Fokker-Planck equation for black holes in thermal potential

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 holes, the energy spectrum is proportional to the temperature of the ensemble, which is an external parameter, and the ground state is nonzero. For BTZ black hole, the energy spectrum only depends on the anti--de Sitter radius, which is the intrinsic parameter. Moreover, the ground state of the BTZ black hole in thermal potential is zero. This also reflects the difference between three-dimensional and four-dimensional gravity.