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 q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

Pattern selection in fingered growth phenomena

When a low-viscosity fluid penetrates a fluid of higher viscosity confined by parallel plates, finger-like patterns propagate at the interface between the two fluids. Experiments now show that tapering the fluid cell can suppress this instability - providing interfacial control via a simple change in geometry.

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Control of interfacial instabilities using flow geometry

Pattern formation during the directional solidification of binary systems

We introduce a simple model of self-organizing hierarchies in animal societies which relies on a basic positive feedback mechanism reinforcing the ability of a given individual to win or lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. If a forgetting strength is included, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual, subcritical or supercritical bifurcations in the formation of the hierarchical structure are observed as the density ϱ of individuals is varied. The nature of the transition is shown to depend on a parameter η, analogous to the inverse of a temperature, defining the amount of determinism in the outcomes of the fights. We therefore observe a dynamical tricritical point in the ϱ-η plane.

Phase diagram of a model of self-organizing hierarchies

An investigation is made of the stability of the shape of a moving planar interface between the liquid and solid phases in the freezing of a dilute binary alloy. A nonlinear model is used to describe an experimental situation in which solidification is controlled so that the mean position of the interface moves with constant speed. The model postulates two-dimensional diffusion of solute and heat such that: 1. Convection in the liquid is negligible. 2. Diffusion of the solute in the solid is negligible. 3. Solute concentration in the liquid is small. 4. The effects of interface attachment kinetics are negligible. 5. The extent of the liquid and solid phases is infinite. 6. c8=c1 where c8(c1) is the specific heat per unit volume of the solid (liquid). 7. (D/Dth) 1, , where D is the diffusion coefficient of the solute in the liquid and Dth is the thermal diffusivity in the liquid. 8. G= G where G is the imposed temperature gradient in the liquid and Gcis the critical value of G at which linear theory predicts the onset of instability. The analysis is expected to be asymptotically valid as G-Gc. It is found that the interface can be unstable to finite amplitude disturbances even when linear stability theory predicts stability to infinitesimal disturbances. Further, cellular structure can be anticipated for certain ranges of parameter values. These results are in accord with relevant experimental evidence.

L. A. Segel

Papers

A nonlinear stability analysis of the freezing of a dilute binary alloy

A nonlinear stability analysis of the freezing of a dilute binary alloy