Exact potentials in multivariate Langevin equations

TL;DR

This paper analyzes how to identify systems with exact potentials in multivariate Langevin equations through differential geometry, enabling the derivation of exact potentials for nonlinear oscillation models.

Contribution

It provides a method to recognize and derive exact potentials in Langevin systems via transformation properties and differential geometry.

Findings

Identified conditions for systems with exact potentials.

Derived exact potentials for nonlinear oscillation models.

Visualized potentials in selected examples.

Abstract

Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation properties of Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models which can be self-consistently defined from given deterministic gradient systems.