An introduction to pattern formation in nonequilibrium systems

Patterns which arise in various nonequilibrium systems are discussed from an elementary but unified point of view. The phenomena of interest range from hydrodynamic flows (Rayleigh- Benard convection, Taylor-Couette flow, parametric waves) to chemical instabilities and morphogenesis in biological systems. A unifying feature is provided by linear stability theory which leads to a classification of patterns depending on the spatial and temporal scales of the instability. Near the instability point a universal amplitude equation is derived and used to elucidate many elementary properties of the system. The simplest solutions of this equation, which we call ideal patterns, are those with maximal symmetry. Some of their properties can be described also away from threshold by means of phase equations or more abstractly in terms of topological concepts. Real patterns, which we consider next, differ from ideal ones through the influence of boundaries and the loss of symmetry caused by spatial disorder. The simplest example of spatial disorder is a single defect in an otherwise regular pattern. Our treatment of real patterns focuses almost exclusively on Rayleigh-Benard convection since this is by far the most studied system. The importance of boundaries and defects is illustrated in the study of wavevector selection and in the description of pattern dynamics near threshold. Numerical and analytical studies of model equations have proved useful in the exploration of these difficult problems, but our present level of understanding is still far from complete.