Analysis of the Fokker-Planck Equation in Schwarzschild Spacetime: A Supersymmetric Connection

Abstract: We have re-analyzed the dynamics of the thermal potential within Schwarzschild spacetime by employing the Fokker-Planck equation. We demonstrate that the Fokker-Planck equation reduces to a simplified form equivalent to a scaled quantum mechanical problem with a harmonic oscillator potential. In this framework, we highlight an interesting correspondence between supersymmetric quantum mechanics (SUSY QM) and the Fokker-Planck dynamics associated with the Schwarzschild metric. Utilizing the isospectral deformation, an intrinsic feature of SUSY QM, we derive a family of one-parameter isospectral potentials. Notably, this new class of potentials exhibits the same energy spectrum as the original harmonic oscillator potential, but with distinct wavefunctions.