[Paper Review] Dichotomous Markov noise: Exact results for out-of-equilibrium systems. A review

Iosif Bena|arXiv (Cornell University)|Jun 5, 2006

stochastic dynamics and bifurcation206 references151 citations

TL;DR

This paper presents exact analytical solutions for nonequilibrium systems driven by dichotomous Markov noise (DMN), a non-Gaussian, colored noise with finite correlation time. It demonstrates that standard long-time results fail when unstable fixed points are crossed, and derives corrected analytical treatments for such cases, revealing nontrivial noise-induced behaviors in systems like ratchets and transport models, thereby establishing DMN as a viable alternative to Gaussian white noise for exact analysis in nonlinear, out-of-equilibrium dynamics.

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.

Motivation & Objective

• To address the long-standing gap in analytical treatment of nonequilibrium systems driven by dichotomous Markov noise (DMN), particularly when unstable fixed points are traversed in the asymptotic regime.
• To correct the standard long-time approximations that break down in the presence of unstable fixed points, which have been overlooked in prior literature.
• To demonstrate that DMN can yield exact analytical results comparable to those obtainable with Gaussian white noise, despite its non-Gaussian and colored nature.
• To illustrate the interplay between noise and nonlinearity in generating complex behaviors such as directed transport and stochastic resonance.
• To provide a rigorous foundation for using DMN as a modeling tool in systems where Gaussian noise approximations fail or are ambiguous.

Proposed method

• Derives the exact time evolution of the Fokker-Planck equation for systems driven by additive and multiplicative DMN, using the master equation approach.
• Applies the stochastic Liouville equation and Van Kampen's lemma to handle functional derivatives of the noise, ensuring consistency in the presence of non-Markovian effects.
• Uses the Shapiro-Longinov formula for differentiation of functionals of the noise to derive exact expressions for moments and probability densities.
• Introduces a modified perturbative approach that accounts for unstable fixed points by redefining the asymptotic behavior of the system's probability distribution.
• Analyzes the system's response through the invariant measure and steady-state probability density, derived from the backward Fokker-Planck equation.
• Demonstrates that the DMN can be mapped to both white shot noise and Gaussian white noise in appropriate limits, validating its use as a bridge between discrete and continuous noise models.

Experimental results

Research questions

• RQ1 Why do standard long-time analytical results for systems driven by dichotomous Markov noise fail when unstable fixed points are crossed?
• RQ2 How can the asymptotic behavior of stochastic systems with unstable fixed points be correctly modeled when driven by DMN?
• RQ3 What are the exact analytical solutions for the first-passage time and steady-state distribution in nonlinear systems driven by DMN?
• RQ4 In what ways does DMN induce nontrivial transport and phase transitions not captured by Gaussian noise approximations?
• RQ5 Can DMN be used as a rigorous alternative to Gaussian white noise in analytically solvable models of nonequilibrium systems?

Key findings

• The standard long-time approximation for systems driven by DMN breaks down when unstable fixed points are crossed, leading to incorrect predictions about the steady-state distribution.
• A corrected analytical treatment is derived that properly accounts for the system's evolution through unstable fixed points, yielding exact expressions for the asymptotic probability density.
• The paper shows that the invariant measure for systems with unstable fixed points under DMN is not a delta function at the fixed point, but a nontrivial distribution with finite width.
• For the hyper-sensitive transport model, the exact solution reveals a non-monotonic dependence of the current on noise parameters, indicating a resonant-like behavior not predicted by linear approximations.
• In the rocking ratchet model, the exact solution confirms the existence of directed transport even in the absence of a net bias, driven purely by the asymmetry of the DMN and system nonlinearity.
• The stochastic Stokes’ drift is shown to be exactly solvable under DMN, with the mean displacement scaling as √D/τc, confirming the non-Gaussian nature of the noise-induced drift.