[Paper Review] Quantum and classical stochastic dynamics: Exactly solvable models by supersymmetric methods

Georg Junker|ArXiv.org|Oct 23, 1998

Quantum chaos and dynamical systems3 citations

TL;DR

This paper extends supersymmetric quantum mechanics to construct exactly solvable classical stochastic systems governed by Fokker-Planck equations. By designing specific drift-potentials on the real and half-lines, the author derives closed-form expressions for decay rates and modes in mono-, bi-, meta-, and unstable systems, enabling exact analytical solutions via supersymmetry-inspired methods.

ABSTRACT

A supersymmetric method for the construction of so-called conditionally exactly solvable quantum systems is reviewed and extended to classical stochastic dynamical systems characterized by a Fokker-Planck equation with drift. A class of drift-potentials on the real line as well as on the half line is constructed for which the associated Fokker-Planck equation can be solved exactly. Explicit drift potentials, which describe mono-, bi-, meta-or unstable systems, are constructed and their decay rates and modes are given in closed form.

Motivation & Objective

• To generalize supersymmetric methods from quantum systems to classical stochastic dynamics described by Fokker-Planck equations.
• To construct explicit drift-potentials on the real line and half-line that yield exactly solvable Fokker-Planck equations.
• To determine decay rates and eigenmodes in conditionally exactly solvable stochastic systems, including metastable and unstable cases.
• To establish a systematic framework for generating exactly solvable classical stochastic models using supersymmetric techniques.
• To provide closed-form analytical solutions for the spectral properties of Fokker-Planck operators in one-dimensional stochastic systems.

Proposed method

• Adapts supersymmetric quantum mechanics techniques to classical stochastic processes by mapping the Fokker-Planck operator to a supersymmetric Hamiltonian structure.
• Constructs a class of drift-potentials on the real line and half-line such that the associated Fokker-Planck equation becomes conditionally exactly solvable.
• Employs shape-invariant potentials and supersymmetry algebra to ensure exact solvability of the Fokker-Planck equation.
• Derives the spectrum of the Fokker-Planck operator, including decay rates and eigenmodes, through the supersymmetric partner potential construction.
• Applies the method to specific cases: monostable, bistable, metastable, and unstable systems, yielding explicit analytical expressions.
• Uses the formalism of supersymmetric quantum mechanics to relate the Fokker-Planck dynamics to isospectral partner systems.

Experimental results

Research questions

• RQ1 Can supersymmetric methods developed in quantum mechanics be extended to construct exactly solvable classical stochastic systems?
• RQ2 What class of drift-potentials on the real and half-lines leads to exactly solvable Fokker-Planck equations?
• RQ3 What are the decay rates and eigenmodes of conditionally exactly solvable stochastic systems, including metastable and unstable cases?
• RQ4 How can the spectral properties of Fokker-Planck operators be systematically derived using supersymmetry?
• RQ5 What is the relationship between the supersymmetric structure and the solvability of the Fokker-Planck equation in one-dimensional stochastic dynamics?

Key findings

• Explicit drift-potentials are constructed on the real line and half-line that yield exactly solvable Fokker-Planck equations through supersymmetric methods.
• Closed-form expressions for decay rates and eigenmodes are derived for monostable, bistable, metastable, and unstable systems.
• The method successfully generalizes supersymmetric quantum mechanics to classical stochastic dynamics, enabling exact spectral analysis.
• The supersymmetric partner potential construction ensures isospectrality and facilitates the derivation of exact solutions.
• The approach provides a systematic way to generate conditionally exactly solvable stochastic models with analytically tractable dynamics.
• The results demonstrate that supersymmetry can be a powerful tool for solving Fokker-Planck equations in one-dimensional stochastic systems.