Stochastic Equations in Black Hole Backgrounds and Non-equilibrium Fluctuation Theorems

TL;DR

This paper applies non-equilibrium fluctuation theorems to black hole thermodynamics by deriving stochastic equations for a scalar field in a black hole background, connecting statistical physics with black hole physics.

Contribution

It introduces stochastic equations for scalar fields in black hole backgrounds and derives generalized second laws, bridging non-equilibrium physics and black hole thermodynamics.

Findings

Derivation of Langevin and Fokker-Planck equations in black hole backgrounds

Application of fluctuation theorems to black hole thermodynamics

Generalized second law derived for black hole systems

Abstract

We apply the non-equilibrium fluctuation theorems developed in the statistical physics to the thermodynamics of black hole horizons. In particular, we consider a scalar field in a black hole background. The system of the scalar field behaves stochastically due to the absorption of energy into the black hole and emission of the Hawking radiation from the black hole horizon. We derive the stochastic equations, i.e. Langevin and Fokker-Planck equations for a scalar field in a black hole background in the $\hbar \rightarrow 0$ limit with the Hawking temperature $\hbar \kappa/2 \pi$ fixed. We consider two cases, one confined in a box with a black hole at the center and the other in contact with a heat bath with temperature different from the Hawking temperature. In the first case, the system eventually becomes equilibrium with the Hawking temperature while in the second case there is an energy flow between the black hole and the heat bath. Applying the fluctuation theorems to these cases, we derive the generalized second law of black hole thermodynamics. In the present paper, we treat the black hole as a constant background geometry. Since the paper is also aimed to connect two different areas of physics, non-equilibrium physics and black holes physics, we include pedagogical reviews on the stochastic approaches to the non-equilibrium fluctuation theorems and some basics of black holes physics.