-invariant metrics include

$$G = (d\Phi - I_a\,dE^a)^2 + \Lambda (E^b I_b)^{2k+1}\, \delta_{ac}\, dE^a\, dI^c,$$

where $\Lambda$ is a nonzero Legendre-invariant function and $k \in \mathbb{Z}$ (Quevedo et al., 2011).

2. Equilibrium Manifold and Metric Induction

The $n$-dimensional equilibrium manifold $\mathcal E$ is defined by specifying a fundamental equation $\Phi=\Phi(E^1,\dots,E^n)$ and an embedding (harmonic map)

$$\varphi: \mathcal{E} \to \mathcal{T}, \quad E^a \mapsto (\Phi(E), E^a, I_a(E)),$$

where the pull-back of the contact form vanishes: $\varphi^*(\Theta) = d\Phi(E) - I_a(E)\,dE^a = 0$. This yields the equilibrium conditions $I_a(E) = \partial\Phi/\partial E^a$ and encodes the first law of thermodynamics. The induced metric on $\mathcal{E}$ is the pullback of $G$: $g = \varphi^*(G).$ For the canonical class with $k=-1$, this reduces to

$g_{ab} = \Lambda\, (E^c\Phi_c)^{-1} \Phi_{ab},$

or more generally,

$$g = \Lambda\,(E^c \partial_c \Phi)^{2k+1} \partial^2_{ab} \Phi\, dE^a\, dE^b.$$

This metric is Legendre-invariant by construction and encodes all thermodynamic information of the fundamental relation (Quevedo et al., 2011).

3. Geodesics, Thermodynamic Processes, and Curvature

Quasi-static thermodynamic processes are identified with geodesics on $(\mathcal{E}, g)$. The geodesic equations

$$\frac{d^2E^a}{d\tau^2} + \Gamma^a_{bc}(E) \frac{dE^b}{d\tau} \frac{dE^c}{d\tau} = 0$$

describe curves that continuously preserve the equilibrium conditions. The second law constrains physically admissible geodesics (directionality in entropy or energy). Of particular importance is the scalar curvature $R[g]$ of the equilibrium manifold:

• $R=0$ corresponds to absence of thermodynamic interaction (ideal systems).
• $R\neq 0$ signals the presence of interactions.
• Singularities of $R[g]$ generically mark phase transition points, including both first and second order transitions [(Quevedo et al., 2011); (Quevedo et al., 2010)].

4. Physical Interpretation and Examples

The GTD formalism recovers classical results and extends them:

• For the ideal gas, with $S(U,V) = (3\kappa/2) \ln U + \kappa \ln V$, the equilibrium metric is flat: $g = dU^2/U^2 + dV^2/V^2$, so $R=0$ (no microscopic interaction).
• The van der Waals gas, $$S(U,V) = (3\kappa/2) \ln (U + a/V) + \kappa \ln (V - b)$$, yields a non-flat metric; curvature singularities appear exactly on the spinodal curve $PV^3 - aV + 2ab = 0$, i.e., the loci of first-order phase transitions. Criticality is encoded in the geometry, and geodesics cannot be extended past singular points, reflecting breakdown of equilibrium (Quevedo et al., 2011).
• In general van der Waals systems, Legendre-invariant GTD metrics distinguish first- and second-order phase transitions by the locus of their respective curvature singularities, revealing full phase-transition structure not captured by Hessian metrics (Quevedo et al., 2021).

5. Generalized Homogeneity, Euler Identity, and Metric Conformal Factor

GTD must account for the homogeneity properties of the fundamental thermodynamic potential. The Euler identity is generalized for quasi-homogeneous systems: $\sum_{a} \beta_a E^a I_a = \beta_\Phi \Phi,$ with $\beta_a$ the scaling exponents of each extensive variable and $\beta_\Phi$ the degree of homogeneity. The Legendre-invariant metric's conformal factor is uniquely fixed by this identity: $g = \beta_\Phi\,\Phi(E)\, \chi_{ab}\, dE^a dE^b,$ where $\chi_{ab}$ sets the signature (e.g., $\delta_{ab}$ for signature $(+,+,\ldots)$ or $\eta_{ab}$ for $(-,+,\ldots)$). This removes ambiguity in the construction and aligns the geometric structure with the thermodynamic scaling laws (Quevedo et al., 2017).

6. Cosmological and Black Hole Applications

GTD generalizes to gravitational and cosmological systems:

• Black hole thermodynamics in higher dimensions: Curvature singularities of the equilibrium manifold coincide precisely with second-order phase transition points, e.g., divergence of heat capacities in Reissner–Nordström and Kerr black holes (Bravetti et al., 2012).
• GTD applied to homogeneous cosmology produces phenomenologically viable models (radiation, matter, dark energy, unified dark sector), with the structure of the equilibrium manifold determining cosmic evolution and connecting non-extensive thermodynamics with cosmic acceleration [(Luongo et al., 2013); (Bravetti et al., 2013); (Aviles et al., 2012)].
• In black hole binaries and systems with additional thermodynamic variables (e.g., spatial distance between horizons), GTD correctly accounts for changes in local thermodynamic stability by zeros of the Hessian determinant and related curvature singularities [(Quevedo et al., 2019); (Bravetti et al., 2013)].

7. Mathematical and Physical Critiques

The formalism of GTD has undergone scrutiny, particularly regarding the role of homogeneity and the choice of conformal factor in the equilibrium metric. If the natural variables are not strictly homogeneous of degree one, as is often the case in black hole applications, then the conformal factor and associated Euler/Gibbs–Duhem identities must be adapted. Failing to do so leads to spurious or missing curvature singularities and potential misidentification of phase transitions. Revised prescriptions clarify the correspondence between metric singularities and genuine thermodynamic critical points and tie the metric structure more closely to the choice of thermodynamic ensemble (Azreg-Aïnou, 2013).

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In summary, geometrothermodynamics provides a mathematically rigorous, Legendre-invariant geometric framework on which the entire structure of equilibrium thermodynamics, phase transitions, and thermodynamic interactions—across ordinary and gravitational systems—can be encoded and analyzed (Quevedo et al., 2011). The central objects are the contact phase structure, the Legendre-invariant metric, the equilibrium submanifold, and the associated curvature. These encodings enable a direct geometric interpretation of thermodynamic behavior and critical phenomena, with clear identification of interaction strengths and phase transitions as geometric invariants.