Exact Results for Spectra of Overdamped Brownian Motion in Fixed and Randomly Switching Potentials

TL;DR

This paper derives exact spectral formulas for overdamped Brownian motion in fixed and randomly switching potentials, revealing phenomena like spectrum narrowing, nonlinear effects, and a new characterization of resonant activation.

Contribution

It provides exact analytical results for spectra in bistable and switching potentials, highlighting novel behaviors and the resonant activation phenomenon.

Findings

Spectrum narrows exponentially with barrier height.

Spectrum narrows with increased switching rate.

Nonmonotonic spectrum behavior at zero frequency.

Abstract

The exact formulae for spectra of equilibrium diffusion in a fixed bistable piecewise linear potential and in a randomly flipping monostable potential are derived. Our results are valid for arbitrary intensity of driving white Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an exponentially rapid narrowing of the spectrum with increasing height of the potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon, which manifests in the narrowing of the spectrum with increasing mean rate of flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero frequency, as a function of the mean rate of switchings, for randomly switching potential. The last feature is a new characterization of resonant activation phenomenon.