Pattern formation outside of equilibrium

Abstract

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector ${q}_{0}$ and frequency ${\ensuremath{\omega}}_{0}$ of the instability. Type ${\mathrm{I}}_{\mathrm{s}}$ systems (${\ensuremath{\omega}}_{0}=0$, ${q}_{0}\ensuremath{\ne}0$) are stationary in time and periodic in space; type ${\mathrm{III}}_{\mathrm{o}}$ systems (${\ensuremath{\omega}}_{0}\ensuremat…