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11 Nonlinear Dynamics, Instabilities, and Chaos INTRODUCTION A new area of condensed-matter physics has been defined as a coherent subfield in the past 10 years. When solid or fluid systems are forced away from equilibrium, they often become strongly nonlinear and as a result exhibit instabilities leading to chaotic or nonperiodic time evolution. In other cases, nonequilibrium systems develop pat- terned but nonperiodic spatial structures. From a general point of view, researchers in this field are concerned with the problem of predictabil- ity in disordered physical systems. Under what conditions can we hope to predict the future behavior of a system that is evolving chaotically, or the spatial structure of one that is highly fractured? Is any predict- ability preserved when a system behaves in an apparently irregular way? The process of obtaining answers to these questions should have a significant impact on fields of science extending far beyond con- densed-matter physics. One focus of experimental activity in this field has been the problem of understanding fluid instabilities (pattern changes) and turbulence, subjects of both fundamental interest and practical importance because of the simplicity of fluids and their ubiquity in nature. However, chaotic dynamics are increasingly being found in other areas of condensed-matter physics, such as superconducting devices and elec- tronic conduction in semiconductors. Phenomena exhibiting complex 215

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216 A DECADE OF CONDENSED-MATTER PHYSICS spatial patterns include the aggregation of smoke particles and the dendritic growth of crystals. One of the defining characteristics of this subfield is a strong interface with the area of mathematics known as dynamical systems theory, which provides a language and set of concepts for understand- ing chaotic dynamics as a consequence of nonlinear models with only a few degrees of freedom. This approach to understanding irregular behavior is different from the conventional viewpoint of statistical mechanics. It represents an expansion of the theoretical tools available to condensed-matter physics. The eventual impact of these new mathematical tools will probably be greater than that of the specific physical systems to which they are currently being applied. Although the discussion in this chapter emphasizes nonlinear dynamics in the context of instabilities and chaotic motion in fluids and other con- densed-matter systems, a much broader approach could be taken, in which the methods of nonlinear dynamics are applied to a wide range of interdisciplinary problems in plasma physics, astrophysics, and even accelerator technology. Physicists working on nonlinear problems sometimes publish in engineering or mathematical journals, and new journals devoted spe- cifically to nonlinear phenomena have been started. The interdiscipli- nary nature of nonlinear dynamics creates both opportunities for collaborations and special problems in structuring and funding re- search. These problems are different from those of the more estab- lished subfields of condensed-matter physics. MAJOR ADVANCES A New Paradigm We now know that chaotic motion in physical systems that dissipate energy can arise from nonlinear effects alone: that is, apparently random behavior arises from the internal dynamics of the system rather than from irregular external influences. The first extensive discussion of chaotic motion of this type in a mathematical model was due to Lorenz in 1963. Since that time, a new mathematical language and way of thinking about nonlinear dynamics has been developed by mathe- maticians and (more recently) physicists. The central concept is that of a state or phase space in which the state of a system is represented by a point, and its evolution appears as an extended trajectory. The trajectories form geometrical shapes or attractors in state space (see the discussion below in the section on

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NONLINEAR D YNAMICS, INSTABl LI TIES, AND CHA OS 2 17 Dynamical Systems Analysis of Experiments) whose forms provide a way of characterizing the actual behavior of the physical system. For example, closed loops in state space describe periodic oscillations; attractors having the topology of a torus describe more complex oscil- lations termed quasi-periodic; and forms known as strange attractors describe chaotic motion. A strange attractor is a set on which trajec- tories wander erratically. They are generally extremely complex; a strange attractor often has the form of an infinitely folded sheet of infinite extent. The description and explanation of the dynamics of chaotic systems is facilitated by a geometrical analysis of the shapes formed by the trajectories in state space. This dynamical systems analysis has been shown to be useful in unifying many diverse physical phenomena, because only a limited number of fundamentally different forms or types of attractors seem to be important experimentally. The develop- ment of tools and language for quantifying the properties of attractors has given physicists a powerful new way of thinking about the dynamics of nonlinear systems. New Experimental Methods New experimental methods were necessary to allow the ideas of nonlinear dynamics to be tested. Most importantly, laboratory com- puter techniques have been essential for experimental studies of complex dynamics. Automated data acquisition was required to obtain the large quantity of data needed for analysis. Computer methods have been needed not only to acquire data but also to analyze it. For example, space and time Fourier transform methods, which reveal the frequency content of the fluctuations, have been used extensively. Numerical methods of measuring the shapes and properties of strange attractors have been devised. Computer-enhanced shadowgraph im- ages reveal flow patterns too feeble to be observed by ordinary photographic or visual techniques (an example is shown in Figure 11.11. These methods are still being refined, and are gradually narrow- ing the gap between the abstract mathematical ideas of nonlinear dynamics and the behavior of physical systems. Routes to Chaos How is a chaotic fluid flow reached as it is forced more and more strongly? The stress on the system is characterized by a dimensionless control parameter (Reynolds number or Rayleigh number, for example,

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218 A DECADE OF CONDENSED-MATTER PHYSICS ''~7~ niU.i3~3-Set ayt_- FIGURE 11.1 Computer-enhanced shadowgraph images of convective flow in a container of circular cross section. The concentric flow state (a) was rendered unstable by reducing the Rayleigh number from 1.2 times the critical value Rt to about 1.1 times R.`.. Part of the resulting evolution of the flow field is given by bed. (Courtesy of G. Ahlers.) depending on the type of flow). Until about 1970, an endless sequence of instabilities was expected, each of which changes the spatial structure (pattern) of the flow or adds a new frequency of oscillation. In this picture, turbulent flow was essentially a complicated superposition of motion at many frequencies simultaneously. No qualitative change separated laminar and turbulent motion in this view. In 1971 a suggestion was made that turbulent flows could actually be modeled by strange attractors having only a few degrees of freedom. The chaotic dynamics of such models are dramatically different from a

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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 219 quasi-periodic flow with several frequencies and also quite different from chaos produced by many interacting variables. Considerable effort has been devoted to the testing of this hypothesis. Over the next few years (1974-1982) many groups explored and characterized the various sequences of instabilities leading to chaotic fluid motion that are obtained as the control parameter is varied. A limited number of well-defined routes to chaos were identified: 1. Period doubling, in which the basic period of oscillation doubles repeatedly at a sequence of thresholds forming a geometric series. This phenomenon is illustrated in Figure 1 1.2. 2. Quasi-periodicity, in which chaos is produced by several inter- acting oscillations at incommensurate frequencies. time. 3. Intermittency, in which laminar and chaotic phases alternate in These basic routes to chaotic motion were found both theoretically in simple models and experimentally in a limited class of fluid systems where the spatial pattern of the flow is fixed and the transition occurs continuously. We use the term chaos (or sometimes weak turbulence) to describe the resulting noisy time-dependent motion. The flow is not necessarily turbulent in the ordinary engineering sense because it may not contain eddies of many different spatial scales. Many of the same routes have also been observed in pen junction oscillators, semicon- ductor transport phenomena, superconducting Josephson junctions, chemical reactions, and elsewhere. These routes to chaos have univer- sal properties that transcend the characteristics of particular systems. The list given above is by no means exclusive; other routes occur in model systems and may be identified experimentally. Dynamical Systems Theory of the Routes to Turbulence The basic sequences of instabilities or transitions found experimen- tally are now fairly well understood theoretically. A set of theoretical methods known as renormalization-group techniques (developed orig- inally to describe continuous phase transitions in equilibrium systems) were crucial in understanding the universal properties of the basic routes to turbulence. The applicability of these methods in the new context depends on the fact that the internal structure of strange attractors often becomes identical (more precisely, self-similar) on all length scales as the transition to chaos is approached. Particularly noteworthy theoretical developments include prediction of the properties of the period doubling, quasi-periodicity, and

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220 A DECADE OF CONDENSED-MATTER PHYSICS R Rc 3.47 3.52 3.62 3.65 0 50 100 150 200 T(sec) FIGURE 11.2 Sequence of period doublings leading to chaos in Rayleigh-B(nard convection, as a function of the Rayleigh number R (proportional to the temperature difference across the fluid layer). The temperature variation is periodic' and the period doubles repeatedly as R is increased. For example, at RIRt. = 3.62, a complete cycle includes four large peaks. (Courtesy of A. Libchaber.) intermittence routes near the onset of chaos and the achievement of an understanding of the effect of external noise on these phenomena. These efforts have revealed new constants of nature and unexpected mathematics. The discovery that the routes to chaos have universal properties is a particularly exciting development and accounts in part for the speed with which nonlinear dynamics (applied especially but not exclusively to fluid motion) has been accepted as a part of condensed-matter physics.

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NONLINEAR D YNAMICS, INSTABILITIES, AND CHAOS 22 1 Dynamical Systems Analysis of Experiments The various routes to turbulence do not directly expose the proper- ties of the strange attractors that characterize chaotic systems. In fact, the goal of understanding strange behavior has not yet been achieved, in part because it has only recently become possible to observe the attractors directly. Whereas earlier work emphasized the use of frequency spectra, one can now use geometrical methods to study the forms of strange attractors in phase space and to measure their properties. An example of the shape of trajectories in phase space near the onset of chaotic fluid flow is shown in Figure li.3. When the FIGURE 11.3 Trajectories in phase space (left) and Poincare sections (right) for a rotating fluid below and above the onset of chaos. The flow is quasi-periodic in the lower panels and chaotic in the middle and upper panels. (Courtesy of H. L. Swinney.)

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222 A DECADE OF CONDENSED-MATTER PHYSICS trajectories are shown in the two-dimensional phase space, they are smooth in the laminar state (lower left) and convoluted in the chaotic state (upper left). By observing the orbits in three dimensions and then allowing them to intersect a plane, the Poincare maps on the right are obtained. The lower one (a closed loop) is the intersection of the torus with a plane. The upper one, where the flow is chaotic, is irregular because the torus has been replaced by a strange attractor. The middle case is just beyond the onset of the chaotic regime. One of the most important properties of a strange attractor is its dimensionality, which can be fractional. This surprising property arises because it often consists of an infinite number of closely spaced surfaces, yielding a dimension greater than that of a single surface but lower than that of a solid object, i.e., a dimension between two and three. A second important property is the rate at which nearby trajectories diverge from each other along certain directions within the attractor. This property of exponential divergence of nearby trajecto- ries or sensitive dependence on initial conditions is responsible for the chaotic behavior. Preliminary measurements of these two properties in experiments have provided solid evidence for the basic correctness of the dynamical systems approach to understanding the onset of chaos in a limited class of fluid-flow transitions. We now know that chaotic motion in a system having 1093 degrees of freedom (i.e., a fluid) can be effectively described in some cases by models having only a very few degrees of freedom (variables). Nonlinear Stability Theory The dynamical systems approach described in the last few sections is particularly useful for systems in which the spatial patterns of the flow are fixed (for example, convection in a layer whose lateral dimensions are small enough). However, in many situations (including those of greatest practical importance) this is not the case: time dependence is accompanied by large variations in the spatial pattern of the flow. One widely applicable theoretical approach that has proven successful in describing pattern changes is known as stability theory. In this method, one examines the linear stability of the static solutions of the nonlinear equations to infinitesimal perturbations. A consider- able amount has been learned about various types of secondary instabilities that modify the primary flow. Encouraging progress has also been made in understanding the elects of the system boundaries on the stability of the motion.

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NONLINEAR D YNAMICS. INSTABlLITlES, AND CHAOS 223 Pattern Evolution Studies of the evolution of spatial flow patterns resulting from instabilities are an important step in understanding the onset of chaotic How. During the past few years, an increasing effort has been devoted to studies of the evolution of flows, especially convective flows, by laser Doppler mapping and other methods. Particular attention has been directed to the effect of defects or dislocations on the evolution of the flow patterns. Theoretical efforts have proceeded in parallel with the experiments. Most of the important qualitative phenomena seem to be predictable on the basis of two-dimensional model equations that are far simpler than the full three-dimensional hydrodynamic equations. These models are used to describe approximately the slow spatial variations of the basic flow structure. Numerical computations based on such model equa- tions have been carried out successfully, and an example is shown in Figure 11.4. One remarkable development is that near the onset of convection? the evolution may possibly be described by the minimiza- tion of a quantity that depends on the concentration of defects, the amount of curvature in the pattern, and the orientation of the flow field with respect to the boundaries. Though evolution toward a minimum is surely not a general property of nonequilibrium systems, the usefulness of equilibrium concepts (approach to a minimum) in this context, even under restricted circumstances, is an important result. Many other systems also show fascinating patterns and pattern selection. The growth of crystals from the melt exhibits complex dendritic or snowflake forms. The growth of soot particles shows somewhat less regular, but fundamentally similar, patterned growth. These spatial structures are, like the attractors of the fluid, strange objects having fractional dimension. Various types of random motions can similarly be described in terms of objects with fractional dimension and can probably be understood using the language of dynamical systems. For example, the invasive percolation of a two-phase fluid in a porous medium, various aggrega- tion processes, and the behavior of electrons lopping in a random material all show dynamically produced structures of fractional dimen- sionality. Instabilities in Other Dissipative Systems Instabilities and chaos occur in many other condensed-matter sys- tems. For example, Josephson junction (superconducting) oscillators

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224 A DECADE OF CONDENSED-MATTER PHYSICS r=50 r = 230 r -1631 r = 13450 FIGURE 11.4 Numerical simulation of convective-pattern evolution based on ~ 2-D model, starting from random initial conditions. The simplification of the flow into pattern of textured rolls as time evolves is at least qualitatively similar to the phenomena observed experimentally. show chaotic dynamics, a fact that may be important in device applications. Cooled extrinsic germanium photoconductors exhibit a series of progressively more complex instabilities with an increasing applied dc electric field. Niobium triselenide shows periodic and chaotic fluctuations in its conductivity that are related to the interac- tion of a charge-density wave with the crystal lattice. The spatial arrangement of adsorbed atoms on a periodic substrate and the

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NONLINEAR D YNAMICS, INS TABl LI TIES, AND CHAOS 225 electronic energy bands of a periodic crystal in a magnetic field both show complex structures that can be understood using the methods of nonlinear dynamics. The dynamics of chemically reacting systems are chaotic in some cases and in fact show a close correspondence with low-dimensional dynamic models. Instabilities are also important in many aspects of the study of phase transitions between various states of matter. Progress has been made in studies of the nucleation of first-order phase transitions (i.e., the thermally activated initiation of a phase transition from a metastable state) and spinodal decomposition (phase transitions occurring from an initially unstable state). These processes are highly nonlinear and have been elucidated recently by light-scattering experiments on systems undergoing phase separation. Nonlinear Dynamics of Conservative Systems The nonequilibrium behavior of condensed-matter systems usually involves the dissipation of energy, and this means that the models of interest have attractors onto which the orbits in phase space converge. However, the nonlinear dynamics of conservative (nondissipative) systems has closely related properties and also gives rise to chaotic behavior. In this section, we briefly note that the history of this field is much older than is generally appreciated. For 200 years after Newton, mathematicians and physicists sought to integrate Newton's equations and generally assumed that all Newton- ian systems would eventually be found to be integrable or separable into equations for each degree of freedom and therefore to behave smoothly and predictably. However, Maxwell pointed out a century ago that the motion of a gas of hard spheres could quickly (in 10-~ second) be observably altered by a small perturbation and therefore could not be predictable. Early in its history, the probabilistic nature of statistical mechanics was realized not to be obviously consistent with an integrable and predictable Newtonian dynamics. Slightly later, in his study of the gravitational N-body problem, Poincare proved that the energy is the only well-behaved constant of the motion for an isolated system, thereby casting grave doubts on the notion that Newton's equations could be integrated. Moreover, the wildly erratic character of the orbits of a system of three bodies led him to remark, "Determinism is a fantasy due to Laplace." The nature of chaotic motion in conservative systems has been elucidated over the years by many distinguished scientists, including Birkhoff, Gibbs, and Hopf. An especially important concept is that of

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226 A DECADE OF CONDENSED-MATTER PHYSICS mixing, in which the flow of trajectories on a surface of constant energy in phase space mixes in the same way as the fluid particles in a stirred martini do. We now know that there are many conservative systems that are not only mixing but also truly chaotic in the sense that the final state is exponentially sensitive to changes in the initial conditions. Newtonian dynamics is thus quite rich; it contains the predictable systems of textbook physics as well as the fully chaotic, random systems envisioned by statistical mechanics. How can one go from one to the other? The much celebrated Kolmogorov-Arnol'd-Moser (KAM) theorem (1954) showed that small perturbations to integrable systems leave most orbits nearly unchanged (with suitable qualifications) while also creating a small set of chaotic orbits. It is this small chaotic set that can grow, as the conditions of the KAM theorem are violated, to yield fully chaotic motion. The final and recent development in conservative dynamics comes via information theory. The basic notion here is that a chaotic system or orbit exhibits much greater variety, and therefore contains much more information, than does an orderly integrable system or orbit. This variety is sometimes quantified using the concept of metric entropy. If finite accuracy measurements of an arbitrary observable of the system, made at regular time intervals from the beginning of time to the present, do not permit precise prediction of the next measurement, the system has positive metric entropy, i.e., information is gained from each measurement. Another concept recently adapted from informa- tion theory is that of algorithmic complexity, which measures the unpredictability of individual orbits. An orbit is said to have maximum algorithmic complexity provided its information content cannot be encoded in an algorithm simpler or shorter than that which simply generates a copy of the orbit. Maximum complexity implies random- ness or chaos, and this concept helps to eliminate the apparent paradox of random behavior in deterministic systems. Many connections have been established between the nonlinear dynamics of conservative systems and the dynamics of dissipative systems. General Remarks The developments described in the preceding sections are only a small sampling of the activities in nonlinear dynamics. Systems with instabilities leading to chaotic behavior have been emphasized. The research has been highly international, with the United States making major contributions but not dominating the field. Although physicists

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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 227 have added something new to the problem of understanding instabili- ties and the transition to weak turbulence, the majority of current research in fluid mechanics is done by the engineering community (and is described in the report of the Panel on the Physics of Plasmas and Fluids). The impact of this research is likely to be substantial because (at the mathematical methods of nonlinear dynamics are helping to solve many previously intractable problems in physics; (b) instabilities and chaotic motion are widespread phenomena with common features; and (c) the growing understanding of the origins of chaotic behavior and the limits of predictability contribute to the foundations of physics. CURRENT FRONTIERS The field of nonlinear dynamics, instabilities, and chaos is undergo- ing a period of rapid expansion. A considerable number of important problems merit attention. Bifurcation Sequences As new systems are investigated, it is likely that new distinct bifurcation sequences will be observed. Furthermore, the universality of the various bifurcation sequences still needs to be explored in some cases. For example, how universal is the breakup of a two-torus in real experiments? Much work remains to be done before the different types of transition not only those that are mathematically possible but also those that occur physically are classified. Are there universal phe- nomena associated with abrupt nonequilibrium transitions? If so, what are they? These phenomena are reminiscent of, but more complicated than, those associated with first-order phase transitions. Little experimental work has been done on systems with two or more control parameters (codimension-two bifurcations). Since theorists have begun to classify the types of bifurcations that can occur in such systems, we anticipate that this will be a fruitful area for further experimental work. The problem of connecting the universal phenomena near the chaotic transition with the nonuniversal behavior farther away is a challenge for the future. We do not yet know how to predict in advance what bifurcation sequence will occur in a given physical situation. At present, we can only say that if there is period doubling (for example), it must have certain asymptotic properties. The effects of sequences of instabilities on heat, mass, and momen

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228 A DECADE OF CONDENSED-MATTER PHYSICS tum transport in fluids have not been adequately studied and could have practical applications (for example, in the problem of lubrica- tion). Patterns There are a large number of general questions associated with the formation of patterns in condensed-matter systems. How are patterns selected? Under what conditions, if any, is there a functional whose extreme describe selected steady states? Under what circumstances are these states sensitive to initial conditions? To boundary conditions? Conversely, how can one understand the apparent inevitability of some patterns, e.g., dendritic structures? How can one understand the selection of quantized modes (for example, in Taylor-Couette flow)? Can group-theoretical methods be useful in understanding or classify- ing possible patterns? What is the role of noise in pattern selection? Can one understand pattern selection in some cases as being caused by competition involving attraction to a large number of steady states? Is the system driven from one state to another by external or dynamical noise? If so, what is the source of this noise? Nonlinear theories quantifying the strength of attraction to different attractors may eventually have to be developed. Pattern formation is a particularly interdisciplinary problem, and physicists can benefit from interaction with metallurgists, geologists, meteorologists, biologists, and chemists. All of these groups of scien- tists deal with the production of intricate patterns and with the pro- gression from patterned to chaotic behavior. Can we begin to see general rules and approaches that cut across disciplines? For example, pattern selection in hydrodynamic systems has conceptual similarities to nucleation in the physics of phase transitions. Can these similarities be used to understand these processes better? In general, the process of pattern selection is likely to become much more recognized than it has been in the past for its discriminatory power to reveal mechanisms. Numerical Simulations Analysis will (probably) continue to provide insight into behavior just beyond the onset of instabilities and will play an important role in understanding pattern selection. However, the difficulty of extending nonlinear stability theory beyond the second instability is clear. Accurate simulations have recently been achieved for Couette flow in

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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 229 the regime beyond the second instability. These simulations are dif- ficult and require about 105 spectral modes. An important problem for the future is the simulation of the onset and growth of turbulence, which will require even more modes and larger computers than those now available. When properly carried out, simulations have the power to help us understand the physics of these difficult nonlinear problems. Numerical studies of low-dimensional models will also be important in the future. Much of what we know about nonlinear dynamics comes from numerical calculations, and this will continue to be the case, especially as systems with several control parameters are investigated. Experimental Methods Now that the usefulness of phase-space methods has been demon- strated, we anticipate that a great deal of attention will be given to the development of reliable ways of measuring such quantities as the dimensionality, Lyapunov exponents, and metric entropy of strange attractors. One important goal of this work is to understand chaotic behavior significantly above its onset and to begin to understand the relationship between chaotic time dependence and spatial variations. Experimenters are finding that increasingly precise methods are required to answer the most interesting questions. Laser Doppler techniques can be made more precise by using correlation techniques; digital imaging methods will make it possible to study time dependence and spatial structure simultaneously. We anticipate that cryogenic methods will continue to be important in this field. For example, dilute solutions of 3He in 4He are already useful in the study of double-diffusive convection (where both concen- tration and temperature are important), and there is some evidence that at low temperatures these solutions may convect as classical single- component fluids with novel properties. Transition from Weak to Fully Developed Turbulence The onset of chaos in large hydrodynamic systems is not well understood. It is important to study the loss of spatial correlation and the evolution of time dependence with increasing Reynolds (or Rayleigh) number. Such studies will require multiple probes, digital imaging techniques, and ever bigger computers. Theory and experiment on fluid instabilities have emphasized the onset of turbulence. There has also been much work (especially in the engineering community) on strong turbulence. However' it is not

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230 A DECADE OF CONDENSED-MATTER PHYSICS known how a weakly turbulent flow at low Reynolds number evolves into fully developed turbulence at high Reynolds number. Is the growth in the dimension of the strange attractor slow or fast? Continuous or discontinuous? How many independent degrees of freedom are re- quired to describe fully developed turbulence (102 or 10'~? At what point does the dynamical systems approach cease to be helpful? These exciting questions will certainly be addressed in the next few years. Eventually, it will be important to understand the asymptotic prop- erties of real turbulence better. Are statistical averages unique at high Reynolds number? Theoreticians will have to learn how to extract information about averages from manifolds of solutions. Combinations of computer simulations and analytical boundary-layer approximations will, it is hoped, bring us closer to an understanding of turbulent fluid flow and turbulent convection in particular. Perhaps theories based on optima or bounded quantities will be fruitful. It will be important to understand the relationship between classical turbulence and quantum turbulence (in superfluid helium). The latter is thought to consist of a tangled mass of quantized vortex lines. Since the circulation about a line is strictly quantized, perhaps quantum turbu- lence may ultimately be easier to understand than classical homoge- neous turbulence. The resulting knowledge is likely to be of engineer- ing importance in the cooling of superconducting devices. Conservative Systems Despite the deep contribution to science made by the study of integrable systems, these are' loosely speaking, no more common than integers on the line. The richness and variety of Newtonian mechanics has only begun to be sampled. The full physical significance of many-body problems that are integrable (or KAM near-integrableJ and that, therefore, do not obey the laws of thermodynamics is ripe for development. The possibility that all Newtonian systems having null metric entropy are analytically solvable (although they may be ergodic and mixing) has only begun to be investigated. The full and rigorous development of the statistical mechanics of conservative and dissipa- tive chaotic systems, including the approach to equilibrium, remains an uncompleted task. The transition from complete integrability to full chaos involves a divided phase space exhibiting chaos surrounding integrable islands of stability. Do these islands decrease in size but never disappear as the energy of the system increases'? Does their presence have physical consequences for the decay of correlations, as recent numerical experiments indicate? What is the ultimate significance of the fact that initial errors in

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NONLINEAR DYNAMICS, INSTABILITIES, AND CHAOS 231 measurement or computation grow exponentially in a chaotic system? To observe the smallest details of a chaotic orbit requires unlimited precision; to predict these details analytically requires the manipula- tion of infinite algorithms, which, strictly speaking, cannot even be defined. One thus begins to suspect that the study of chaos will provide an especially deep physical meaning to Godel's theorem and the limitation it implies for human logical systems, including science. Although a final resolution to these matters lies in the future, algorith- mic complexity theory has already established that most real numbers in the continuum are, humanly speaking, undefinable. The problem of long-time prediction for chaotic systems is especially interesting. Using numerical as well as analytic arguments, numerous scientists have suggested that long-time prediction is impossible. Time complexity theory is currently being used to ask how quickly a solution algorithm can predict the future of a dynamical system, i.e., how much time is required to compute the behavior, in order to obtain insight into this question of predictability. Finally, there is the little explored question of whether chaos remains a meaningful concept in quantum mechanics. At present, there is a growing amount of convincing evidence indicating that the quantum mechanics of finite bounded systems contains no chaotic time depen- dence. This leads to the deep suspicion that the quantum description of a classical chaotic system does not tend to the proper classical limit; much worse, it may be that quantum mechanics was tacitly constructed on the notion of intergrability. If so, the inclusion of chaos may require fundamental changes in the foundations of quantum mechanics. Nonequilibrium Systems There are a number of important fundamental questions having to do with the fact that systems undergoing instabilities are generally away from thermodynamic equilibrium. How does one characterize and classify the steady states that are obtained under constant external conditions away from thermodynamic equilibrium? (It is generally assumed that such steady states have simpler behavior than arbitrary nonequilibrium states.) Are there general techniques for classifying the spatial and temporal patterns that emerge in such systems? What are the essential similarities and differences between externally driven nonequilibrium steady states, such as cellular convection or oscillatory chemical reactions, and freely equilibrating systems such as those undergoing nucleation or spinodal decomposition (i . e ., phase changes)? There is still no really fundamental understanding of the mechanisms

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232 A DECADE OF CONDENSED-MATTER PHYSICS that control solidification patterns in, for example, snowflakes, quenched alloys, or directionally solidified eutectic mixtures. Some progress has been made in developing a stability-based theory of free dendritic growth in pure substances and dilute solutions, and it seems possible that a firm mathematical basis is being established for work along these lines. The problems of predicting periodicities of cellular solidification fronts or lamellar eutectics appear to have much in common with pattern-selection problems in Rayleigh-Benard convec- tion or chemical-reaction diffusion systems. Despite a long history of metallurgical investigation, the solidification problem is relatively underexplored experimentally. There is a great need for precise measurements of morphological response to varying growth conditions in simple, carefully characterized model systems. In the case of phase transitions involving nucleation or spinodal decomposition, the major opportunity now is to use neutrons, synchro- tron radiation, and other modern methods to make real-time observa- tions of nonequilibrium processes in alloy solids, polymers, and similar materials. How do states of frozen equilibrium, like glasses and spin (magnetic) glasses, fit into the picture? Can they be understood via dynamical systems methods? New Directions It is important to recognize that this field is evolving rapidly and will find application to many other systems. Instabilities in semiconductors are important for device applications. The nonlinear current-voltage characteristics of a great variety of materials can lead to dynamical instabilities. Chaos is known to occur in liquid-crystal instabilities. The properties of magnetic fluids, liquid metals, and dielectric fluids are also likely to result in important nonlinear phenomena. The theoretical methods of nonlinear dynamics have already found application in models of systems with competing length scales (commensurate and incommensurate phases) in condensed-matter systems. Although sys- tems studied in the past have been at equilibrium, nonequilibrium systems with competing periodicities are also important and are now being investigated for the first time. Finally, there are many astrophys- ical problems (for example, pulsating stars and geomagnetic reversals) for which the methods of nonlinear dynamics may prove useful. There are a number of important biological applications of nonlinear dynamics, especially to neural networks. The idea that memories can be represented by attractors in a dynamical system is an appealing concept that may be capable of explaining some of the important

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NONLINEAR DYNAMICS. INSTABILITIES, AND CHAOS 233 properties of memory, such as the ability to recall a great deal of information starting with a small part of it. Another neurological application of nonlinear dynamics is the explanation of certain types of neural excitations (e.g., hallucinations) in terms of symmetry-breaking instabilities in the neural network. Research on networks is closely related to advances in computer science, where efforts are being made to apply nonlinear dynamics to the behavior of computing machines. Phenomena in which fractal structures occur in real space rather than in phase space are being actively studied. Examples include diffusion-limited aggregation and the behavior of a two-phase fluid in a porous network. These structures have patterns on all length scales and can be understood by methods that exploit this scale invariance.

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Appendixes