Holographic Heat Engines

TL;DR

The paper extends black hole thermodynamics to include a dynamical pressure $p=-\Lambda/8\pi$ and conjugate volume $V$, identifying the mass with enthalpy $M=U+pV$ and analyzing how cycles in the $p$–$V$ plane can harvest work. It shows that for static AdS black holes the Carnot cycle between temperatures $T_H$ and $T_C$ coincides with a Stirling cycle due to $C_V=0$ and the $S$–$V$ relationship, providing explicit expressions for heat flows and efficiencies, including an isobar–isochoric engine with high-temperature limits. An explicit example with charged AdS$_4$ black holes derives the equation of state, the enthalpy form $H(S,p)$, and confirms the vanishing heat capacity at constant volume, while discussing small/large black hole phase transitions and their impact on cycle shapes. The work then develops a holographic RG-based interpretation: dynamical pressure corresponds to flowing between dual field theories with different $N$ via RG flows, reconciling the gravity and field theory pictures and suggesting that holographic heat engines can serve as tools to study gauge theories through renormalization-group engineering and conjugate-volume-like quantities.

Abstract

It is shown that in theories of gravity where the cosmological constant is considered a thermodynamic variable, it is natural to use black holes as heat engines. Two examples are presented in detail using AdS charged black holes as the working substance. We notice that for static black holes, the maximally efficient traditional Carnot engine is also a Stirling engine. The case of negative cosmological constant supplies a natural realization of these engines in terms of the field theory description of the fluids to which they are holographically dual. We first propose a precise picture of how the traditional thermodynamic dictionary of holography is extended when the cosmological constant is dynamical and then conjecture that the engine cycles can be performed by using renormalization group flow. We speculate about the existence of a natural dual field theory counterpart to the gravitational thermodynamic volume.

Figures (4)

• Figure 1: The heat engine flows.
• Figure 2: Our Carnot engine, which for static black holes is also a Stirling Engine.
• Figure 3: Sample (uncorrected) isotherms. Values chosen were $q=0.05$ and $T$ from 0.4 to 1.6 in intervals of 0.2. Lower curves are at lower temperatures. The multi--valued parts of the low temperature curves are removed by first order phase transitions, also removing the unphysical negative pressure. See text.
• Figure 4: Our other engine.