Spatial pattern formation induced by Gaussian white noise

TL;DR

This paper reviews how Gaussian white noise can induce spatial patterns in dynamical systems through stochastic mechanisms involving local dynamics, environmental noise, and spatial coupling, supported by analytical and numerical methods.

Contribution

It provides a comprehensive overview of stochastic mechanisms for pattern formation driven by Gaussian noise, combining analytical and numerical approaches.

Findings

Gaussian noise can induce ordered spatial states.

Analytical tools confirm noise-induced pattern formation.

Numerical simulations support theoretical predictions.

Abstract

The ability of Gaussian noise to induce ordered states in dynamical systems is here presented in an overview of the main stochastic mechanisms able to generate spatial patterns. These mechanisms involve: (i) a deterministic local dynamics term, accounting for the local rate of variation of the field variable, (ii) a noise component (additive or multiplicative) accounting for the unavoidable environmental disturbances, and (iii) a linear spatial coupling component, which provides spatial coherence and takes into account diffusion mechanisms. We investigate these dynamics using analytical tools, such as mean-field theory, linear stability analysis and structure function analysis, and use numerical simulations to confirm these analytical results.