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Detailed balance and the quantum-mechanical Fokker-Planck equations

H. J. Carmichael, Daniel F. Walls

Physics Letters AMay 1, 19753 citations

Stochastic quantization and path integral formulation of Fokker-Planck equation

L. Garrido, D. Lurié, Maxi San Miguel

Physics Letters AAug 1, 19784 citations

Quantum Fokker–Planck Dynamics

Louis E. Labuschagne, W. A. Majewski

arXiv:2106.05718Annales Henri PoincaréNov 22, 20211 citation

Nonreversible Gauge Fields in Fokker--Planck Dynamics: Supersymmetric Hamiltonians and Learned Finite Forces

Masayuki Ohzeki

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 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: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.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 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: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.

cond-mat.dis-nnCross-listedarXiv:2606.06412Jun 5, 2026

Relation between canonical and fokker-planck dynamics

L. Garrido, José María Sancho

Physics Letters AOct 1, 19781 citation

Generalized quantum Fokker–Planck theory and its application to laser driven intramolecular hydrogen transfer reactions in condensed phases

Oliver Kühn, Yi Dong Zhao, Feng Shuang +1 more

A generalized quantum Fokker–Planck theory is proposed to treat the correlated dynamics of coherent driving and Markovian dissipation. The resulting formulation is applicable to arbitrary external time-dependent driving fields and satisfies the detailed balance condition at arbitrary temperatures. Analyzed are also the formal relations among the Caldeira–Leggett quantum Fokker–Planck equation, the Bloch–Redfield theory, and the present formulation. The approach is numerically implemented to study the intramolecular hydrogen transfer reaction dynamics in a one-dimensional model system. Different forms of external pulsed driving fields are exploited and their ability to compete with concurring relaxation processes is investigated. Energy relaxation and pure dephasing are shown to have rather different influences on the reaction yield.A generalized quantum Fokker–Planck theory is proposed to treat the correlated dynamics of coherent driving and Markovian dissipation. The resulting formulation is applicable to arbitrary external time-dependent driving fields and satisfies the detailed balance condition at arbitrary temperatures. Analyzed are also the formal relations among the Caldeira–Leggett quantum Fokker–Planck equation, the Bloch–Redfield theory, and the present formulation. The approach is numerically implemented to study the intramolecular hydrogen transfer reaction dynamics in a one-dimensional model system. Different forms of external pulsed driving fields are exploited and their ability to compete with concurring relaxation processes is investigated. Energy relaxation and pure dephasing are shown to have rather different influences on the reaction yield.

The Journal of Chemical PhysicsApr 8, 200030 citations

Stochastic quantization and detailed balance in Fokker-Planck dynamics

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