Random perturbations of dynamical systems

Random perturbations of dynamical systems

Dynamical systems are often subject to random perturbations or noise. Even when the noise amplitude is small, it has a profound influence on the dynamics on the appropriate time-scale. At a first sight one might think that this influence happens at such a long time-scale that it is rarely of practical importance. That this is an incomplete view may be understood by noticing that most physical processes will not happen at zero temperature when thermal noise is absent. In reality nature presents one with a very wide range of temporal scales. The small noise induces events which are rare with respect to the internal clock of the system, but this clock can be very fast, and the processes of interest to our daily lives are mostly on the order of seconds or longer. This leaves plenty of room for rare events caused by thermal noise to make their appearance. In fact, phenomena like nucleation events during phase transitions, chemical reactions, conformation changes of biomolecules, bistable behaviors in genetic switches, or regime changes in climate are just a few examples of rare events among many others.

Traditionally the methods of choice for a quantitative understanding of the effect of noise has been Monte Carlo or direct simulation of Langevin equations. When the noise is small, which is the case of interest here, these methods become prohibitively expensive, due to the presence of two disparate time-scales: the time-scale of the deterministic dynamics and the time-scale between the rare events caused by the noise. Noticing this difficulty, alternative theories and numerical methods have been proposed.

This section contains our papers in the context of dynamical phase transformations; papers in which limiting dynamics are derived for various systems subject to random perturbations; and papers on self-induced stochastic resonance (SISR), a phenomenon similar to standard stochastic resonance but in which the external forcing is replaced by some internal reset mechanism.

Dynamical phase transformations In the papers below we study noise-induced phase transitions in various mean-field models such as the Allen-Cahn equation or the Ginzburg-Landau-Gilbert equation in micromagnetism.

Breakup of universality in the generalized spinodal nucleation theory, with C. B. Muratov, J. Stat. Phys. 114, 605-623 (2004).
Magnetic elements at finite temperature and large deviation theory, with. R. V. Kohn and M. Reznikoff, J. Nonlinear Sci. 15, 223-253 (2005).
Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation, with R. V. Kohn, F. Otto and M. Reznikoff, to appear in: Comm. Pure App. Math

Limiting dynamics In these papers, limiting dynamical models are derived from various systems subject to random perturbations in specific limits (large number of degrees if freedom, large volume limit, small noise, etc.).

Statistical description of contact-interacting Brownian walkers on the line, with I. Fatkullin, J. Stat. Phys. 111, 565-679 (2003).
Coarsening by diffusion-annihilation in a bistable system driven by noise, with I. Fatkullin, unpublished.
Stochastic models for selected slow variables in large deterministic systems, with A. J. Majda and I. Timofeyev, Nonlinearity 19 769-794 (2006).
A strong limit theorem in Kac-Zwanzig model, with G. Ariel, submitted to: Comm. Math. Phys.

Self-induced stochastic resonance Stochastic resonance is a well-known phenomenon by which the effect of small noise can be amplified via nontrivial interplay with an external deterministic forcing. The papers below study a generalization of stochastic resonance which is self-induced in that no external forcing is required: the amplification of the effects of the noise arise due to the interplay between this noise and the deterministic dynamics of the system. Self-induced stochastic resonance (SISR) is relevant in the context of excitable media, where it leads to unexpected limit-cycle behavior and in the context of motor proteins, where it leads to regularity and synchrony.

SISR in excitable media

Self-induced stochastic resonance in excitable systems, with W. E and C. B. Muratov, Physica D 210, 227-240 (2005).
Two distinct mechanisms of coherence in randomly perturbed dynamical systems, with R. E. Lee DeVille and C. B. Muratov, Phys. Rev. E 72, 031105 (2005).
Non-meanfield deterministic limits in chemical reaction kinetics, with R. E. Lee DeVille and C. B. Muratov, J. Chem. Phys. 124, 231102 (2006).
Wavetrain response of an excitable medium to local stochastic forcing, with R. E. Lee DeVille, to appear in: Nonlinearity.
Noise can play an organizing role for the recurrent dynamics in excitable media, with W. E and C. B. Muratov, submitted to: Proc. Nat. Acad. USA.

SISR in molecular motors

A nontrivial scaling limit for multiscale Markov chains, with R. E. Lee DeVille, to appear in: J. Stat. Phys
Regularity and synchrony in motor proteins, with R. E. Lee DeVille, submitted to: Bull. Math. Bio.