Effective Free Energy Landscapes and Black Hole Thermodynamic Phase Transitions

TL;DR

The paper investigates how size-dependent, multiplicative noise alters black hole thermodynamic phase transitions within the Gibbs free energy landscape framework. By formulating a Langevin dynamics with multiplicative noise and deriving the corresponding Fokker-Planck equation, the authors obtain an effective free energy $G(r,{\mathcal T})$ that governs the stochastic dynamics, reducing to the familiar $G_0$ in the additive-noise limit. They show that, in general, multiplicative noise makes black hole formation harder unless $G$ and the original free energy share extrema, and they demonstrate that specific noise profiles can create metastable transient states and modify phase structures for Schwarzschild-AdS and RN-AdS black holes. The work clarifies how size-dependent fluctuations influence Hawking-Page-type transitions and van der Waals-like transitions, offering a framework to assess stochastic effects in black hole thermodynamics with potential implications for quantum gravity phenomenology.

Abstract

A recent interesting development in the dynamics of black hole phase transitions has been the so-called Gibbs free energy landscape approach. In this formalism, it is assumed that there exists a canonical ensemble of a series of black hole spacetimes with arbitrary horizon radius at a given ensemble temperature. An off-shell Gibbs free energy is defined for every spacetime state in the ensemble, with the horizon radius treated as the order parameter. The minima (maxima) of this function correspond to the various stable (unstable) black hole states. This off-shell Gibbs free energy is then treated as a classical effective drift potential of an associated Fokker-Planck equation used to study the stochastic dynamics of black hole phase transition under thermal fluctuations. Additive noise, which is independent of the black hole size, is assumed in obtaining the Fokker-Planck equation. In this work we extend the previous treatment by considering the effects of multiplicative noise, namely, noise that could scale with black hole size. This leads to an effective free energy function that can be used to study the modification of the thermodynamic phase transition of a black hole system. It is realized that it is generally difficult to form black holes under a multiplicative noise, unless the effective and the original free energy become extremal at the same horizon radius. For this latter situation some theoretical noise profiles which are monotonically increasing/deceasing functions of the horizon radius are considered. It is found that stronger noise disfavors the formation of black hole

Figures (10)

• Figure 1: Plot of the Hawking temperature $T_H$ in Eq. (\ref{['data']}).
• Figure 2: Plot of the Gibbs free energy $G_0(r,T)$ for the Schwarzschild-AdS system at different ensemble temperatures. Here $T_m=\sqrt{3}/2\pi=0.27567$ and $T_{HP}=1/\pi=0.31831$.
• Figure 3: Plot of the effective Gibbs free energy $G(r, T_{HP})$ in Eq. (\ref{['Ge']}) for the Schwarzschild-AdS system, labeled by $a$, at $T_{HP}=1/\pi$ (for $a=0$) with noise profile $g(r)=1+a - sign(a) \sqrt{a^2-(r-|a|)^2}$ for different values of $a$.
• Figure 4: Plot of the effective Gibbs free energy $G(r,T)$ for the Schwarzschild-AdS system vs $r$ for different ensemble temperatures with $g(r)$ defined by $a=0.2$. Here $T_m=\sqrt{3}/2\pi=0.27567$ and $T_{HP}=1/\pi=0.31831$. The noise function $g(r)>1$ differs from the additive noise ($g(r)=1$) only on the left side of the vertical dotted line $x=a$. Extremum on the left side of $r=a$ corresponds to thermal transient state, whilst the extrema on the right side correspond to unstable small/stable large black hole solutions of the Einstein equation.
• Figure 5: Plots of the phase structures of the Schwarzschild-AdS black spacetime in terms of $G-T$ diagrams for the case with the additive noise ($a=0$) and the multiplicative case (for $a=0.2$). The dotted/solid curves represent unstable small black holes/stable large black hole phases, respectively. These two curves bifurcate at $T_m=\sqrt{3}/2\pi=0.27567$ in both plots (shown by the $y$-axis for $a=0$ plot and the dotted vertical line for $a=0.2$ plot). The dashdotted horizontal line shows the thermal AdS radiation phase for $a=0$ case, and the lower long-dashed curve corresponds to the metastable thermal transient phase for $a=0.2$ case.