Overcoming Barriers: Kramers' Escape Rate Analysis of Metastable Dynamics in First-Order Multi-Phase Transitions

TL;DR

The paper addresses whether Kramers' escape rate can reliably describe first-order phase transitions in black holes that exhibit multiple coexisting phases and critical points. By embedding the dynamics in an overdamped Fokker–Planck framework and utilizing a generalized Gibbs free-energy landscape $G_L = M - T S$, the authors analyze a 4D Einstein–gravity black hole with nonlinear electrodynamics that supports multicritical behavior. They fix specific nonlinear couplings and pressure, compute the Gibbs landscape $G_L(r)$ and the temperature $T$, and evaluate the Kramers escape rate $K_{ ext{er}}(r)$ across several temperature points to map the transition kinetics. The main findings show that $K_{ ext{er}}$ accurately marks the initiation and completion of first-order transitions even when multiple local minima are present, with transitions proceeding in a sequential, stepwise fashion rather than direct bypasses of intermediate states. This work highlights the utility of kinetic, stochastic descriptions for elucidating the energetic landscapes and dynamical pathways of black-hole phase transitions, with potential implications for broader gravitating systems and multi-phase thermodynamics.

Abstract

The expanding application of classical thermodynamic methods to black hole physics has yielded significant advances in characterizing phase transition behavior. Among these approaches, thermodynamic analysis -- particularly kinetic formulations like the Kramers' escape rate -- provides a robust framework for probing black hole phase transitions with minimal relativistic constraints. This study investigates the kinetics and dynamic evolution of first-order phase transitions in black holes exhibiting multiple critical points, employing a particle-based escape rate model. The distinct free energy landscapes inherent to multi-critical systems, which can simultaneously support multiple local minima under specific thermodynamic conditions (temperature and pressure) within a given reference frame, raise fundamental questions regarding transition pathways. We rigorously assess whether the Kramers' escape rate retains its predictive validity in these complex multi-minima systems, as established for conventional single-minimum configurations. Furthermore, we examine whether transitions proceed via a sequential, stepwise mechanism between adjacent minima, or if pathways exist that bypass intermediate states through direct descent to the global minimum. Our analysis of black holes undergoing multiphase transitions reveals both parallels and significant deviations from single-transition models. Crucially, we demonstrate that the Kramers' escape rate remains a quantitatively reliable indicator of first-order phase transitions in black holes, even within multi-critical frameworks. This approach offers deeper insights into the governing energetic landscapes and kinetic processes underlying these phenomena.

Figures (10)

• Figure 1: The general form of potential U
• Figure 2: The T diagram for 4D-Einstein-nonlinear electrodynamics Black hole
• Figure 3: Behavioral sequence of energy in terms of r for different temperatures
• Figure 4: 4a: The T diagram , 4b: The graph of $G_{L}$ against r for the free chosen parameters and the FPT points, 4c: Kramers escape rate during the FPT with respect to $r$ , 4d: comparing $K_{e-r}$ with respect to r for$(\alpha \rightsquigarrow \gamma)$ and $(\gamma \rightsquigarrow \alpha)$.
• Figure 5: Kramers escape rate during $\alpha\rightsquigarrow\gamma$ transition with respect to $r$