Statistical dynamics with thermal noise

Summary

Abstract

We consider how to add noise to a non-linear system in a way that obeys the laws of thermodynamics. We treat a class of dynamical systems which can be expressed as a (possibly non-linear) motion through the set of probability measures on a sample space. Thermal noise is added by coupling this random system to a heat-particle distributed according to a Gibbs state. The theory is illustrated by the Brussellator, where it is shown that the noise converts a limit cycle into a global attractor. In the linear case it is shown that every Markov chain with transition matrix close to the identity is obtained by coupling to thermal noise with a bistochastic transition matrix.

Motivation

Suppose that we have a dynamical system whose (non-linear) equations depend on a parameter λ, and that there is a bifurcation to periodicity or chaos as we increase λ up to a critical value λ₀. It is then interesting to add noise to the system, and to ask whether this induces the system to bifurcate earlier, at some λ < λ₀, or whether it delays the onset until λ > λ₀. It might on the other hand change the nature of the criticality, of destroy it altogether.

Here we report on a study (made with L. Rondoni) of the chemical system known as the Brussellator modified to follow the activity-led law of mass action. For certain values of the rate constants the system exhibits a limit-cycle, which is converted into a global attractor when the system is coupled to thermal noise in a particular way.