Pattern Formation in Nonequilibrium Physics
Total Page：16
File Type：pdf, Size：1020Kb
Haverford College Haverford Scholarship Faculty Publications Physics 1999 Pattern formation in nonequilibrium physics Jerry P. Gollub Haverford College J. S. Langer Follow this and additional works at: https://scholarship.haverford.edu/physics_facpubs Repository Citation J.P.Gollub and J.S. Langer, "Pattern Formation in Nonequilibrium Physics", in the Centennial Issue of Reviews of Modern Physics, 71, S396-S403 (1999). This Journal Article is brought to you for free and open access by the Physics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact [email protected]. Pattern formation in nonequilibrium physics J. P. Gollub Haverford College, Haverford, Pennsylvania 19041 and Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 J. S. Langer Department of Physics, University of California, Santa Barbara, California 93106 Remarkable and varied pattern-forming phenomena occur in ﬂuids and in phase transformations. The authors describe and compare some of these phenomena, offer reﬂections on their similarities and differences, and consider possibilities for the future development of this ﬁeld. [S0034-6861(99)04702-9] I. INTRODUCTION for the present purposes are spatially extended dynami- cal systems with many degrees of freedom, for which The complex patterns that appear everywhere in na- partial differential equations are needed. Corresponding ture have been cause for wonder and fascination physical systems include ﬂuids subjected to heating or throughout human history. People have long been rotation, which exhibit sequences of increasingly com- puzzled, for example, about how intricate snowﬂakes plex spatiotemporal patterns as the driving forces can form, literally, out of thin air; and our minds boggle change. These are all purely deterministic pattern- at the elegance of even the simplest living systems. As forming systems. Understanding how they behave has physicists, we have learned much about natural pattern been a crucial step toward understanding deterministic formation in recent years; we have discovered how rich chaos—one of the most intriguing and profound scien- this subject can be, and how very much remains to be tiﬁc concepts to emerge in this century. understood. Our growing understanding of the physics At the center of our modern understanding of pattern of pattern formation has led us to speculate—so far with formation is the concept of instability. It is interesting to only limited success—about a more general science of note that the mathematical description of instabilities is complexity, and to pose deep questions about our ability strikingly similar to the phenomenological theory of to predict and control natural phenomena. phase transitions ﬁrst given by Landau (Landau and Lif- Although pattern formation—i.e., morphogenesis— shitz, 1969). We now know that complex spatial or tem- has always been a central theme in natural philosophy, it poral patterns emerge when relatively simple systems has reemerged in mainstream nonequilibrium physics are driven into unstable states, that is, into states that only in the last quarter of the 20th Century. This has will deform by large amounts in response to inﬁnitesi- happened, in part, as an outgrowth of physicists’ and mally small perturbations. For example, solar heating of materials scientists’ interest in phase transitions. Many the earth’s surface can drive Rayleigh-Be´nard-like con- of the most familiar examples of pattern formation oc- vective instabilities in the lower layer of the atmosphere, cur in situations in which a system is changing from one and the resulting ﬂow patterns produce fairly regular phase to another—from a liquid to a geometrically pat- arrays of clouds. At stronger driving forces, the convec- terned solid, for example, or from a uniform mixture of tion patterns become unstable and turbulence increases. chemical constituents to a phase-separated pattern of Another familiar example is the roughness of fracture precipitates. As scientists have learned more about the surfaces produced by rapidly moving cracks in brittle equilibrium aspects of phase transitions, many have be- solids. When we look in detail, we see that a straight come interested in the non-equilibrium processes that crack, driven to high enough speeds, becomes unstable accompany them. This line of investigation has led di- in such a way that it bends, sends out sidebranching rectly to questions of pattern formation. cracks, and produces damage in the neighboring mate- Another direction from which physicists have ap- rial. In this case, the physics of the instability that leads proached the study of pattern formation has been the to these irregular patterns is not yet known. theory of nonlinear dynamical systems. Mechanical sys- After an instability has produced a growing distur- tems that can be described by ordinary differential equa- bance in a spatially uniform system, the crucial next step tions often undergo changes from simple to complex be- in the pattern-forming process must be some intrinsi- havior in response to changes in their control cally nonlinear mechanism by which the system moves parameters. For example, the periodically forced and toward a new state. That state may resemble the un- damped pendulum shows chaotic motion for certain in- stable deformation of the original state—the convective tervals of the forcing amplitude, as well as periodic win- rolls in the atmosphere have roughly the same spacing as dows within the chaotic domains—a temporal ‘‘pattern’’ the wavelength of the initial instability. However, in with considerable complexity. This is a simple case, how- many other cases, such as the growth of snowﬂakes, the ever, with only a few degrees of freedom. More relevant new patterns look nothing like the linearly unstable de- S396 Reviews of Modern Physics, Vol. 71, No. 2, Centenary 1999 0034-6861/99/71(2)/396(8)/$16.60 ©1999 The American Physical Society J. P. Gollub and J. S. Langer: Pattern formation in nonequilibrium physics S397 formations from which they started. The system evolves The best way to visualize the solutions of such equa- in entirely new directions as determined by nonlinear tions is to think of them as points in a multidimensional dynamics. We now understand that it is here, in the non- mathematical space spanned by the dynamical variables, linear phase of the process, that the greatest scientiﬁc that is, a ‘‘phase space.’’ The rules that determine how challenges arise. these points move in the phase space constitute what we The inherent difﬁculty of the pattern-selection prob- call a ‘‘dynamical system.’’ One of the most important lem is a direct consequence of the underlying (linear or developments in this ﬁeld has been the recognition that nonlinear) instabilities of the systems in which these dynamical systems with inﬁnitely many degrees of free- phenomena occur. A system that is linearly unstable is dom can often be described by a ﬁnite number of rel- one for which some response function diverges. This evant variables, that is, in ﬁnite-dimensional phase means that pattern-forming behavior is likely to be ex- spaces. For example, the ﬂow ﬁeld for Rayleigh-Be´nard tremely sensitive to small perturbations or small changes convection not too far from threshold can be described in system parameters. For example, many patterns that accurately by just a few time-dependent Fourier ampli- we see in nature, such as snowﬂake-like dendrites in so- tudes. If we think of the partial differential equations as lidifying alloys, are generated by selective ampliﬁcation being equivalent to ﬁnite sets of coupled ordinary differ- of atomic-scale thermal noise. The shapes and speeds of ential equations, then we can bring powerful mathemati- growing dendrites are also exquisitely sensitive to tiny cal concepts to bear on the analysis of their solutions. crystalline anisotropies of surface energies. As we shall emphasize in the next several sections of Some important questions, therefore, are: Which per- this article, dynamical-systems theory provides at best a turbations and parameters are the sensitively controlling qualitative framework on which to build physical models ones? What are the mechanisms by which those small of pattern formation. Nevertheless, it has produced valu- effects govern the dynamics of pattern formation? What able insights and, in some cases, has even led to predic- are the interrelations between physics at different length tion of novel effects. It will be useful, therefore, to sum- scales in pattern-forming systems? When and how do marize some of these general concepts before looking in atomic-scale mechanisms control macroscopic phenom- more detail at speciﬁc examples. An introductory discus- ena? At present, we have no general strategy for an- sion of the role of dynamical systems theory in ﬂuid me- swering these questions. The best we have been able to chanics has been given by Aref and Gollub (1996). do is to treat each case separately and—because of the In dynamical-systems theory, the stable steady solu- remarkable complexity that has emerged in many of tions of the equations of motion are known as ‘‘stable these problems—with great care. ﬁxed points’’ or ‘‘attractors,’’ and the set of points in the In the next several sections of this article we discuss phase space from which trajectories ﬂow to a given ﬁxed the connection between pattern formation and nonlin- point is its ‘‘basin of attraction.’’ As the control param- ear dynamics, and then describe just a few speciﬁc ex- eters are varied, the system typically passes through ‘‘bi- amples that illustrate the roles of instability and sensitiv- furcations’’ in which a ﬁxed point loses its stability and, ity in nonequilibrium pattern formation. Our examples at the same time, one or more new stable attractors ap- are drawn from ﬂuid dynamics, granular materials, and pear. An especially simple example is the ‘‘pitchfork’’ crystal growth, which are topics that we happen to know bifurcation at which a stable ﬁxed point representing a well.