Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
5 What Happens Far from Equilibrium and Why? Isolated systems evolve toward equilibrium, a special state in which properties do not change with time. Yet much of the richness of the world around us arises from conditions far from equilibrium. Phenomena such as turbulence, earthquakes, fracture, and life itself only occur far from equilibrium. Subjecting materials to conditions far from equilibrium leads to otherwise-unattainable properties. For example, rapid cooling is a key process in the manufacture of the strongest me- tallic alloys and toughest plastics. Processes that occur far from equilibrium also create some of the most intricate structures known, from snowflakes to the highly organized structures of life. While much is understood about systems at or near equilibrium, scientists are just beginning to uncover the basic principles govern- ing systems far from equilibrium. Breakthroughs in this area of condensed-matter and materials physics (CMMP) research will affect virtually every discipline in the physical sciences, life sciences, and engineering. The Importance of Far-From-Equilibrium Phenomena We live in a world of evolving structures and patterns. When energy is continu- ally supplied to systems with many interacting constituents, the outcome gener- ally differs strikingly from the unchanging state that characterizes equilibrium. From the molecular processes on the nanoscale that form the basis of life, to the dynamically changing climate on this planet, to the clustering of matter within the universe as a whole, a myriad of phenomena owe their existence to being not just slightly away from equilibrium, but far from it (Figure 5.1). Far-from- 91
92 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s a b c FIGURE 5.1â (a) Swarming schools of fish, (b) swirling storms, and (c) galaxies are all examples of systems formed and evolving far from equilibrium. SOURCES: (a) Department of the Interior. (b)Â Laboratory for Atmospheres, National Aeronautics and Space Administration. (c) National Space Science Data Center, National Aeronautics and Space Administration. 5-1 a,b,c equilibrium conditions also significantly alter the behavior of ordinary fluids and solids. Dramatic examples occur when fluid flow turns turbulent or when solids give way and fracture (Figure 5.2). Both turbulence and fracture generate patterns of amazing complexity that not only completely change the materials properties but also redistribute energy across a whole hierarchy of nested structures, ranging from the microscopic to the macroscopic scale. Far-from-equilibrium processes span a similarly immense range of timescales, from electronic transitions at the subnanosecond scale, to glassy relaxation too slow to measure with any technique, to the age of the universe. Far-from-equilibrium behavior is not confined to special conditions or certain types of materials. Instead, it arises across the entire spectrum of condensed-matter and materials physics in a host of problems of fundamental interest. Far-from- equilibrium phenomena also benefit and plague us in technology and in everyday life. Indeed, some of the most complex outcomes of behavior far from equilibrium
W h a t H a pp e n s F a r from Equilibrium and Why? 93 a b FIGURE 5.2â The need to control far-from-equilibrium behavior. (Left) Turbulent airflow produced by the wingtips of a small airplane (visualized by red smoke). (Right) Disastrous effect of materials fatigue and eventual fracture. SOURCES: (Left) Langley Research Center, National Aeronautics and Space Administration. (Right) Hawaii State Archives. emerge in situations familiar in everyday experience. For example, we can see turbulence in cloud patterns as well as in a bathtub; we take advantage of glassy behavior in nearly all plastics but suffer from ita, traffic jams; we exploit the break- 5-2 in b ing up of a stream of fluid into droplets with fuel injection and ink-jet printing but also find it in every leaky faucet. The reach of far-from-equilibrium phenomena extends even farther, to many systems of profound societal importance. In the past decade, CMMP researchers have begun to tackle far-from-equilibrium behavior governing the workings of systems ranging from the economy to ecosystems and the environment. Key Themes Defining the Scope of the Challenge Two important themes define the scope of the challenge. They run as persistent motifs through a description of the current status of CMMP far from equilibrium. The first theme is that far-from-equilibrium behavior is ubiquitous. The breadth of phenomena investigated makes the study of far-from-equilibrium systems an inherently interdisciplinary field that forges connections between the CMMP com- munity and researchers in biology, chemistry, applied mathematics, geology, me- teorology, and engineering. Far-from-equilibrium physics is connected intimately to both fundamental scientific challenges and cutting-edge materials processing. And, far-from-equilibrium physics underlies a wide range of phenomena outside the traditional boundaries of CMMP, including earthquakes, hurricanes, galaxy formation, and consciousness. As a result, breakthroughs in the area have the po- tential for far-reaching impact across many scientific disciplines. The second key theme is that, despite its importance, far-from-equilibrium
94 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s behavior still remains largely uncharted territory. Far-from-equilibrium behavior is not a simple extension of equilibrium or near-equilibrium physics. ÂInstead, it corre- sponds to qualitatively different types of behavior and response, typically associated with crossing some threshold into a new regime. In some Âspecific cases, Âresearchers have been able to unearth the microscopic origins of far-from-Âequilibrium phe- nomena, but still lacking is the understanding necessary to Â develop more com- prehensive frameworks. The reasons why far-from-equilibrium Âphenomena often resist understanding are described below. This is followed by a discussion of problems for which robust features have been identified, both in experiment and in theory, that can serve as starting points for work over the next decade. Finally, critical needs and recommendations for achieving progress over the next decade are discussed. What CMMP Brings to the Table Condensed-matter and materials physics is uniquely positioned to spearhead progress in the field of far-from-equilibrium behavior. As one of the forefront areas of interdisciplinary research, CMMP has long been a focal point for new approaches that bring together ideas from physics and other science and engineer- ing disciplines and that connect basic science with applied research. CMMP also specializes in developing new theoretical, numerical, and experimental tools and techniques (Chapter 11) for systems of many interacting constituents. Experimen- tal techniques that have been especially useful for probing far-from-equilibrium behavior include novel imaging tools and spectroscopic and particle-tracking methods. Many powerful theoretical and numerical techniques for studying the emergent behavior of many-particle systems near equilibrium have been general- ized to systems far from equilibrium; for example, techniques originally developed for studying magnets have been extended to the flocking of birds. Perhaps the fieldâs most valuable characteristic, however, is its penchant for searching for com- monalities in wildly disparate systems. This focus led to the spectacular success of CMMP in realizing that the enormous variety of equilibrium phase transitions can be understood in terms of a few classes of behavior. This history motivates CMMP researchers to search for similar organizing principles in the even vaster array of far-from-equilibrium phenomena. Far-from-equilibrium behavior is an important component in several of the other CMMP grand challenge areas discussed in this report. It underlies many emergent phenomena (Chapter 2) in systems ranging from the nanoscale (Chapter 6) to the macroscale, and it plays an essential role in the physics of living systems (Chapter 4). Because many far-from-equilibrium phenomena require energy in order to be driven, they are also inevitably implicated in energy consumption and conversion (Chapter 3). In quantum computing (Chapter 7), the challenge is to
W h a t H a pp e n s F a r from Equilibrium and Why? 95 prepare qubits in prescribed pure quantum states. Such systems are necessarily far from equilibrium. Finally, because far-from-equilibrium phenomena are so com- mon in everyday life and underlie so many societal concerns, they provide a rich context for education and learning, for the next generation of scientists as well as for the general public (Chapter 8). How do systems reach the far-from-equilibrium regime and what makes far-from-equilibrium physics difficult? One way to keep a system from its natural state of rest and to push it into the far-from-equilibrium regime involves continual and sufficiently strong forcing. For example, the energy that continually strikes Earth from the Sun gives rise to far-from-equilibrium behavior ranging from chaotic weather patterns to the stag- gering diversity of life. If solar energy were no longer supplied, many systems on Earth would revert to equilibrium. Driven systems such as these not only give rise to rich and unanticipated phenomena but are also of tremendous importance to technological applications. For example, in molecular or nanoscale electronics, new phenomena arise from the response to large electromagnetic fields, currents, and mechanical stresses. As one scales the physical dimensions of matter to the nanometer scale, the applied fields that drive the system away from its equilibrium state are amplified, while the scattering that allows relaxation back to equilibrium is suppressed. As a result, such devices often operate in the far-from-equilibrium regime, unlike conventional semiconductor devices at the micron scale, which typically operate much closer to equilibrium. Conditions far from equilibrium also provide a route for controlling a larger variety of patterns and for assembling structures from the nanoscale on up at growth rates much faster than would be possible with near-equilibrium approaches. Importantly, far-from-equilibrium processes can achieve structural and dynami- cal richness even with the simplest of ingredients, such as the intricate dendritic growth realized in snowflakes (Figure 5.3). Other systems are trapped far from equilibrium because they simply cannot relax back to equilibrium even after all driving forces have been removed. This happens for many materials vital to industrial society, including glasses, powders, foams, and most plastics, which attain their properties from being intrinsically caught in far-from-equilibrium states (Figure 5.4). These materials exhibit struc- tural properties that, under equilibrium conditions, would identify them as liquids; yet they can behave like solids. Processes occurring far from equilibrium are beginning to force researchers to rethink some of the foundations of condensed-matter and materials physics. Yet even today, most of the knowledge about how systems with many constituent par- ticles behave and evolve is based on considerations valid only close to equilibrium.
96 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s FIGURE 5.3â Far-from-equilibrium growth in nature: the snowflake. SOURCE: Image courtesy of K Â enneth Libbrecht, SnowCrystals.com. a b c FIGURE 5.4â Glasses and foams are examples of important materials that are generically in states far from equi- librium. (Left to right) Molten glass freezing into a solid, soap foam, and open-cell aluminum foam. SOURCES: (Left) Savannah River National Laboratory, Department of Energy. (Middle) D. Durian, University of Pennsylvania. (Right) D.C. Curran. 5.4 a, b, c
W h a t H a pp e n s F a r from Equilibrium and Why? 97 CMMP researchers know much more about systems near equilibrium and have developed a powerful formalism, statistical mechanics, to predict the emergent, collective behavior of many-particle systems. This framework has allowed CMMP researchers to understand a large number of phases of matter, the origins of many of their properties, and the nature of transitions between them. However, this framework applies only to situations in which a system is thermally and mechani- cally in balance with its surroundings, and thus it covers only a small subset of the phenomena observed around us and confronted in applications. One conceptual difficulty posed by systems far from equilibrium thus arises from the absence of established theoretical frameworks. However, by virtue of being far from equilibrium, such systems also pose additional challenges. They are typi- cally nonlinear: that is, their response to perturbation is often not proportional to the magnitude of the perturbation, as for systems near equilibrium. Such systems are often disordered: that is, their structure is typically not crystalline, as equilibrium solids generally are. Finally, such systems are often non-ergodic: that is, they do not necessarily explore a large subset of the states available to them, as equilibrium systems must. As a result, even characterizing their behavior and structure leads one onto largely unfamiliar ground from the standpoint of most of CMMP. Far-from-Equilibrium Materials Certain classes of materials almost always exist under conditions far from equi- librium. Many materials investigated by researchers in the area of soft Âcondensed- matter physics fall into this category, including glasses, foams, granular materials, and dense colloidal suspensions. In all of these examples, the thermal energy s Â upplied by the surroundings is too small to allow the systems to explore many con- figurations. Instead, they are trapped in configurations that structurally resemble a liquid (they are dense and highly disordered), but are unable to flow and thus behave as solids. This glassy behavior, a hallmark of many far-from-equilibrium materials, is observed for constituents ranging from molecules in glass-forming liquids to grains of sand in dunes. Since these materials cannot relax to equilib- rium, they typically retain a memory of the preparation or processing conditions, a key for many technological innovations such as molded plastic parts and shape memory polymers. Transitions from far-from-equilibrium glassy states to near- equilibrium crystalline states are the basis for chalcogenide glass optical disks and phase-change memory devices. Over the past decade, granular matter has emerged as a key prototype of a far-from-equilibrium material (see Figure 1.4 in Chapter 1). In its simplest form, granular matter consists of nothing more than a large number of noncohesive, macroscopic hard spheres interacting only at contact; yet it exhibits all the charac- teristics of far-from-equilibrium behavior, as discussed in the subsequent sections.
98 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s Furthermore, several ideas developed originally within the context of granular materials have by now been successfully âexportedâ into other areas; for example, the concept of jamming gives insight into glassy phenomena. Similarly, ideas about avalanche statistics in driven dissipative systems, investigated early on in sandpiles, have been applied to earthquakes and have led to renewed interest in flux-bundle motion in superconducting magnets. Beyond fundamental research, a large number of industrial processes depend on the handling and transport of granular matter, from seeds and fertilizer pellets in agriculture, to ore and gravel in mining operations, to powders and pills in the pharmaceutical industry. Yet the inherently far-from-equilibrium behavior of these materials is still poorly understood and controlled. For example, in North America, new plants designed for processing granular materials initially operate at only about 50 to 60 percent of design capacity, while those designed for the handling of liquids immediately operate at nearly full efficiency. Investment in this area of CMMP would not only raise the level of fundamental understanding needed for innovative solutions to pervasive industrial problems but would also increase the pool of scientifically trained people who can contribute to the understanding of materials-processing industries. Far-from-Equilibrium Processing and Assembly Many materials-processing techniques exploit far-from-equilibrium conditions for the growth and manufacture of materials that otherwise could not be fabricated. Many high-strength alloys are formed by the same rapid dendritic growth that un- derlies the formation of snowflakes. Some of the very strongest materials available are metallic alloy glasses, made by rapid cooling into amorphous states far from equilibrium (Figure 5.5). Lightweight, strong, and tough plastics for car bumpers and aircraft are produced by similar processes. The understanding and control of out-of-equilibrium behavior are also important for interface growth processes, as in those used to produce the huge, essentially defect-free single crystals of silicon used in the semiconductor industry. Far-from-equilibrium processing conditions can be used to drive a system toward unique final configurations in very efficient and speedy ways. On the nanoscale this offers new advantages. For example, certain polymers (diblock co- polymers) spontaneously organize themselves into extended patterns with repeat spacings in the 10 nm to 50 nm range. Such spacings are desirable for applications such as high-density magnetic storage but difficult to achieve with conventional lithographic methods (see Chapter 7). In equilibrium, these polymeric structures are typically fairly disordered and contain a large number of defects. If the systems are sheared far from equilibrium, however, the defects can be removed and the structures can order over extremely large distances. Another advantage of far-
W h a t H a pp e n s F a r from Equilibrium and Why? 99 FIGURE 5.5â Far-from-equilibrium processing produces some of the toughest, highest-strength materi- als (glassy metal alloys). The abscissa corresponds to the yield strain, while the ordinate corresponds to the Youngâs modulus of the materials identified in the figure. SOURCE: Courtesy of William L. Johnson, California Institute of Technology. from-equilibrium conditions is that small differences in the physical or chemical properties of neighboring regions in a material can be amplified; in equilibrium, diffusion tends to smooth out such differences. This property can be exploited to aggregate inorganic components, such as metallic, magnetic, or semiconducting particles, on selective polymer domains. The resulting configuration faithfully duplicates the domain pattern instead of assuming the uniform coverage found in equilibrium. What determines behavior far from equilibrium? In equilibrium, minimization of a free energy determines the preferred state, and the system reaches this state independent of the initial conditions. Far from equilibrium, systems typically exhibit a very rich set of characteristic behaviors that are not generally described by a minimization principle. What physics governs the state that a system chooses? Physicists have made considerable progress in a number of specific cases. This section discusses advances in the areas of fluids and dynami- cal systems and looks at the use of singularities in understanding and controlling far-from-equilibrium behavior.
100 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s Systems with Hydrodynamic Equations of Motion In many cases, far-from-equilibrium systems exhibit a convenient separation of length scale and timescale. In order to understand many fluid-flow problems, such as the vortex of a tornado, it is not necessary to describe the motions of individual molecules. The experience of CMMP with equilibrium systems has taught that it is often fruitful to focus on the long-length-scale, long-timescale behavior. This so- called hydrodynamic approach has been the basis of success in describing a number of systems far from equilibrium. Once the basic differential equations that describe the long-length-scale and -timescale behavior are known, such as the Navier-Stokes equation for fluid flow, an astounding range of far-from-equilibrium, nonlinear behaviors can be tackled. These behaviors include the erratic fluttering of flags in the wind (Figure 5.6), the flapping of a birdâs wings, and the breaking of water waves on a beach. Similar descriptions also apply to complex fluids under flow, a frontier area that is only beginning to be explored. Finally, the hydrodynamic approach can be applied to a wide range of phe- nomena not associated with fluids at all, such as the braking of gravity waves on a collapsing white dwarf, the flocking of birds and other organisms, and the devel- opment of single-celled amoebae into multicellular organisms. Another example is found in semiconductor heterostructures in which electron density waves are confined to the sample edge. There, strong electronic correlations are predicted to produce shock waves that resemble roll clouds in the atmosphere. Many more examples are provided by the physical, chemical, and biological systems that exhibit pattern formation, in which a uniform system develops patterns in space and/or time by being driven out of equilibrium. The idea that the dynamics of a system with many degrees of freedom can be dominated by the interaction of only a few (such as those at long length scale and timescale) is an important CMMP contribution that motivates the study of simple dynamical models in order to gain insight into complex phenomena. Models such as the Lorenz model and other climate models include only a few degrees of free- dom, yet successfully capture qualitatively many features of Earthâs climate. Similar approaches are used to gain insight into the origin of Earthâs magnetic field, mantle convection, Jupiterâs red spot, and the cycle of solar flares. The challenges in tackling this class of problems lie in the identification of the few crucial degrees of freedom that must be retained and in the complexity of the resulting equations of motion. Much progress has come from a close coupling of analytic theory with large-scale computer simulations, informed by experiments; access to the fastest supercomputers will become increasingly important over the next decade. As the understanding of complex interacting dynamical systems and the power of computational resources increase, one great challenge will be to apply
W h a t H a pp e n s F a r from Equilibrium and Why? 101 b a FIGURE 5.6â Modeling the fluttering of flags in the wind: the transition from steady (left) to flutter- ing (right) motion, visualized by imaging fluid flow around a filament tied to a post (circle at the top). SOURCE: J. Zhang, S. Childress, A. Libchaber, and M. Shelley, âFlexible Filaments in a Flowing fig 5-6 a, b Soap Film as a Model for One-Dimensional Flags in a Two-Dimensional Wind,â Nature 408, 835-839 (2000).
102 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s this knowledge and computing power to global warming, as CMMP researchers are beginning to do. Turbulence and Fracture Many far-from-equilibrium phenomena pose special challenges because they involve a multitude of length scales and timescales that interact and thus all be- come important. Large-scale turbulence is connected directly to flow behavior at scales many orders of magnitude smaller; macroscopic fracture patterns depend intimately on the local configuration of molecular bonds in front of the crack tip (Figure 5.7). In problems such as turbulence, hydrodynamic equations apply but become impossible to solve. Theoretical techniques used in CMMP to study equi- librium critical phase transitions, such as the renormalization group, can be useful here. These techniques are designed to understand how physics at small length scales or timescales affects behavior at somewhat larger length scales or timescales, a b FIGURE 5.7â Far-from-equilibrium behavior often involves processes interacting over a large range of length scales and timescales, leading to characteristic patterns such as the ones observed in the fracture of a glass (left) and in turbulent cloud formations (right). SOURCES: (Left) John C. Barry, PicaMS Pty Ltd. (Right) Visible Earth, National Aeronautics and Space Administration. 5-7
W h a t H a pp e n s F a r from Equilibrium and Why? 103 and so on, ultimately leading to an understanding of how a wide range of length scales or timescales interact with one another. The focus here is on turbulence, which is one of the most common far-from- e Â quilibrium phenomena in the environment and in industrial processes. Turbu- lence produces complex flow structures that modify the transport of momentum, mass, and heat, thereby creating a wide variety of both wanted and unwanted effects: a means for rapid mixing of reagents in industrial processes but also parasitic drag in pipe flow and, on a larger scale, catastrophic weather patterns such as hurricanes. Very similar unstable flow structures are produced during the extrusion of polymers or pastes through an orifice, in slow flows of complex fluids such as polymer solutions, and in slow sedimentation of particles at high concentration in a fluid. Thus, the mechanisms underlying turbulence appear to be remarkably general. Ideas from turbulence have even been applied to finance. Despite the ubiquity and importance of turbulence, however, how it develops is not understood well enough to control or prevent it in many cases. The onset flow rate and the nature of the onset of turbulence are still puzzling; turbulence often sets in gradually, in stages, but in many cases, including simple pipe flows, turbulence sets in prematurely for reasons that remain vexingly elusive. Finally, despite much progress during the past decade or two, the nature of the fully turbulent state still poses many open problems. In this state, long-lived, long-length-scale coherent structures play an important but still poorly understood role. In the next decade, new particle-tracking techniques for imaging fluid elements during turbulent flow should shed light on many of these long-standing questions. Singularities In many circumstances, especially under extreme mechanical loading or shear- ing conditions, materials are driven so far from equilibrium that they change their shape irreversibly. This happens every time a liquid splashes and breaks up into droplets, a piece of glass fractures, a sheet of paper crumples, or a car crashes. Such catastrophic events are typically connected with deformations or failure modes that act at the smallest possible scales and yet affect the overall shape. Consider a slowly dripping faucet with water that is just about to pinch off into a drop. What sets the shape of drop and of the neck by which it hangs just before breaking off? It turns out that these shapes are controlled completely and at every stage by only one spot along the neckânamely, where the neck is thinnest. This type of behavior is scale invariantâan image of a neck gives no clue as to the overall size of the neck. In other words, the breaking apart into a drop is controlled by a local singularity, in this case the divergence of the neck curvature. Similarly, the overall behavior of a crumpled piece of paper is determined by a small number of local spots, sharp points of very high curvature connected by a network of ridges. Such singular spots
104 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s instantly transform an otherwise floppy sheet into a structure that can bear loads and absorb shocks (Figure 5.8, left panel). Similar scale invariance occurs at singularities such as those at critical phase transitions in equilibrium systems. Over the past decade, CMMP researchers have built on the foundation of equilibrium phase transitions to identify and tackle far-from-equilibrium materials under extreme conditions. These systems were previously intractable precisely because of their singularities; the triumph in the past decade has been to exploit singularities in order to understand how they con- trol the behavior of such systems over a broad range. Extensions of this approach have demonstrated how the unique behavior in the vicinity of a singularity can be used to achieve unprecedented levels of processing control, which can be used, for example, to uniformly encapsulate live cells prior to transplantation (Figure 5.8, right panel). The extreme mechanics associated with singularities are likely to become increasingly important. They also are prime examples of how, far from equilibrium, the evolution of structure and dynamics are often inseparable. Robustness as a Design Principle In the past decade, ideas from engineering and biology have led CMMP re- searchers to explore a mechanism of state selection very different from equilibrium free-energy minimization. Many far-from-equilibrium systems have been designed, either by deliberate engineering or through evolution and natural selection, to be a b FIGURE 5.8â Using singularities to control materialsâ properties. (Left) Crumpling a piece of paper stiffens it and allows it to absorb shock. (Right) Particles are entrained, for encapsulation purposes, into the near-singular flow when an interface between two fluids (here, oil and water) is deformed by extruding the oil with a pipette. SOURCES: (Left) T.A. Witten, University of Chicago. (Right) S.R. Nagel, University of Chicago.
W h a t H a pp e n s F a r from Equilibrium and Why? 105 robust to perturbations. For example, cars are now designed with complicated internal networks involving many components, backup mechanisms, and adaptive feedback loops to ensure reliable operation under a wide range of environmental conditions. Likewise, biological networks, such as those that enable white blood cells to pursue invading bacteria, have evolved to be insensitive to biochemical changes in their components. In the past decade, CMMP researchers have realized that maximization of robustness can be viewed as a mechanism of state selection in interacting networks. This opens up a vast array of systems that can be studied using the tools of CMMP, ranging from circadian clocks to the Internet and from the human immune system to financial markets (Figure 5.9). One interesting common feature of systems designed for robustness is that their complexity renders them vulnerable to rare, unexpected perturbations. For example, the network of interconnected species in the worldâs oceans has adapted over millennia to be remarkably stable despite the vast number of perturbations that can occur. Yet a small change of acidity in ocean waters produced by increased carbon dioxide in the atmosphere may trigger mass extinctions of species. Even far- from-equilibrium systems, such as materials under stress, which have not evolved or been specifically designed, can exhibit similar vulnerabilities, such as fracture, owing to the history of their formation and the complexity of interactions among the many atoms or molecules that constitute them. a b FIGURE 5.9â Examples of evolving network structures far from equilibrium. (Left) Map of interacting yeast proteins. (Right) Internet nodes. SOURCES: (Left) H. Jeong, S. Mason, A.-L. BarabÃ¡si, and Z.N. Oltvai, âLethality and Centrality in Protein Networks,â Nature 411, 41-42 (2001). (Right) Barrett Lyon, The Opte Project.
106 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s Predictability and Control: What Can We Learn from Fluctuations? For systems composed of many particles in or near equilibrium, statistical me- chanics says that fluctuations of observable quantities around their average values tend to be small and to have a Gaussian (bell-shaped) distribution. For systems far from equilibrium, there is no general framework such as statistical mechanics, and fluctuations tend to be distributed rather differently. The distributions are often broader than Gaussian, for example with power laws, so that large and catastrophic, but rare, events can dominate behavior. This is the case in avalanches involving sudden magnetic domain reorientations or flux-bundle motion in superconducting magnets. Similar avalanches occur in granular materials, as in landslides or mud- slides, or during earthquakes (Figure 5.10). Turbulence and spatiotemporal chaos b a c FIGURE 5.10â Large and often-catastrophic fluctuations, such as avalanches, are characteristic of many systems far from equilibrium. (a) Snow avalanche, (b) granular avalanche, (c) flux-bundle avalanche in a superconductor. SOURCES: (a) See http://www.avalanche.org; photo by Bradley White. (b) S.R. Nagel and H.M. Jaeger, University of Chicago. (c) Cynthia Reichhardt, Los Alamos National Laboratory. figure 5-10 a,b,c
W h a t H a pp e n s F a r from Equilibrium and Why? 107 also produce characteristic fluctuations in the measured quantities. The spectrum of fluctuations thus can serve as a signature of far-from-equilibrium behavior. One of the most important questions that one can ask about a many-particle system is how it will respond to perturbations. For systems in thermal equilibrium, the fluctuation-dissipation theorem provides the answer: If the perturbation is small, the system will respond just as it does to naturally occurring fluctuations. The relationship between correlation and response depends on temperature; tem- perature measures the size of fluctuations relative to the response, which quantifies how hard it is to create a fluctuation. For systems far from equilibrium, temperature no longer plays such a role. However, in analogy to the thermal case, it is possible, in some cases, to define an effective temperature from the relationship between correlation and response. For certain classes of driven dissipative systemsâsuch as sheared glasses, foams, or fluidized granular materials such as vibrated or gas- fl Â uidized granular bedsâthere is evidence that the notion of an effective tempera- ture can be useful in predicting behavior. Important CMMP issues are to elucidate the conditions under which effective temperatures provide a reasonable description and to determine the extent of the analogy to ordinary temperature. Formal Theoretical Developments One of the great challenges of far-from-equilibrium systems is to develop a theoretical framework, akin to equilibrium and near-equilibrium thermodynam- ics and statistical mechanics, for tackling these systems. In the past decade, sub- stantial progress has been made in generalizing thermodynamics and statistical mechanics to far-from-equilibrium systems. Steady-state thermodynamics takes into account the heat that is continually generated in steadily driven systems to generalize the second law of thermodynamics. Other approaches generalize the concept of entropy to zero-temperature systems, while still others generalize the fluctuation-dissipation theorem to far-from-equilibrium systems. A new thermo- dynamic result has made it possible to extract equilibrium free-energy differences from far-from-equilibrium processes. As a result of these developments, the field of nonequilibrium thermodynamics and statistical mechanics is gathering additional momentum. Getting (Un-)Stuck: Jammed States and Jamming Transitions The prototypical example of a jammed state is a glass, a state that has both fluid- and solid-like attributes: It has the amorphous structure of a liquid, yet re- sponds to an applied stress as a solid responds. All liquids will form glasses upon cooling if crystallization can be avoided (for example, by cooling rapidly enough), and for complex fluids such as polymers, the transition to a glass (plastic) is nearly
108 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s impossible to avoid. As a liquid is cooled, the time required to reach equilibrium, the relaxation time, increases, and the response of the system to perturbations becomes more and more sluggish until it is immeasurably slow. At this point, the system is called a glass. The increase of relaxation time is continuous, but it occurs over an incredibly narrow range of temperature, so that lowering the temperature by 10 to 20 kelvin can increase viscosity and relaxation time by 10 orders of mag- nitude. Because the relaxation time exceeds any measurable timescale as the glassy state is approached, a glass by definition is a system far from equilibrium. Similar glassy states are found not only in ordinary liquids but in many elec- tronic systems in the presence of disorder, including interacting electron spin sys- tems (spin glasses) or systems of interacting magnetic flux bundles (vortex glasses). They also occur whenever particles of any size congregate at sufficiently high concentrations. For example, micelles or colloids in dense suspensions, lubricants trapped between surfaces, bubbles in foams, and candies in a jar all get trapped in glassy states (Figure 5.11). The onset of glassy behavior is easily observed in an hourglass filled with sand: A fluid-like stream of grains falling through the central neck is rapidly quenched into a solid-like heap that retains the streamâs amorphous structure but, unlike a fluid, supports a finite angle of repose. However, once the particles become macroscopic as in the case of sand grains, temperature is no longer effective in facilitating escape from the glassy state. Instead, mechanical fields such a b FIGURE 5.11â In systems at or near the jamming transition, network-like structures are formed dy- namically, such as chains of particles experiencing high contact forces in slowly sheared granular systems (left) and strings of particles whose motion is correlated in a gas-fluidized granular bed (right). SOURCES: (Left) D. Howell and R.P. Behringer, Duke University. (Right) A.S. Keys, A.R. Abate, S.C. Glotzer, and D.J. Durian, âMeasurement of Growing Dynamical Length Scales and Prediction of the Jamming Transition in a Granular Material,â Nat. Phys. 3, 260-264 (2007). 5.11 a, b
W h a t H a pp e n s F a r from Equilibrium and Why? 109 as stress or vibration can take over this role and unjam the system. The suggestion that temperature and stress can act similarly in systems close to the onset of rigidity has led to the introduction of a more general framework, the concept of jamming. This concept describes the cooperative phenomenon of jamming in terms of the interplay of three key parameters: random thermal motion, applied forcing, and geometrical constraints (Figure 5.12). The idea of a general jamming transition, applying to both thermal and non- thermal systems, has put the spotlight on some of the most long-standing problems in condensed-matter physics, such as the glass transition. Because the jammed state is out of equilibrium, even the most basic questions about any jamming transition remain intensely controversial. Is there a true thermodynamic transition, at which the relaxation time diverges? Or is there a dynamical transition to the jammed state, where the relaxation time diverges with no thermodynamic signature? Or is there no transition at all, so that the relaxation time only truly diverges at zero temperature or mechanical driving? It is because these fundamental questions FIGURE 5.12â Jamming phase diagram, delineating the conditions under which a multitude of systems become rigid and solid-like. Inside the jammed region (grey), these systems are far from equilibrium. SOURCE: A.J. Liu and S.R. Nagel, âNonlinear Dynamics: Jamming Is Not Just Cool Any More,â Nature 396, 21-22 (1998).
110 C o n d e n s e d - M at t e r and M at e r i a l s P h ys i c s remain unresolved that the onset of glassy behavior is generally considered one of the most intriguing unsolved problems in CMMP. The concept of jamming has fueled an explosion of new interactions and cross- cutting research between previously separate communities working on the glass transition, gelation, granular materials, foams, and dense colloidal suspensions. It also has driven much fruitful interaction between condensed-matter physicists and engineers in this field. The Next Decade Far-from-equilibrium behavior is emerging as one of the major challenges within CMMP and beyond. The importance of making progress in this field is underlined by several key facts. First, far-from-equilibrium behavior is not rare but ubiquitous, occurring from the nanometer scale on up, in daily life as well as in high-technology applications. Second, it connects directly to critical, national needs for the next decade, affecting a large fraction of the manufacturing base as well as the U.S. economy, climate, and environment. The committee emphasizes that far-from-equilibrium behavior cannot be understood simply through small modifications of equilibrium physics. Because it differs so strikingly and at the same time represents largely uncharted intellectual territory, it provides exciting opportunities for major scientific breakthroughs. CMMP researchers are tackling ever-bigger and -broader problems in far- from-equilibrium phenomena. This expansion drives critical needs. Currently, research on far-from-equilibrium phenomena is fragmented into small subfields. These are typically divided along the types of materials or specific phenomena studiedâfor example, fracture in solids or turbulence in fluids. The field of far- from-Âequilibrium physics is vast, and it is unlikely that any one organizing principle will work for all far-from-equilibrium systems. Nonetheless, there is great value in identifying classes of systems that might have common underlying physics or that might be tackled by common methods. There have been few incentives to adopt such broader approaches, but this will be increasingly required in order to make progress. Recent work within the CMMP community has set the stage for fresh ap- proaches to long-standing problems concerning far-from-equilibrium behavior by introducing new model systems such as granular matter, new unifying paradigms such as jamming, new organizing principles such as robustness, and new formal approaches such as steady-state thermodynamics. The community is also finding important connections to a wide range of other fields, both within and outside physics, connections that are likely to amplify the impact of CMMP even further. Over the next decade it will be critical to find ways to stimulate new links and nurture crosscutting approaches.