Dichotomous Markov noise: Exact results for out-of-equilibrium systems. A review

TL;DR

This review discusses exact analytical results for nonequilibrium systems driven by dichotomous Markov noise, highlighting recent advances in handling unstable fixed points and illustrating applications like hypersensitive transport and ratchets.

Contribution

It reveals how to modify calculations for systems crossing unstable fixed points, expanding the analytical understanding of dichotomous noise in nonequilibrium dynamics.

Findings

Standard long-time results fail when unstable fixed points are crossed.

Modified calculations provide accurate descriptions in these cases.

Dichotomous noise can be analyzed as effectively as Gaussian white noise.

Abstract

Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on {\em analytically-solvable} situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications -- the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on a par with Gaussian white noise as far as obtaining analytical results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications.