Geometrothermodynamics: A Geometric Approach

• Geometrothermodynamics is a formalism that unifies differential geometry with thermodynamics through Legendre invariant metric structures in an extended phase space.
• Its equilibrium manifold is defined via a pullback metric that encodes thermodynamic response functions and identifies phase transitions through curvature singularities.
• The framework assigns geometric meaning to thermodynamic interactions, linking flat curvature to ideal systems and divergent curvature to phase criticality in both ordinary and black hole thermodynamics.

Geometrothermodynamics (GTD) is a formalism that unifies differential geometry and thermodynamics through the construction of Legendre invariant metric structures on an extended thermodynamic phase space. GTD generalizes the geometric approach to thermodynamics by promoting Legendre invariance as a fundamental principle, thereby ensuring that the thermodynamic description is independent of the choice of potential. The curvature properties of the equilibrium manifold in GTD give geometric meaning to thermodynamic interaction and encode singularities at phase transition points. This framework provides a geometric diagnostic that unifies the paper of both ordinary and black hole thermodynamics, rigorously connecting geometric singularities to physical critical phenomena.

1. Legendre Invariant Geometric Structure

The foundational element of GTD is the construction of a $(2n+1)$-dimensional phase space $\mathcal{T}$ endowed with a contact 1-form $\Theta$ and a Legendre invariant Riemannian metric $G$. The standard contact structure is given in local coordinates as

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

where $\Phi$ is a thermodynamic potential, $E^a$ are extensive variables, and $I^a$ are their conjugate intensive variables. Legendre transformations, which exchange pairs $(E^a, I^a)$ and modify $\Phi$ accordingly, leave both $\Theta$ and $G$ invariant. This invariance is realized by constructing $G$ to have the generic form: $$G = (\Theta)^2 + (\xi_{ab} E^a I^b)(\chi_{cd} dE^c dI^d)$$ with suitable choices for the constant matrices $\xi$ and $\chi$ to match the desired type of phase transitions. The requirement of Legendre invariance ensures that the thermodynamic interpretation is independent of representation, aligning the geometric formulation with the physical principle that thermodynamic potentials can be freely exchanged without physical consequence (Quevedo et al., 2010, Quevedo et al., 2011, Bravetti et al., 2013).

2. Equilibrium Manifold and Pullback Metric

The full thermodynamic description is encoded in the equilibrium manifold $\mathcal{E}$, an $n$-dimensional submanifold of $\mathcal{T}$ specified via the smooth embedding map

$$\varphi: \{ E^a \} \mapsto \{ \Phi(E^a), E^a, I^a(E^a) \}$$

together with the pullback condition $\varphi^*(\Theta) = 0$. This condition enforces the first law of thermodynamics and the Gibbs relations

$$d\Phi = I_a dE^a, \qquad I_a = \frac{\partial\Phi}{\partial E^a}.$$

A metric $g$ is induced on $\mathcal{E}$ via the pullback: $g = \varphi^*(G),$ which explicitly incorporates both the structure of the underlying thermodynamic potential and the choice of Legendre invariant metric. For instance, with the pseudo-Euclidean signature appropriate for second order phase transitions, the induced metric becomes

$$g = (\xi_{ab} E^a \frac{\partial \Phi}{\partial E^b}) (\chi_a^c \frac{\partial^2 \Phi}{\partial E^c \partial E^d} dE^a dE^d).$$

This geometric construction ensures all thermodynamic response functions and critical behavior are encoded in the curvature properties of $g$ (Quevedo et al., 2010, Quevedo et al., 2011, Bravetti et al., 2012).

3. Curvature and the Geometric Signature of Thermodynamic Interaction

The geometric interpretation of thermodynamic interaction in GTD is founded on the curvature of the equilibrium manifold. A vanishing scalar curvature characterizes noninteracting (ideal) systems, as demonstrated for the ideal gas, where the induced metric is explicitly flat after coordinate transformation: $g = dU^2/U^2 + dV^2/V^2.$ Nonzero curvature indicates the presence of effective microscopic interactions, with curvature singularities corresponding to phase transitions:

• In black hole thermodynamics (e.g., Kerr-Newman, Reissner-Nordström), the scalar curvature $R$ computed from $g$ is of the structure

$R = \frac{N}{D},$

where $D$ is a function whose zeros coincide with the divergence points of the heat capacity or other response functions, directly marking second order phase transitions.

• For ordinary fluids (like the van der Waals gas), GTD captures both first and second order transitions by adopting appropriate metric signatures; the curvature diverges at points determined by the equations governing phase coexistence or instability (e.g., $PV^3 - 3V + 2 = 0$ for van der Waals) (Quevedo et al., 2010, Quevedo et al., 2021).

This close identification between curvature singularities and phase transitions is a central achievement of GTD, confirming that these geometric features serve as robust indicators of thermodynamic criticality.

4. Classification of Phase Transitions via Metric Structure

GTD distinguishes first and second order phase transitions by the choice of the Legendre invariant metric:

• Pseudo-Euclidean Metrics: Appropriately encode second order transitions, as required for black holes and other systems where the heat capacity diverges.
• Euclidean Metrics: Specially constructed to capture first order transitions, as found in realistic fluids.
• Curvature Singularities: In both cases, the singular locus of the scalar curvature is shown to coincide with the loci where classical thermodynamic criteria for phase transitions are satisfied (e.g., divergence of heat capacity for second order, non-analyticity of the free energy for first order).

For concrete systems, the explicit forms of the denominators in the curvature expressions are always functions whose zeros indicate phase criticality: $$D = [2M^6 - 3M^4 Q^2 - 6M^2 J^2 + Q^2 J^2 + 2(M^4 - M^2 Q^2 - J^2)^{3/2}]^2$$ in the Kerr-Newman case, with its vanishing marking the phase transition point (Quevedo et al., 2010).

5. Applications: Black Hole Thermodynamics and Ordinary Systems

GTD provides a unified geometric analysis of a diverse array of systems:

• Black Holes: The formalism is rigorously applied to Reissner-Nordström, Kerr-Newman, and other black holes for both second and first order transitions, using the metric structure to identify the phase transitions and the nature of microscopic interactions. All critical points identified geometrically correspond exactly to the points of divergent heat capacity or classical instability (Quevedo et al., 2010, Quevedo et al., 2011, Bravetti et al., 2012).
• Ordinary Fluids: For van der Waals fluids and the 1D Ising model, the induced curvature singularities recover the full phase transition behavior, whereas the ideal gas metric remains flat, serving as a geometric separatrix between interacting and noninteracting systems (Quevedo et al., 2010, Quevedo et al., 2011).
• Thermodynamic Representation Invariance: Both entropy and energy representations yield consistent results for phase transitions, a consequence of Legendre invariance in metric construction.

These applications demonstrate that GTD's geometric characterization is not restricted to any particular class of systems, but is robust across gravitational and standard thermodynamic contexts.

6. Broader Implications and Extensions

By establishing the geometric meaning of thermodynamic interaction and phase transitions through Legendre invariant structures, GTD offers the following implications:

• Unified Criterion for Interaction and Criticality: Curvature provides a universal geometric marker of interaction strength (vanishing for the ideal gas, nonzero and singular at transitions in interacting systems).
• Phase Space Geometry and Information Encoding: Choice of pseudo-Euclidean versus Euclidean metric structures encodes information about the order of possible phase transitions; thus, the type of geometric signature mirrors physical critical behavior.
• Extension to Non-equilibrium: Although GTD, as formulated, addresses equilibrium thermodynamics, curvature singularities signal breakdowns at the boundary of equilibrium, indicating potential for extension to non-equilibrium systems.
• Invariance and Robustness: Legendre invariance ensures that geometric diagnostics remain unaltered under changes of thermodynamic potential, aligning with fundamental principles of thermodynamic consistency.

The geometric paradigm of GTD thus unifies and extends the analytic criteria for phase transitions, providing a correspondence between curvature features of the equilibrium space and microscopic thermodynamic structure. The geometric language supports both theoretical analysis and practical computation of phase diagrams, stability, and interaction strength in a wide variety of physical systems (Quevedo et al., 2010, Quevedo et al., 2011, Bravetti et al., 2012, Quevedo et al., 2011).