[Paper Review] Kramer's Escape Rate and Phase Transition Dynamics in AdS Black Holes

Mohammad Ali S. Afshar, Saeed Noori Gashti+2 more|arXiv (Cornell University)|Apr 27, 2024

Black Holes and Theoretical Physics6 citations

TL;DR

The paper studies dynamic phase transitions of AdS black holes influenced by dark structures using Kramers’ escape rate across different free energy landscapes, comparing radius- and temperature-based analyses for two dark-structure black-hole models.

ABSTRACT

Traditional static methods in phase transition studies, provide good insights into the thermodynamics of black holes. However, they practically lose sight of the dynamic aspects and temporal sequence of events. The Kramer's escape rate, central to our research, offers a somewhat dynamic approach to phase transition. We examine the free energy landscapes for black holes under the influence of 'dark' and 'stringy dark' structures, assessing how additional parameters affect the escape rates and dynamics of the transition during the first-order phase transition from small to large black holes. In our analysis, we consider the escape rate as a function of the black hole radius and study its variations. We will observe that, on one hand, the escape rate well represents our assumption based on the movement from zero, increasing to a maximum point, and then decreasing back to zero as reactive structures become active during the phase transition interval. However, the critical point in this method is the encounter with a specific and distinct point. This is where the diagram of the direct process (escape rate from small to large black holes) intersects with the reverse process (large to small black holes), becoming equally probable (contact point). The point, which seems improbable at the onset of the phase transition or very negligible, gains more significance as the process progresses. This increase indicates the dominance of a region where the escape rate from larger black holes to smaller ones prevails. The predominance of the reverse process, which increases as we approach the end of the process and is necessarily accompanied by a variation in radius, may be considered as a natural reaction of the black hole against the 'phase change' action. A reaction which attempting to prevent any uncontrolled radial growth that could jeopardize the stability of the black hole.

Motivation & Objective

• Motivate and model black hole phase transitions as stochastic processes using free energy landscapes in the presence of dark structures.
• Compare three free energy formalisms (Gibbs free energy landscape, Landau free energy, and thermal potential) in canonical ensembles for AdS black holes.
• Analyze two dark-structure AdS black hole models (NLM-C-Q-PFD and EGB-YM-CS) to assess how dark parameters alter transition dynamics.
• Compute Kramers’ escape rates to understand rates and directions of small-to-large and large-to-small black hole transitions.

Proposed method

• Derive and compare Gibbs free energy landscape G_L, Landau free energy L, and thermal potential U for the black hole models.
• Apply the Kramers escape rate formula r_k ~ exp(-(U_max-U_min)/D) to quantify transition rates.
• Examine escape rates as functions of the horizon radius r to identify a maximum and subsequent decrease, indicating a reaction mechanism.
• Examine escape rates as functions of temperature (pressure) to identify critical points where transition directions change.
• Analyze two specific models: non-linear magnetic-charged AdS black hole with quintessence in perfect fluid dark matter background and 4D AdS Einstein-Gauss-Bonnet-Yang-Mills black hole with a cloud of strings.

Figure 1: The possibility of leakage due to Brownian motion or thermal fluctuation

Experimental results

Research questions

• RQ1 How do dark structures (dark energy and dark matter components) influence the Kramers escape rate between small and large AdS black holes?
• RQ2 What are the similarities and differences among Gibbs free energy landscape, Landau free energy, and thermal potential in capturing black hole phase-transition dynamics under thermal fluctuations?
• RQ3 How do temperature and pressure changes affect the preferred direction of transition (small-to-large vs large-to-small) in the presence of dark structures?
• RQ4 Do the two dark-structure black hole models exhibit distinct dynamical behaviors in their escape-rate landscapes, and what does this imply about the role of dark parameters?

Key findings

• The escape rate increases with radius up to a maximum, then decreases due to a reaction mechanism, with escape from smaller to larger black holes generally exceeding the reverse in the radius-based view.
• As temperature (and corresponding pressure) rises, the probability diagrams converge to a point where transitions can occur between small and large black holes with equal likelihood, after which the models diverge.
• In the nonlinear model, an inversion occurs where larger-to-smaller transitions become dominant beyond a certain temperature/pressure, though the inversion region is reduced compared to the charged BH model, indicating dark structures dampen transition likelihoods.
• In the string model, increasing pressure enhances large-to-small transitions more than the reverse, but decreasing pressure can suppress small-to-large transitions, highlighting distinctive dark-structure effects.
• Across the studied models, adding dark parameters generally minimizes the occurrence of certain transitions, signaling subtle influences on phase-transition dynamics.
• The three free-energy representations (Gibbs, Landau, and thermal potential) yield practically similar landscape shapes in the canonical ensemble, supporting their interchangeable use for qualitative dynamics.