Holography, Thermodynamics and Fluctuations of Charged AdS Black Holes

TL;DR

The paper analyzes charged Reissner–Nordström–AdS black holes across dimensions within a holographic framework, refining the thermodynamic phase structure in both canonical and grand canonical ensembles. It demonstrates a striking liquid–gas–like behavior with a swallowtail free-energy and a cusp equation of state, identifying a second-order critical point described by an $A_3$ Landau–Ginzburg potential and universality across dimensions. Using intrinsic counterterm subtraction, it derives explicit expressions for Gibbs and Helmholtz free energies, Clapeyron-type coexistence relations, and stability criteria, then connects macroscopic fluctuations to the underlying microscopic (dual field theory) degrees of freedom, showing they scale with the inverse number of degrees of freedom and diverge at criticality. The results extend to higher dimensions, preserving the qualitative phase structure and illuminating how holography encodes microscopic dynamics in thermodynamics and phase transitions of strongly coupled field theories on brane worldvolumes.

Abstract

The physical properties of Reissner-Nordstrom black holes in (n+1)-dimensional anti-de Sitter spacetime are related, by a holographic map, to the physics of a class of n-dimensional field theories coupled to a background global current. Motivated by that fact, and the recent observations of the striking similarity between the thermodynamic phase structure of these black holes (in the canonical ensemble) and that of the van der Waals-Maxwell liquid-gas system, we explore the physics in more detail. We study fluctuations and stability within the equilibrium thermodynamics, examining the specific heats and electrical permittivity of the holes, and consider the analogue of the Clayperon equation at the phase boundaries. Consequently, we refine the phase diagrams in the canonical and grand canonical ensembles. We study the interesting physics in the neighbourhood of the critical point in the canonical ensemble. There is a second order phase transition found there, and that region is characterized by a Landau-Ginzburg model with A_3 potential. The holographically dual field theories provide the description of the microscopic degrees of freedom which underlie all of the thermodynamics, as can be seen by examining the form of the microscopic fluctuations.

Figures (16)

• Figure 1: Plots of the equation of state of $\Phi$vs.$Q$, showing isotherms above and below the critical temperature $T_{\rm crit}$. For $T{<}T_{\rm crit}$, there is only one branch of solutions, while for $T{>}T_{\rm crit}$, there are three branches. The values of $T$ for the isotherms plotted are (top down) $T{=}0,0.8,T_{\rm crit},1.0,1.2$. The central (dotted) curve is at the critical temperature.
• Figure 2: Plots of the Gibbs potential $W[\Phi,T]$ in three dimensions.
• Figure 3: Slices of the Gibbs potential $W[\Phi,T]$, for $\Phi{=}0$, $\Phi{=}0.7$ and $\Phi{=}1$.
• Figure 4: The free energy vs. temperature for the fixed charge ensemble, in a series of snapshots for varying charge, for values $Q{=}0,0.15$ and $Q{=}0.299$. Note that $Q_{\rm crit}{=}0.289$, so in the last plot, the bend (near $T_{\rm crit}{=}0.943$, is in the neighbourhood of the critical point of second order.
• Figure 5: The free energy vs. charge for the fixed charge ensemble, in a series of snapshots for varying temperature, for values $T{=}0.943,0.997,1.00,1.10$, and (for the "Zorro" plot) $T{=}T_{\rm HP}{=}1.154$, and finally $T{=}1.20$. Note that $T_{\rm crit}{=}0.943$, and so in the first plot, the bend (near $Q_{\rm crit}{=}0.289$), is in the neighbourhood of the critical point of second order.