Exact potentials in multivariate Langevin equations

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. We show how to identify such systems by understanding their transformation properties arising from the chain rule, and we use this basic technique to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations.