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  • Open Access

Nonreversible gauge fields in Fokker-Planck dynamics: Supersymmetric Hamiltonians and learned finite forces

Masayuki Ohzeki*

  • Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan; Department of Physics, School of Science, Institute of Science Tokyo, Tokyo 152-8551, Japan; Research and Education Institute for Semiconductors and Informatics, Kumamoto University, Kumamoto 860-8555, Japan; and Sigma-i Co., Ltd., Tokyo 108-0075, Japan

  • *Contact author: mohzeki@tohoku.ac.jp
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Phys. Rev. Research 8, 033170 – Published 10 August, 2026

DOI: https://doi.org/10.1103/tg4c-7cnr

Abstract

We formulate stationary-density-preserving nonreversible perturbations of Fokker-Planck dynamics as gauge fields that deform relaxation spectra while leaving the invariant state fixed. When detailed balance holds, a similarity transformation maps the reversible Fokker-Planck operator to the scalar sector of a Witten-Laplacian-type supersymmetric Hamiltonian; nonreversible gauges then appear as non-Hermitian perturbations that preserve the zero mode but modify the excited spectrum. This operator viewpoint gives a common language for relaxation gaps, circulating probability currents, hypocoercive acceleration, and finite control costs. We represent admissible gauge currents by antisymmetric tensor fields and identify the detailed-balance-violating Ohzeki-Ichiki force as a constant symplectic example whose infinite-strength limit is Hamiltonian dynamics. The continuous-time spectral gap alone does not select a finite gauge strength, so we introduce a finite-time regularized objective and an actor-critic procedure for learning the gauge. An exactly solvable anisotropic Gaussian Ornstein-Uhlenbeck benchmark separates the spectral transition from the finite-time optimum and shows that the learned gauge recovers the Lyapunov-equation optimum. A double-well benchmark then illustrates the same constrained selection in a nonconvex metastable landscape. Stochastic gradient methods enter this framework as physically relevant Fokker-Planck systems: Minibatch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.

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Phys. Rev. Research 8, 033170– Published 10 August, 2026
  • Received 4 June 2026
  • Accepted 14 July 2026
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