Black hole chemistry: thermodynamics with Lambda

TL;DR

The article reviews the development of black hole thermodynamics in extended phase space, treating the cosmological constant as a pressure and the mass as enthalpy to define a thermodynamic volume. It highlights how this framework yields rich phase structure analogous to chemical systems, including Van der Waals transitions, reentrant behavior, and triple points, and discusses extensions to Lovelock and Born–Infeld theories and holographic interpretations. The review also covers the AdS/CFT dictionary in this extended setting, entanglement entropy, and applications to Lifshitz and de Sitter spacetimes, while outlining open questions about volume definitions and microscopic interpretations. Overall, the work establishes black holes as chemical systems with deep ties to holography, quantum information, and beyond-AdS geometries, while charting future directions for the field.

Abstract

We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequality - an inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.

Figures (17)

• Figure 1: Super-entropic black hole: horizon embedding. The horizon geometry is embedded in $\mathbb{E}^3$ for the following choice of parameters: $l=1$, $r_+=\sqrt{10}$ and $\mu = 2\pi$.
• Figure 2: Hawking--Page transition.Left. The Gibbs free energy of a Schwarzschild-AdS black hole is displayed as a function of temperature for fixed pressure $P=1/(96\pi)$. The upper branch of small black holes has negative specific heat and is thermodynamically unstable. For $T>T_{\hbox{\tiny HP}}$ the lower branch of large black holes (with positive specific heat) has negative Gibbs free energy and corresponds to the globally thermodynamically preferred state. At $T_{\hbox{\tiny HP}}$ we observe a discontinuity in the first derivative of the radiation/black hole Gibbs free energy characteristic of the first order phase transition. Right. The $P-T$ phase diagram has a coexistence line of infinite length and is reminiscent of the solid/liquid phase portrait.
• Figure 3: Analogue of Van der Waals behavior.Left. A characteristic swallowtail behavior of the Gibbs free energy of a charged-AdS black hole is displayed for fixed $Q=1$. Right. The $P-T$ phase diagram shows SBH/LBH phase transition reminiscent of the liquid/gas phase transition. The coexistence line terminates at a critical point where the phase transition is of the second order.
• Figure 4: Further analogies with Van der Waals fluid.Left. The figure schematically displays Maxwell's equal area law describing the liquid/gas phase transition of the Van der Waals fluid: the 'oscillating' (dashed) part of the isotherm $T<T_c$ is replaced by an isobar, such that the areas above and below the isobar are equal one another. Similar law holds for the SBH/LBH phase transition of the charged AdS black hole in the $P-V$ diagram with the specific heat of the fluid $v$ replaced by the thermodynamic volume $V$ of the black hole. Right. The ratio of micromolecular densities, $\eta=(n_s-n_l)/n_c$, is displayed as a function of $T/T_c$.
• Figure 5: Reentrant phase transition in nicotine/water mixture. The diagram displays possible phases of the mixture dependent on the temperature and percentage of the nicotine. 'Outside of the bubble' the mixture is in a homogeneous state. Inside, two layers of nicotine and water exist separately: in the upper half the nicotine layer is above the water layer while the layers swap in the bottom half of the bubble. Fixing the percentage of nicotine at, for example, $40\%$ and increasing the temperature from low to high, we observe the following phases: homogeneous mixture (low temperatures), water above nicotine (bottom half of the bubble) nicotine above water (upper half of the bubble), homogeneous mixture (high temperatures). Since the initial and final states are macroscopically similar, this is an example of an RPT. Reproduced from ref. Hudson:1904 ( C. Hudson, Die gegenseitige lslichkeit von nikotin in wasser Z. Phys. Chem. 47 (1904) 113) with permission from De Gruyter.