Spectral properties of a time-periodic Fokker-Planck equation

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Abstract
The Floquet spectrum of the time-periodic Fokker-Planck equation for a driven Brownian rotor is studied. We show that the Fokker-Planck equation can be transformed to a Schrödinger-like equation, with the same set of eigenvalues, whose dynamics is governed by a time-periodic Hamiltonian in which the diffusion coefficient plays a role analogous to Planck’s constant. For a small diffusion coefficient, numerical calculations of the spectrum starting from the Schrödinger-like equation are more convergent than those starting from the Fokker-Planck equation. When the Hamiltonian exhibits a transition to chaos, those decay rates affected by the chaotic regime exhibit level repulsion. This level repulsion of decay rates, in turn, changes the behavior of a typical mean first passage time in the problem. The size of the diffusion coefficient determines the extent to which the stochastic dynamics is affected by the transition to chaos in the underlying Hamiltonian.

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