Probabilistic Evolution of Black Hole Thermodynamic States via Fokker-Planck Equation

Abstract: Employing the generalized free energy landscape and solving the associated Fokker-Planck equation, we obtain the time-dependent probability evolution of the order parameter for the RN-AdS black hole phase transitions. Our analysis reveals two distinct kinetic regimes, namely relaxation dynamics initialized at the unstable maximum and phase transition from the metastable state. Furthermore, we characterize the non-equilibrium irreversibility and macroscopic uncertainty using the entropy production rate and the Shannon entropy. The results demonstrate that the phase transition synchronizes exactly with a prominent peak in the entropy production rate, identifying the barrier crossing event as a process fundamentally driven by maximum thermodynamic dissipation.

• The paper develops a Fokker-Planck framework to capture the non-equilibrium evolution of RN-AdS black hole phase transitions.
• It employs a generalized free energy landscape to identify kinetic trapping and barrier-crossing scenarios through numerical solutions.
• The findings link gravitational thermodynamics to statistical mechanics via measurable entropy production rates and fluctuation-induced phenomena.

Probabilistic Evolution of RN-AdS Black Hole Thermodynamics via the Fokker-Planck Framework

Introduction

This paper presents a comprehensive investigation of the dynamic, stochastic nature of black hole phase transitions, focusing on the four-dimensional Reissner-Nordström anti-de Sitter (RN-AdS) black hole. By modeling the horizon radius as a stochastic order parameter driven by thermal fluctuations, the authors analyze phase transitions on the generalized free energy landscape through explicit solution of the overdamped Fokker-Planck equation. This approach moves beyond conventional equilibrium thermodynamics and static swallowtail diagrams by detailing the non-equilibrium, time-dependent probability evolution during thermodynamic state changes.

Generalized Free Energy Landscape and Black Hole Thermodynamics

The formalism starts from the extended and restricted phase space perspectives, building upon the identification of the cosmological constant as pressure and the inclusion of central charge as a thermodynamic variable. The RN-AdS black hole under consideration exhibits Van der Waals-type behavior, with first-order phase transitions between small and large black hole solutions, as inferred from the presence of a swallowtail structure in the Gibbs free energy versus temperature diagram for subcritical pressures.

The core tool is the generalized free energy $\mathcal{G}(r;T,P)$ as a function of (off-shell) horizon radius $r$,

$\mathcal{G}(r;T,P) = M(r,P) - T S(r)$

which, for fixed temperature and pressure in the first-order regime, develops a characteristic double-well structure. Local minima correspond to small and large black hole states, separated by an intermediate unstable maximum.

Stochastic Dynamics and Fokker-Planck Equation

The evolution of the horizon radius is described as an overdamped Langevin process:

$$\frac{dr}{dt} = -\frac{1}{\zeta} \frac{\partial \mathcal{G}(r)}{\partial r} + \xi(t)$$

with a corresponding Fokker-Planck equation for the time-dependent probability distribution $P(r, t)$:

$$\frac{\partial P}{\partial t} = \frac{\partial}{\partial r} \left( \frac{\partial \mathcal{G}}{\partial r} P \right) + D \frac{\partial^2 P}{\partial r^2}$$

where $D$ is the diffusion coefficient (Einstein relation $D = T/\zeta$), and macroscopic probability flux is rigorously tracked.

Two primary scenarios for the initial condition are analyzed:

1. Evolution from the metastable minimum: Kinetic trapping at low diffusion ($\Delta\mathcal{G}\gg D$) causes exponentially suppressed phase transition rates, quantified by the Kramers escape time. In the high-diffusion regime, thermal fluctuations are sufficient for barrier crossing, producing a dynamically realized phase transition.
2. Evolution from the unstable maximum: Negative curvature in the landscape results in deterministic repulsive evolution, with the probability packet bifurcating into adjacent wells, followed by relaxation into the globally stable state.

Numerical solutions to the Fokker-Planck equation display the continuous nature of the evolution: relaxation within wells, barrier crossing signaled by distinct bimodal distributions, and ultimate settling into stationary Gaussian-like states. The framework accommodates both linearized (local harmonic approximation) and fully nonlinear dynamics.

Quantification of Uncertainty and Irreversibility

Non-equilibrium thermodynamics are quantitatively characterized using two functionals of the probability evolution:

• Shannon Entropy:

$S(t) = -\int P(r, t) \ln P(r, t) dr$

measures macroscopic uncertainty of the order parameter distribution and captures the broadening and contraction of $r$0 during relaxation and transition.

• Entropy Production Rate:

$r$1

quantifies irreversibility and thermodynamic dissipation throughout the process.

A key result is that the critical phase transition event—crossing the potential barrier—coincides precisely with a sharp global peak in the entropy production rate $r$2, identifying the dissipative, irreversible nature of barrier crossing as an essential feature of non-equilibrium stochastic evolution in black hole thermodynamics.

Numerical Results and Dynamical Regimes

Distinct dynamical regimes are rigorously dissected:

• In the kinetic trapping regime (low $r$3), probability is confined near the metastable state, with entropy and entropy production rate relaxing to constant and zero, respectively.
• In the thermally activated regime (sufficient $r$4), a broadening distribution enables barrier crossing, leading to transient bimodal distributions, peaks in Shannon entropy, and pronounced entropy production signaling maximal irreversibility.
• Relaxation from the unstable maximum is characterized by rapid and symmetric broadening, high initial entropy production, and split probability packets, with final redistribution into stationary states.

A notable finding is the divergence of the stationary variance $r$5 of fluctuations as the temperature approaches the spinodal limits, corresponding to vanishing curvature of the free energy landscape and complete loss of local mechanical stability.

Implications and Future Directions

This work reframes black hole phase transitions as continuous, stochastic, and highly dissipative processes governed by the interplay of deterministic forces (from the free energy landscape) and intrinsic thermal fluctuations. The adoption of stochastic thermodynamics concepts such as entropy production and irreversibility links gravitational thermodynamics to statistical mechanical frameworks already well-developed in condensed matter and chemical physics.

The practical implications extend to the study of dynamical universality classes in holographic duals, non-equilibrium microstructure of horizons, and fluctuation-induced phenomena (e.g., critical slowing down, noise-driven transitions), providing a basis for further exploration of quantum effects, non-linear response, and possible experimental signatures in analog gravity systems. The methodology also invites extension to higher curvature gravity, variable coupling, and non-Markovian dynamics.

Conclusion

The paper provides a technical and systematic account of black hole phase transitions within the Fokker-Planck statistical mechanics paradigm. By computing the time-resolved evolution of horizon radius probability distributions, and leveraging measures of uncertainty and dissipation, it explicitly demonstrates the kinetic, stochastic, and fundamentally dissipative nature of such processes in gravitational thermodynamics. This establishes a rigorous connection between the geometric features of the free energy landscape and the temporal signature of non-equilibrium phenomena in black hole physics, laying a foundation for further studies at the intersection of gravity, non-equilibrium statistical mechanics, and information theory.