Extended Phase Space Thermodynamics

• Extended Phase Space Thermodynamics is an approach that reinterprets the cosmological constant and other couplings as dynamic variables, redefining black hole mass as enthalpy.
• It produces equations of state analogous to Van der Waals fluids, enabling the study of first-order transitions, reentrant phase transitions, and critical phenomena.
• The framework extends to include nonlinear electrodynamics and higher curvature corrections, offering insights into black hole stability and complex phase structures.

Extended phase space thermodynamics is a framework that generalizes classical black hole thermodynamics by promoting the cosmological constant (Λ) and other Lagrangian coupling constants to dynamical thermodynamic variables. In this approach, the black hole mass is interpreted as the enthalpy rather than internal energy, additional conjugate quantities are introduced for new variables, and the equations of state exhibit rich analogies to standard fluid systems, such as the Van der Waals liquid-gas system. This formalism allows for rigorous analysis of phase transitions, critical phenomena, and stability in gravitational systems, especially in the presence of nonlinear matter fields, higher-curvature interactions, and modifications such as massive gravity.

1. Theoretical Framework of Extended Phase Space

The core ingredient is the promotion of the (typically negative) cosmological constant Λ to a thermodynamic pressure,

$P = -\frac{\Lambda}{8\pi}$

with the conjugate thermodynamic volume defined by

$$V = \left(\frac{\partial M}{\partial P}\right)_{\text{other variables}}$$

where $M$ is the ADM mass, now interpreted as the enthalpy $H = U + PV$ of the system.

The differential form of the first law for black holes then becomes

$$dM = T dS + \Phi dQ + V dP + \sum_{i} \mathcal{A}_i da_i + \cdots$$

where $T$ is the Hawking temperature, $S$ is the Bekenstein-Hawking entropy, $\Phi$ and $Q$ are the conjugate electric potential and charge, and $a_i$ represent other dimensionful couplings with their conjugate quantities $\mathcal{A}_i$. For theories with nonlinear electrodynamics (e.g., Born-Infeld, rational NED), the nonlinearity parameter (e.g., $b$, $\beta$) also enters as a thermodynamic variable, with its conjugate interpreted as a vacuum polarization term. This structure is required to satisfy both the extended first law and a generalized Smarr relation derived from dimensional (Euler) scaling.

The extended phase space approach can be summarized by the following table:

Thermodynamic Quantity	Definition (General)	Typical Example
Pressure ($P$)	$-\Lambda/(8\pi)$	$P = 3/(8\pi \ell^2)$ ($\ell$ = AdS radius)
Volume ($V$)	$(\partial M/\partial P)$	$V = (4\pi/3) r_+^3$ (Schwarzschild/AdS)
Nonlinear param. ($b$/$\beta$)	Lagrangian coupling	$b$ in Born-Infeld, $\beta$ in NED
Conjugate ($\mathcal{B}$)	$(\partial M/\partial b)$, $(\partial M/\partial \beta)$	Vacuum polarization term

This extension enables a direct mapping of black hole thermodynamics to standard thermodynamic systems and underpins the analogy with liquid-gas transitions.

2. Equation of State and Critical Phenomena

With $P$ and $V$ promoted to thermodynamic variables, the Hawking temperature can be rewritten to yield an equation of state,

$P = f(T, v; \text{charges, couplings})$

where $v$ is a "specific volume" typically identified via $v = 2 r_+$ (equivalently, related to $V$ by a geometric factor).

For instance, Born–Infeld-AdS black holes have

$$P = \frac{(D-2)T}{4 r_+} - \frac{(D-2)(D-3)k}{16\pi r_+^2} - \frac{b^2}{4\pi} \left[1 - \sqrt{1 + \frac{16\pi^2 Q^2}{b^2 \Sigma_k^2 r_+^{2D-4}}} \right]$$

and, by introducing the specific volume,

$v = \frac{4 r_+}{D-2},$

one obtains van der Waals–like $P$–$v$ isotherms where the presence of inflection points

$$\left(\frac{\partial P}{\partial v}\right)_T = \left(\frac{\partial^2 P}{\partial v^2}\right)_T = 0$$

signals a critical point. The near-critical equation of state can be expanded into the standard Landau form,

$$p = 1 + a_{10} t + a_{11} t \omega + a_{03} \omega^3 + \cdots,$$

with $p = P/P_c$, $t = (T-T_c)/T_c$, $\omega = (v-v_c)/v_c$, leading to mean-field (van der Waals) critical exponents: $\alpha = 0$, $\beta = 1/2$, $\gamma = 1$, $\delta = 3$ (Zou et al., 2013).

Dimensionality and horizon topology play pivotal roles: four-dimensional Born-Infeld-AdS black holes can exhibit reentrant phase transitions (multiple transitions upon monotonic parameter variation), while $D\ge5$ restricts to standard first-order transitions, and $D=3$ admits no criticality at all.

3. Inclusion of Nonlinear Matter and Higher Curvature Corrections

The extended phase space formalism effectively generalizes to gravitational theories with nonlinear matter sectors or higher curvature interactions.

• Nonlinear electrodynamics (NED): For Lagrangians of the form $$\mathcal{L}(\mathcal{F}) = -\mathcal{F}/(4\pi (1+2\beta\mathcal{F})^\sigma)$$, new thermodynamic quantities (e.g., $\beta$, $\mathcal{B}$) are introduced (Kruglov, 21 May 2024, Kruglov, 10 Jul 2024). The mass function, thermodynamic conjugates, and generalized Smarr relation include these variables:

$$dM = T dS + V dP + \Phi dq + \mathcal{B} d\beta, \qquad M = 2ST - 2PV + q\Phi + 2\beta\mathcal{B}.$$

The phase structure remains van der Waals–like, but with modified critical parameters and existence of additional phenomena such as finite field strengths at $r=0$ and regularized self-energies.

• Higher curvature gravities (e.g., Lovelock, massive gravity): The first law and Smarr formula are augmented with conjugates to Lovelock coefficients ($\alpha$) or massive graviton couplings ($c_i$) (Hendi et al., 2018). Rich phase structures are found, including multiple critical points, reentrant transitions, and triple points especially in hyperbolic and Ricci-flat topology. The generalized first law reads

$$dM = T dS + V dP + \mathcal{A} d\alpha + \sum_i \mathcal{C}_i dc_i$$

with corresponding Smarr scaling.

• Modified theories of gravity (e.g., $f(R)$, scalar-tensor): Interpretation ambiguities for pressure and volume arise, especially under conformal transformations. The extended variables are only frame-invariant if identified with the bare cosmological constant without additional conformal factors (Bhattacharya, 2021).

4. Phase Structure, Stability, and Classification

Phase diagrams in extended phase space reveal a spectrum of behaviors, closely paralleling those of classical fluids:

• First-order transitions between large/small black holes, with Gibbs free energy profiles exhibiting swallowtail structures, are generic.
• Zeroth-order transitions (finite jumps in free energy) and second-order critical points (divergence of specific heat) can occur, often intertwined in reentrant transitions as the temperature or pressure is varied monotonically (Zou et al., 2013, Guo et al., 2021).
• Multicriticality and triple points are possible in higher-order/higher-curvature theories (Hendi et al., 2018).
• Stability is dictated by the sign of specific heat at fixed charge and pressure, $C_{Q,P}$. Thermodynamically favored branches are those minimizing the Gibbs free energy with $C_{Q,P}\ge0$; however, branches may be electrically unstable (e.g., negative isothermal permittivity) even if thermally stable (Wang et al., 2018).
• Joule–Thomson expansion and inversion phenomena are calculable in these settings, with inversion temperatures, isenthalpic and inversion curves can be explicitly derived (see, e.g., (Kruglov, 2022, Kruglov, 2022)), yielding an additional diagnostic for phase structure and cooling/heating regimes.

5. Extended First Law, Smarr Relation, and Conjugate Quantities

A general extended first law in these frameworks is

$$dM = T dS + V dP + \Phi dQ + \sum_i \mathcal{A}_i da_i.$$

The presence of additional variables (such as NED parameters or higher curvature couplings) is essential for the validity of the first law and the Smarr relation, which is always enforced by Euler's theorem for homogeneous functions: $$M = n_1 TS - n_2 PV + n_3 Q\Phi + \cdots + \sum_i n_i a_i \mathcal{A}_i,$$ with weights $n_i$ fixed by the scaling dimensions of the respective quantities.

These conjugates have direct physical interpretation:

• $V$: thermodynamic volume (always nontrivially related to $r_+$, may differ from geometric volume in presence of hair or nontrivial topologies).
• $\mathcal{B}$: vacuum polarization or conjugate to nonlinear coupling, encoding a response to variations in the fundamental parameters of the theory.
• $\mathcal{A}$, $\mathcal{C}_i$: conjugates to curvature or massive gravity coefficients, physically corresponding to "pressure-like" work due to the variations of gravitational couplings.

In all cases, detailed cross-checks of the first law and Smarr relation confirm the internal consistency of the extended phase space formulation (Wang et al., 2018, Kruglov, 21 May 2024, Kruglov, 10 Jul 2024, Kruglov, 2022).

6. Geometric and Statistical Thermodynamics in Extended Phase Space

Riemannian geometric methods (Ruppeiner geometry, scalar curvature of thermodynamic parameter space) have been adapted to extended phase space, but certain caveats appear. In the limit where the specific heat at constant volume vanishes (typical for Reissner–Nordström–AdS), the scalar curvature becomes proportional to the thermodynamic volume and the formal applicability of geometric criticality diagnostics is limited to regions near criticality (Bairagya et al., 2020). Furthermore, normalization is required to compare with standard fluid systems due to the nonextensive nature of black hole entropy and vanishing $c_v$.

Notably, the conjecture that the correlation lengths of coexisting phases are equal near the phase transition point (as measured by normalized scalar curvatures) appears to hold in charged AdS black hole backgrounds.

7. Applications, Analogies, and Broader Implications

Extended phase space thermodynamics not only yields insights into black hole microphysics and stability but also enables a precise mapping to classical thermodynamic systems:

• Van der Waals analogy: The equations of state, critical points, and critical exponents for a wide class of black holes (standard charged/rotating, nonlinear, higher-curvature, and even "hairy" black holes) match those of a van der Waals fluid.
• Hawking–Page and reentrant transitions: Competing phases (e.g., thermal AdS vs. large/small black holes) and complex transitions, including zeroth- and second-order, directly parallel observable fluid transitions.
• Cavity thermodynamics: By treating the boundary radius (or cavity wall) as a thermodynamic variable, further analogies are established between black holes in a cavity and their AdS counterparts (Wang et al., 2020).
• Dyonic, magnetized, massive gravity, and higher curvature cases: Each extension introduces novel critical phenomena—including multiple critical points, reverse van der Waals behavior, triple points, and region-dependent phase structure—further reinforcing the versatility of the extended framework.

These theoretical advances generate a robust platform for black hole chemistry, influence holographic studies of strongly coupled gauge theories, and support the search for universal structures in gravitational thermodynamics.