Non-Equilibrium Physics of Thermodynamicized Black Holes

Abstract: This work presents a non-equilibrium framework for thermodynamicized black holes, inspired by the entropy-functional interpretation of emergent gravity and by residue-based methods in black hole thermodynamics. The main idea is to unify three components: an entropy functional principle for selecting physical on-shell backgrounds, a Euclidean and contour-based description of the horizon temperature through simple pole singularities, and a topological residue classification of multi-horizon black hole configurations. On this basis, the paper introduces a quasi-stationary non-equilibrium partition functional in which irreversible entropy production appears as an additional contribution to the singular action. The formalism reproduces the standard equilibrium relations in the adiabatic limit, while also extending them to dynamically driven black-hole systems with matter, charge, and rotational fluxes. The framework is then applied to Kerr Newman type black holes in constant curvature f(R) gravity, where the equilibrium entropy remains weighted by the derivative of f at the background curvature, while non-equilibrium corrections arise from flux-induced deformations of the effective thermodynamic action. The analysis further shows that the outer and inner horizons carry opposite topological orientations, so the non-extremal Kerr Newman family stays in the topological class W = 0 unless a horizon bifurcation or merger changes the singularity structure. Finally, several function plots are provided to illustrate the behavior of equilibrium and non-equilibrium free energy, the Kerr Newman temperature curve, and the entropy production law.

• The paper develops a residue-based formulation of horizon temperature and uses an entropy-functional approach to establish local equilibrium conditions.
• It combines analytic contour integrals with topological indices to classify multi-horizon states and delineate mechanisms of irreversible entropy production.
• The study extends its framework to f(R) gravity and cosmological scenarios, uniting reversible gravitational dynamics with dissipative processes.

Non-Equilibrium Physics and Thermodynamicization of Black Holes

Entropy-Functional Selection, Residue Thermodynamics, and Horizon Topology

The paper "Non-Equilibrium Physics of Thermodynamicized Black Holes" (2604.21166) elaborates a rigorous framework for describing black hole thermodynamics beyond static equilibrium, integrating entropy-functional background selection, complex-analysis residue prescriptions for horizon temperature, and topological classification of multi-horizon states. The approach builds upon the viewpoint that gravitational dynamics near horizons admit a macroscopic thermodynamic interpretation, compatible with entropy-functional criteria for background selection and emergent-gravity principles.

An auxiliary vector field $\xi^\mu$, typically associated with horizon generators or local Killing vectors, enters as the core variable in an entropy-functional $S_\xi[g, \xi]$. Stationarity of this entropy-functional yields a local equilibrium condition $R_{\mu\nu}\xi^\mu\xi^\nu=0$, restricting admissible backgrounds to those where the thermodynamic interpretation is meaningful, without imposing the full gravitational field equations.

Horizon temperature is formulated as a residue of the analytically continued lapse function $f(r)$, specifically via $\beta_h = 4\pi\,\text{Res}_{r_h}(1/f(r))$, connecting the temperature to the residue at the horizon's simple zero. This residue representation is robust and insensitive to bulk details away from the horizon. Such a technical bridge ensures that the singularity structure directly encodes thermodynamic data.

For black holes in $f(R)$ gravity, the entropy acquires a Wald-type correction weighted by $f'(R_0)$, where $R_0$ is the constant background curvature. The Kerr--Newman family is generalized via modifications to the horizon function $\Delta(r)$ and its roots, and the entropy and temperature formulas preserve their geometric dependence, but with $f(R)$-specific couplings.

Quasi-Stationary Non-Equilibrium Formalism

The extension to non-equilibrium is realized by introducing a quasi-stationary partition functional and a generalized singular action formalized through contour integrals. Here, irreversible entropy production $S_\xi[g, \xi]$0 appears additively in the thermodynamic action, aligning with the dissipative sector of black-hole physics.

The partition functional maintains the structure

$S_\xi[g, \xi]$1

where $S_\xi[g, \xi]$2, $S_\xi[g, \xi]$3, $S_\xi[g, \xi]$4, $S_\xi[g, \xi]$5 are extensive quantities, and $S_\xi[g, \xi]$6, $S_\xi[g, \xi]$7, $S_\xi[g, \xi]$8 are their conjugates. The entropy balance law generalizes the first law:

$S_\xi[g, \xi]$9

with $R_{\mu\nu}\xi^\mu\xi^\nu=0$0 denoting the irreversible entropy production rate, ensuring compatibility with the generalized second law. The additive structure separates reversible equilibrium dynamics from irreversible flux-driven contributions, harmonizing with the standard adiabatic limit.

Topological Classification and Horizon Branch Structure

The topological classification of horizon branches is constructed using an index $R_{\mu\nu}\xi^\mu\xi^\nu=0$1, with $R_{\mu\nu}\xi^\mu\xi^\nu=0$2 depending on branch orientation: outer horizon ($R_{\mu\nu}\xi^\mu\xi^\nu=0$3), inner ($R_{\mu\nu}\xi^\mu\xi^\nu=0$4). For the Kerr--Newman family, $R_{\mu\nu}\xi^\mu\xi^\nu=0$5 as the contributions cancel, and weak $R_{\mu\nu}\xi^\mu\xi^\nu=0$6 corrections or mild non-equilibrium deformations do not alter $R_{\mu\nu}\xi^\mu\xi^\nu=0$7 unless a qualitative change (e.g., horizon merger or bifurcation) occurs. This ensures that dissipative processes, as long as they do not modify the singularity structure, preserve the topological class—a robust feature aligned with recent studies on thermodynamic topology (Chen, 28 Dec 2025).

Explicit Analytic Models and Entropy Production

The paper utilizes analytic function plots to illustrate key aspects of the formalism:

• Equilibrium and non-equilibrium branches of the free-energy function display how dissipative dressing lifts the effective potential.
• Kerr--Newman horizon temperature curves with non-equilibrium phenomenological deformation parameter $R_{\mu\nu}\xi^\mu\xi^\nu=0$8 visualize temperature suppression due to weak dissipation.
• A quadratic entropy production law $R_{\mu\nu}\xi^\mu\xi^\nu=0$9 (with $f(r)$0 as the flux variable) encodes irreversible entropy generation, consistent with coarse-grained horizon viscosity or transport.

Such models are intended for analytical control and visualization; they are not numerical solutions of the full field equations but serve to elucidate how dissipation modifies black-hole thermodynamics in a tractable manner.

Gravity-Thermodynamization and Constitutive Closure

A significant portion is devoted to thermodynamizing the field equations, emphasizing that for $f(r)$1 gravity and curvature-corrected models, an equilibrium formulation is inadequate. A non-equilibrium entropy-production term is required, both in local Rindler horizon settings and in cosmological (FRW) contexts. The Wald entropy is used as the geometric entropy, and effective matter fluxes, temperature assignments, and irreversible terms are incorporated into the entropy balance.

Constitutive closure relations are introduced to establish positive-definite entropy production, including terms proportional to horizon expansion ($f(r)$2), shear, and scalaron flux. This closure guarantees the generalized second law and enables a variational principle for gravity-thermodynamized models, coupling reversible and irreversible sectors.

In cosmological settings, the first law at the apparent horizon is rewritten to accommodate geometric corrections as either additional entropy production or as effective energy-momentum contributions, enabling a dual equilibrium/non-equilibrium description depending on the adopted coarse-graining.

Implications and Future Directions

The formalism provides a unified mathematical framework with the following implications:

• Universality of residue calculus: Horizon temperature remains encoded by the local singularity structure of the lapse function, allowing for universal thermodynamic prescriptions in both equilibrium and slowly driven non-equilibrium settings.
• Topological protection of horizon branch index: Thermodynamic classes are robust under perturbative corrections, supporting the prospect of classifying black hole states via their singularity topology.
• Quasi-stationarity as a regime of applicability: The approach is valid for slowly driven black holes. Full time-dependent, strongly non-equilibrium phenomena would require extension to real-time formulations (e.g., Schwinger–Keldysh).
• Thermodynamicization extends gravity beyond field equations: Real geometric entropy production becomes essential for higher-derivative theories and non-equilibrium evolution, reinforcing the view that gravitational dynamics have an intrinsic thermodynamic structure.
• Analytical tractability: The presented models offer pathways for future numerical and analytic investigations into multi-horizon configurations, AdS backgrounds, and transport-theoretic derivations of dissipation coefficients.

Conclusion

This paper establishes a mathematically explicit non-equilibrium framework for thermodynamicized black holes, integrating entropy functionals, residue-based temperature prescriptions, and topological classification. The main results include encoding equilibrium and non-equilibrium dynamics by contour residues and additive entropy production, characterizing the topological stability of horizon configurations under flux-driven deformations, and rigorously reformulating gravity as a thermodynamic system with both reversible and irreversible processes. The framework is suitable for generalization to more complex black-hole spacetimes, for connecting to microscopic entropy models, and for systematic study of dissipative horizon dynamics in gravitational theories with nontrivial coupling and curvature corrections.