Supersymmetric Fokker-Planck strict isospectrality

TL;DR

This paper explores how supersymmetric quantum mechanics techniques can be applied to the nonstationary Fokker-Planck equation, revealing space-dependent damping effects and enabling the normalization of solutions for physical interpretation.

Contribution

It introduces the concept of strictly isospectral damping in Fokker-Planck solutions and demonstrates how to normalize these solutions for physical relevance using supersymmetric methods.

Findings

Space-dependent (modulational) damping of Fokker-Planck solutions.

Method to normalize nonstationary solutions into probability densities.

Potential applications to transient physical processes.

Abstract

I report a study of the nonstationary one-dimensional Fokker-Planck solutions by means of the strictly isospectral method of supesymmetric quantum mechanics. The main conclusion is that this technique can lead to a space-dependent (modulational) damping of the spatial part of the nonstationary Fokker-Planck solutions, which I call strictly isospectral damping. At the same time, using an additive decomposition of the nonstationary solutions suggested by the strictly isospectral procedure and by an argument of Englefield [J. Stat. Phys. 52, 369 (1988)], they can be normalized and thus turned into physical solutions, i.e., Fokker-Planck probability densities. There might be applications to many physical processes during their transient period