Exact potentials in multivariate Langevin equations

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 analy...

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Bibliographic Details
Published in:Chaos (Woodbury, N.Y.) N.Y.), 2022-12, Vol.32 (12), p.123146-123146
Main Authors: Pedergnana, Tiemo, Noiray, Nicolas
Format: Article
Language:English
Subjects:
Differential geometry
Gradient flow
Jacobi matrix method
Jacobian matrix
Multivariate analysis
Stochastic models
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Summary: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 rules for 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.
ISSN:1054-1500
1089-7682
DOI:10.1063/5.0124031