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.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The synchronization of chaotic systems

The theory of phase-ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase, is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, for example vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.

Theory of phase-ordering kinetics

The goal of this survey is to review recent developments in the hydro­ dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. We wish to dem­ onstrate how these notions can be used effectively to obtain a qualitative and quantitative description of the spatio-temporal dynamics of open shear flows, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, etc. In this review, we only consider open flows where fluid particles do not remain within the physical domain of interest but are advected through downstream flow boundaries. Thus, for the most part, flows in "boxes" (Rayleigh-Benard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc.) are not discussed. Further­ more, the implications of local/global and absolute/convective instability concepts for geophysical flows are only alluded to briefly. In many of the flows of interest here, the mean-velocity profile is non-

/pdf/local-and-global-instabilities-in-spatially-developing-flows-ja4x14nrk6.pdf

Local and global instabilities in spatially developing flows

A general scheme is presented to evaluate the mobility tensors of an arbitrary number of spheres, immersed in a viscous fluid, in a power series expansion in R-1, where R is a typical distance between spheres. Some general properties of these (translational and rotational) mobility tensors are discussed. Explicit expressions are derived up to order R-7. To this order, hydrodynamic interactions between two, three and four spheres contribute.

Many-sphere hydrodynamic interactions and mobilities in a suspension

In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ``wave number'' that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -\ensuremath{\gamma} ${\ensuremath{\partial}}^{4}$\ensuremath{\varphi}/\ensuremath{\partial}${x}^{4}$ to it. I predict that for \ensuremath{\gamma}(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form \ensuremath{\varphi}(x-vt) that propagates with the marginal stability velocity. For \ensuremath{\gamma}g(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts \ensuremath{\varphi}(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

/pdf/front-propagation-into-unstable-states-marginal-stability-as-3vpf0drhs0.pdf

Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection

Uniformly translating solutions of the one-dimensional complex Ginzburg-Landau equation are studied near a subcritical bifurcation. Two classes of solutions are singled out since they are often produced starting from localized initial conditions: moving fronts and stationary pulses. A particular exact analytic front solution is found, which is conjectured to control the relative stability of pulses and fronts. Numerical solutions of the Ginzburg-Landau equation confirm the predictions based on this conjecture.

/pdf/pulses-and-fronts-in-the-complex-ginzburg-landau-equation-29uh1hzp33.pdf

Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation.

On montre que la transition de la stabilite lineaire marginale a la stabilite non lineaire marginale peut etre decrite dans une representation dynamique de la propagation du front dans des etats instables. L'approche resultante donne une interpretation physique et une extension des resultats d'Aronson et Weinberger et de Ben-Jacob

/pdf/front-propagation-into-unstable-states-ii-linear-versus-340v8ay8q3.pdf

Front Propagation into Unstable States II : Linear versus Nonlinear Marginal Stability and Rate of Convergence

http://www.lorentz.leidenuniv.nl/~saarloos/Papers/chaoscgl.pdf

W. van Saarloos

Papers

Many-sphere hydrodynamic interactions and mobilities in a suspension

Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection

Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation.

Front Propagation into Unstable States II : Linear versus Nonlinear Marginal Stability and Rate of Convergence

Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation