Weiss mean-field approximation for multicomponent stochastic spatially extended systems

TL;DR

This paper develops a mean-field approach and numerical method for analyzing multicomponent stochastic spatial systems, specifically applied to a chemical reaction model, revealing complex solution behaviors near bifurcation points.

Contribution

It introduces a multivariate nonlinear self-consistent Fokker-Planck equation and a finite-difference numerical method for spatially extended stochastic systems, applied to the stochastic brusselator.

Findings

Multiple solution types near Turing bifurcation with increased noise

Existence of transient bimodality and probability density repumping

Behavior of the order parameter indicating transition dynamics

Abstract

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method. Numerical study of the nonlinear self-consistent Fokker-Planck equation solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple repumping of probability density through bimodality. Additionally we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.