Fokker-Planck equation for black holes in thermal potential

TL;DR

This paper models black hole thermodynamics within a thermal potential using the Fokker-Planck equation, revealing discrete energy spectra and differences between Schwarzschild and BTZ black holes related to their geometric and thermodynamic properties.

Contribution

It introduces a novel approach to analyze black hole thermodynamics via stochastic processes and solves the Fokker-Planck equation to find energy spectra for different black holes.

Findings

Schwarzschild black hole energy spectrum proportional to temperature

BTZ black hole energy spectrum depends on AdS radius

Ground state of BTZ black hole is zero, unlike Schwarzschild

Abstract

We construct a kind of thermal potential and then put the black hole thermodynamic system in it. In this regard, some thermodynamic properties of the black hole are related to the geometric characteristics of the thermal potential. Driven by the intrinsic thermodynamic fluctuations, the behavior of the black hole in the thermal potential is stochastic. With the help of solving the Fokker-Planck equation analytically, we obtain the discrete energy spectrum of Schwarzschild and Banados-Teitelboim-Zanelli (BTZ) black holes in the thermal potential. For Schwarzschild black hole, the energy spectrum is proportional to the temperature of the ensemble, which is an external parameter, and the ground state is non-zero. For BTZ black hole, the energy spectrum only depends on the AdS radius, which is the intrinsic parameter. Moreover, the ground state of BTZ black hole in thermal potential is zero. This also reflects the difference between three-dimensional gravity and four-dimensional gravity.