The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   35

Pattern Formation Out of Equilibrium

by M. C. Cross and P. Hohenberg

Reviews of Modern Physics 65 (1993): 851--1112
(Technically, this is not a book, but an article in Reviews of Modern Physics; but anything that weighs in at two hundred and sixty pages and masses about a kilo is book enough for this Review.)

Equilibrium pattern formation is a standard part of the physics curriculum; as the spontaneous breaking of spatial symmetries in thermodynamic equilibrium, it's covered in many good books on statistical mechanics and condensed matter physics (and, naturally, even more bad books on those same subjects). Pattern formation away from thermodynamic equilibrium is a much less well understood subject, where competing formalisms and opinions bordering on metaphysics (or even over the border) proliferate. Cross and Hohenberg's book is one of the best available on the subject, and a sign that some light is beginning to emerge, as well as heat.

To begin with, the authors cover all the main experimental systems used to study physical pattern formation, and their phenomenological descriptions; there is even some theoretical material on biological morphogenesis. (Fairly little attention is given to scaling considerations or power-law behavior, which the authors dismiss as uncommon outside the laboratory.) Typically, one can construct a dimensionless parameter showing just how far one of these systems is from equilibrium; below a certain critical value of this parameter, the system remains homogeneous, but above it spatial symmetries are broken spontaneously and patterns form. This suggests of course phase transitions in equilibrium systems, and a large and vocal body of theorists have tried to capitalize on this analogy. The products of this "ponderous industry of theoretical elaboration" (Arthur Winfree) are given about half a dozen paragraphs here and then ignored, since they have contributed almost nothing to our substantive understanding of the phenomena. Cross and Hohenberg do not, however, merely present an eclectic collection of models, though their unifying principles are rather different.

Near the threshold one is dealing with initially small disturbances from a stationary, homogeneous state. This suggest representing the inhomogeneities and oscillations as sums of plane waves, and finding an equation for the relative growth rates of the different Fourier components, ignoring their phases (which is why this approach is called the "envelope formalism"). This can be done through perturbation theory, starting with the "microscopic" equations for the system in question, and it is gratifying and somewhat surprising that the resulting "amplitude equations" almost always take one of a small number of forms, depending on the spatial dimension of the system and what wave-vector the most unstable mode has. (This is suggestive of both the universality classes induced by the renormalization-group in equilibrium phase transitions, and more directly of the normal forms of bifurcation theory; some connection to the latter has already been established, but there are few formal results on this.)

Far from the threshold, in what is sometimes called the fully nonlinear regime, the approach is to find a uniform solution to the nonlinear field equations, and then apply a modulation to it, i.e. give it a phase or phases. The result is known, naturally enough, as a "phase equation," and it describes the time-evolution of the phase field. This reliance on a continuous phase field throughout the system runs into two problems. One is mathematical: in general, there are singularities in the phase field. (What is the longitude of the poles?) The other is physical, and has to do with the fact that pattern formation arises from symmetry-breaking; typically, different parts of the system break into different degenerate states; the places where they have irreconcilable differences are the places where the assumptions behind the phase equations break down, and form the singularities of the phase field. (It is only the phase field which is singular, of course; the microscopic variables are always continuous.) This is familiar from the theory of equilibrium phase transitions, where, following crystallographic usage, the points of breakdown are known as "defects," and take various forms --- dislocations, grain boundaries, wedges, twists, screws, etc.; all of these have analogs in non-equilibrium pattern formation. These are persistent entities whose form and dynamics can be worked out from the microscopic equations, and which structure the phase field in their vicinity. They can in fact be viewed as localized particles interacting with one another through the phase field, though owing to the complication of the equations there are very few cases where matters have been reduced to a field theory of the sort we all know and love. It hardly seems fair to call such useful, resilient and strangely familiar things "defects," so the rather more PC names "coherent structures" and "organizing centers" are also used.

Such is the modeling framework; it works, in a most gratifying manner, particularly in the best-studied experimental systems, like Rayleigh-Bernard convection and Taylor-Couette flow. One searches the works of Prigogine and his ilk in vain for the most attractive pictures in science, graphs where the solid black line of theory has experimental data-points growing on it like lichens; Cross and Hohenberg are able to include many such beauties, usually one-parameter fits. All of this is obtained from the so-called microscopic equations --- physically realistic equations like Navier-Stokes --- through well-controlled approximations and perturbative techniques, without invoking uncontrolled term-discarding and slaving, or strange extensions to thermodynamics, or dubious variational principles. Those of us an empirical temper could hardly ask for more --- which is not of course to say that non-equilibrium pattern formation is a closed subject!

Cross and Hohenberg's writing is colorless but clear. They assume familiarity with many areas of physics (thermodynamics, statistical mechanics, electromagnetism, hydrodynamics) but not expertise; also a good deal of mathematical competence. It is probably beyond the range of any undergraduate, but should be accessible to beginning graduate students in physics willing to work; more advanced graduate students can be lazier. An index would have been exceedingly helpful, though the highly detailed table of contents partially makes up for its absence. Considering its range, depth and quality of coverage, and not least its price, it is a volume which belongs on the shelf of anyone studying physical pattern formation.

262 pp., black and white illustrations, analytical table of contents, bibliography, no index
Condensed Matter / Physics Self-Organization, Complexity, etc.
20 March 1998 (updated link to article, 26 December 2010)