Vol. 410, No. 6825 (8 March 2001).
The science of complexity, as befits its name, lacks a simple definition. It has been used to refer to the study of systems that operate at the 'edge of chaos' (itself a loosely defined concept); to infer structure in the complex properties of systems that are intermediate between perfect order and perfect disorder; or even as a simple restatement of the cliché that the behaviour of some systems as a whole can be more than the sum of their parts.
241 Complex systems
242 Crackling noise
JAMES P. SETHNA, KARIN A. DAHMEN &
CHRISTOPHER R. MYERS
251 Noise to order
TROY SHINBROT AND FERNANDO J. MUZZIO
259 Supercooled liquids and the glass transition
PABLO G. DEBENEDETTI AND FRANK H. STILLINGER
268 Exploring complex networks
STEVEN H. STROGATZ
Synchronization and rhythmic processes in
The science of complexity, as befits its name, lacks a (Image: D. Massonnet/ CNES/SPL.) simple definition. It has been used to refer to the study
of systems that operate at the 'edge of chaos' (itself a
loosely defined concept); to infer structure in the
complex properties of systems that are intermediate
between perfect order and perfect disorder; or even as
a simple restatement of the cliché that the behaviour of
some systems as a whole can be more than the sum of
Notwithstanding these difficulties over formal
definition, the study of complex systems has seen
tremendous growth. Numerous research programmes, institutes and scientific journals have been established
under this banner. And the new concepts emerging Cover illustration from these studies are now influencing disciplines as A computer simulation of the ground disparate as astronomy and biology, physics and displacement due to the 1992
finance. The richness of the field and the diversity of Landers earthquake — an example of
one of the many systems that show its application lends itself naturally to the Insight complex dynamical behaviour. format, although our choice of themes to review is
necessarily somewhat eclectic.
We begin by considering systems in which the microscopic properties and processes can be immensely complex and seemingly noisy, yet on larger scales they exhibit certain classes of simple behaviour that seem insensitive to the mechanistic details. On page 242, Sethna et al.
show that the seemingly random, impulsive events by which many physical systems evolve exhibit universal — and, to some extent, predictable — behaviour. Shinbrot and Muzzio on
page 251 offer a different perspective on noise, describing how order and patterns can emerge from intrinsically noisy systems.
We then shift our focus to systems where both the properties of the individual components and the nature of their interactions are reasonably well understood, yet the collective behaviour of the ensemble can still defy simple explanation. On page 259, Debenedetti and
Stillinger show how recent theoretical progress on describing the dynamics of systems of many identical interacting particles — in the form of a multidimensional 'energy landscape'
— is shedding light on the age-old phenomena of supercooling and glass formation. And for extensive networks of simple interacting systems, Strogatz shows on page 268 how
network topology can be as important as the interactions between elements.
But complex systems do not always lend themselves to such easy (if qualitative) categorization. For example, the many complex rhythms encountered in living organisms arise not just from intrinsic stochastic or nonlinear dynamical processes, but also from their interaction with an external fluctuating environment. Yet, according to Glass on page 277,
decoding the essential features of these rhythms might ultimately be of benefit to medicine, even in the absence of a simple mathematical interpretation.
As should be clear from these articles, the science of complexity is in its infancy, and some research directions that today seem fruitful might eventually prove to be academic cul-de-sacs. Nevertheless, it seems reasonable to suppose that the general principles emerging from these studies will help us to better understand the complex world around us.
JAMES P. SETHNA*, KARIN A. DAHMEN† & CHRISTOPHER R. MYERS‡
* Laboratory of Atomic and Solid State Physics, Clark Hall, Cornell University, Ithaca, New York 14853-2501, USA email@example.com † Department of Physics, 1110 West Green Street, University of Illinois at Urbana-Champaign, Illinois 61801-3080, USA firstname.lastname@example.org ‡ Cornell Theory Center, Frank H. T. Rhodes Hall, Cornell University, Ithaca, New York 14853-3801, USA email@example.com
Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behaviour over a huge range of sizes, their behaviour is likely to be independent of microscopic and macroscopic details, and progress can be made by the use of simple models. The fact that these models and real systems can share the same behaviour on many scales is called universality. We illustrate these ideas by using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses, and we highlight some continuing challenges in this still-evolving field.
In the past decade or so, science has broadened its purview to include a new range of 1-5phenomena. Using tools developed to understand second-order phase transitions in the 67-91960s and 70s, stochastic models of turbulence in the 1970s, and disordered systems in
the 1980s, scientists now claim that they should be able to explain how and why things crackle.
Many systems crackle; when pushed slowly, they respond with discrete events of a variety 10of sizes. The Earth responds with violent and intermittent earthquakes as two tectonic 1112, plates rub past one another (Fig. 1). A piece of paper (or a candy wrapper at the cinema13) emits intermittent, sharp noises as it is slowly crumpled or rumpled. (Try it, but preferably not with this page.) A magnetic material in a changing external field magnetizes
14, 15. These individual events span many orders of magnitude in size — in a series of jumps
indeed, the distribution of sizes forms a power law with no characteristic size scale. In the past few years, scientists have been making rapid progress in developing models and theories for understanding this sort of scale-invariant behaviour in driven, nonlinear, dynamical systems.
Figure 1 The Earth crackles. Full legend
High resolution image and legend (58k)
Interest in these sorts of phenomena goes back several decades. The work of Gutenberg and 10Richter in the 1940s and 1950s established the well-known frequency– magnitude
relationship for earthquakes that bears their names (Fig. 1). A variety of many-degree-of-16-28freedom dynamical models, with and without disorder, have been introduced in the
years since to investigate the nature of slip complexity in earthquakes. More recent impetus for work in this field came from the study of the depinning transition in sliding charge-29-35density wave (CDW) conductors in the 1980s and early 1990s. Interpretation of the
CDW depinning transition as a dynamic critical phenomenon sprung from Fisher's early 29, 30work, and several theoretical and numerical studies followed. This activity culminated 32in the renormalization-group solution by Narayan and Fisher and the numerical studies by 3334Middleton and Myers, which combined to provide a clear picture of depinning in CDWs and open the doors to the study of other disordered, non-equilibrium systems.
36, 37Bak, Tang and Wiesenfeld inspired much of the succeeding work on crackling noise.
They introduced the connection between dynamical critical phenomena and crackling noise, and they emphasized how systems may end up naturally at the critical point through a process of self-organized criticality. (Their original model was that of avalanches in 38, 39growing sandpiles — sand has long been used as an example of crackling noise, but we 40, 41now know that real sandpiles do not crackle at the longest scales.)
Researchers have studied many systems that crackle. Simple models have been developed 4243to study bubbles rearranging in foams as they are sheared, biological extinctions (where 44, 45the models are controversial — they ignore catastrophic external events like asteroids), 46-51fluids invading porous materials and other problems involving invading fronts (where 46, 4752-54the model we describe was invented), the dynamics of superconductors and 55, 5657superfluids, sound emitted during martensitic phase transitions, fluctuations in the 58, 596061, 62stock market, solar flares, cascading failures in power grids, failures in systems 63-6566designed for optimal performance, group decision-making, and fracture in disordered 67-72materials. These models are driven systems with many degrees of freedom, which
respond to the driving in a series of discrete avalanches spanning a broad range of scales —
what in this paper we term crackling noise.
There has been healthy scepticism by some established professionals in these fields to the sometimes-grandiose claims by newcomers claiming support for an overarching paradigm. But often confusion arises because of the unusual kind of predictions the new methods provide. If such models apply at all to a physical system, they should be able to predict most behaviour on long scales of length and time, independent of many microscopic details of the real world. But this predictive capacity comes at a price: the models typically do not make clear predictions of how the real-world microscopic parameters affect the behaviour at long length scales.
1-5Here we provide an overview of the renormalization group used by many researchers to
understand crackling noise. Briefly, the renormalization group discusses how the effective evolution laws of a system change as measurements are made on longer and longer length scales. (It works by generating a coarse-graining mapping in system space, the abstract space of possible evolution laws.) The broad range of event sizes are attributed to a self-similarity, where the evolution laws look the same at different length scales. This self-similarity leads to a method for scaling experimental data. In the simplest case this yields power laws and fractal structures, but more generally it leads to universal scaling functions — where we argue the real predictive power lies. We will only touch upon the complex analytical methods used in this field, but we believe we can explain faithfully and fully both what our tools are useful for, and how to apply them in practice. The renormalization group is perhaps the most impressive use of abstraction in science.
Why should crackling noise be comprehensible?
Not all systems crackle. Some respond to external forces with many similar-sized, small events (for example, popcorn popping as it is heated). Others give way in one single event (for example, chalk snapping as it is stressed). In broad terms, crackling noise is in between these limits: when the connections between parts of the system are stronger than in popcorn but weaker than in the grains making up chalk, the yielding events can span many size scales. Crackling forms the transition between snapping and popping.
Figure 1b presents a simple relationship between earthquake number and magnitude. We expect that there ought to be a simple, underlying reason why earthquakes occur on all different sizes. The properties of very small earthquakes probably depend in detail on the kind of dirt (fault gouge) in the crack. The very largest earthquakes will depend on the geography of the continental plates. But the smooth power-law behaviour indicates that something simpler is happening in between, independent of either the microscopic or the macroscopic details.
There is an analogy here with the behaviour of a fluid. A fluid is very complicated on the microscopic scale, where molecules are bumping into one another: the trajectories of the molecules are chaotic, and depend both on exactly what direction they are moving and what they are made of. However, a simple law describes most fluids on long time and size scales. This law, the Navier–Stokes equation, depends on the constituent molecules only through a few parameters (the density and viscosity). Physics works because simple laws emerge on
large scales. In fluids, these microscopic fluctuations and complexities disappear on large scales: for crackling noise, they become scale-invariant and self-similar. How do we derive the laws for crackling noise? There are two approaches. First, we can calculate analytically the behaviour on long time and size scales by formally coarse-graining over the microscopic fluctuations. This leads us to renormalization-group 1-5, which we discuss in the next section. The analytic approach can be challenging, methods
but it can give useful results and (more important) is the only explanation for why events on all scales should occur. Second, we can make use of universality. If the microscopic details do not matter for the behaviour at long length scales, why not make up a simple model with the same behaviour (in the same universality class) and solve it?
15, 46, 47, 73-77The model we focus on here is a caricature of a magnetic material. A piece of
iron will 'crackle' as it enters a strong magnetic field, giving what is called Barkhausen noise. We model the iron as a cubic grid of magnetic domains S, whose north pole is either i
pointing upwards (S=+1) or downwards (S=-1). The external field pushes on our domain ii
with a force H(t), which will increase with time. Iron can be magnetized because
neighbouring domains prefer to point in the same direction: if the six neighbours of our cubic domain are S, then in our model we let their force on our domain be JS (where we jjj
set the coupling J=1). Finally, we model dirt, randomness in the domain shapes, and other kinds of disorder by introducing a random field h, different for each domain and chosen at i
random from a normal distribution with standard deviation R, which we call the disorder.
The net force on our domain is thus
The domains in our model all start with their north pole pointing down (-1), and flip up as soon as the net force on them becomes positive. This can occur either because H(t)
increases sufficiently (spawning a new avalanche), or because one of their neighbours flipped up, kicking them over (propagating an existing avalanche). (Thermal fluctuations are ignored: a good approximation in many experiments because the domains are large.) If the disorder R is large, so the h are typically big compared to J, then most domains flip i
independently: all the avalanches are small, and we get popping noise. If the disorder is small compared to J, then typically most of the domains will be triggered by one of their neighbours: one large avalanche will snap up most of our system. In between, we get crackling noise. When the disorder R is just large enough so that each domain flip on
average triggers one of its neighbours (at the critical disorder R), then we find avalanches c
on all scales (Figs 2, 3).
73-76Figure 2 Magnets crackle. Full legend
High resolution image and legend (60k)
Figure 3 Self-similarity. Full legend
High resolution image and legend (202k)
What do these avalanches represent? In nonlinear systems with many degrees of freedom, there are often large numbers of metastable states. Local regions in the system can have multiple stable configurations, and many combinations of these local configurations are possible. (A state is metastable when it cannot lower its energy by small rearrangements. It is distinguished from the globally stable state, which is the absolute lowest energy possible for the system.) Avalanches are the rearrangements that occur as our system shifts from one metastable state to another. Our specific interest is in systems with a broad distribution of avalanche sizes, where shifting between metastable states can rearrange anything between a few domains and millions of domains.
There are many choices we made in our model that do not matter at long time and size 78, 79scales. Because of universality, we can argue that the behaviour would be the same if
we chose a different grid of domains, or if we changed the distribution of random fields, or if we introduced more realistic random anisotropies and random coupling constants. Were this not the case, we could hardly expect our simple model to explain real experiments. The renormalization group and scaling 1-5, 78, 80To study crackling noise, we use renormalization-group tools developed in the study
of second-order phase transitions. The word renormalization has roots in the study of quantum electrodynamics, where the effective charge changes in size (norm) as a function of length scale. The word group refers to the family of coarse-graining operations that are basic to the method: the group product is composition (coarsening repeatedly). The name is unfortunate, however, as the basic coarse-graining operation does not have an inverse, so that the renormalization group does not have the mathematical structure of a group.
The renormalization group studies the way the space of all physical systems maps into itself under coarse-graining (Fig. 4). The coarse-graining operation shrinks the system and
removes degrees of freedom on short length scales. Under coarse-graining, we often find a fixed point S*: many different models flow into the fixed point and hence share long-3wavelength properties. Figure 3 provides a schematic view of coarse-graining: the 1,000 3cross-section looks (statistically) like the 100 section if you blur your eyes by a factor of
ten. Much of the mathematical complexity of this field involves finding analytical tools for computing the flow diagram in Fig. 4. Using methods developed to study thermodynamical 232phase transitions and the depinning of charge-density waves, we can calculate for our 78-80model the flows for systems in dimensions close to six (the so-called -expansion,
where =6-d, with d being the dimension of the system). Interpolating between dimensions may seem a surprising thing to do. In our system it gives good predictions even in three dimensions (that is, =3), but it is difficult and is not discussed here. Nor will we discuss 1real-space renormalization-group methods or series-expansion methods. We focus on the
relatively simple task of using the renormalization group to justify and explain the universality, self-similarity and scaling observed in nature.
Figure 4 Renormalization-group flows. Full legend
High resolution image and legend (37k)
Consider the 'system space' for disordered magnets. There is a separate dimension in system space for each possible parameter in a theoretical model (disorder, coupling, next-neighbour coupling, dipolar fields, and so on) or in an experiment (for example, temperature, annealing time and chemical composition). Coarse-graining, however one implements it, gives a mapping from system space into itself: shrinking the system and ignoring the shortest length scales yields a new physical system with identical long- distance physics, but with different (renormalized) values of the parameters. We have abstracted the problem of understanding crackling noise in magnets into understanding a dynamical system acting on a space of dynamical systems.
Figure 4 represents a two-dimensional cross-section of this infinite-dimensional system space. We have chosen the cross-section to include our model (equation (1)): as we vary the
disorder R, our model sweeps out a straight line (red) in system space. The cross- section also includes a fixed point S*, which maps into itself under coarse-graining. The system S*
looks the same on all scales of length and time, because it coarse-grains into itself. We can picture the cross-section of Fig. 4 either as a plane in system space (in which case the
arrows and flows depict projections, as in general the real flows will point somewhat out of the plane), or as the curved manifold swept out by our one-parameter model as we coarse-
grain (in which case the flows above our model and below the maroon curved line should be ignored).
The flow near S* has one unstable direction, leading outwards along the maroon curve (the unstable manifold). In system space, there is a surface of points C which flow into S* under
coarse- graining. Because S* has only one unstable direction, C divides system space into
two phases. To the left of C, the systems will have one large, system-spanning avalanche (a snapping noise). To the right of C, all avalanches are finite and under coarse-graining they
, our model goes through a all become small (popping noise). As it crosses C at the value Rc
Our model at R is not self-similar on the shortest length scales (where the square lattice of c
domains still is important), but because it flows into S* as we coarse-grain we deduce that
it is self-similar on long length scales. Some phase transitions, such as ice melting into water, are abrupt and do not exhibit self-similarity. Continuous phase transitions like ours almost always have self-similar fluctuations on long length scales. In addition, note that our model at R will have the same self-similar structure as S* does. Indeed, any experimental c
or theoretical model lying on the critical surface C will share the same long-wavelength
critical behaviour. This is the fundamental explanation for universality. The flows in system space can vary from one class of problems to another: the system space for some earthquake models (Fig. 5a) will have a different flow, and its fixed point
will have different scaling behaviour (yielding a different universality class). In some cases, a fixed point will attract all the systems in its vicinity (no unstable directions; Fig. 6).
Usually at such attracting fixed points the fluctuations become unimportant at long length scales: the Navier–Stokes equation for fluids described earlier can be viewed as a stable 81, 82fixed point. The coarse-graining process, averaging over many degrees of freedom, naturally smoothens out fluctuations, if they are not amplified near a critical point by the unstable direction. Fluctuations can remain important when a system has random noise in a conserved property, so that fluctuations can die away only by diffusion: in these cases, the 83, 84whole phase will have self-similar fluctuations, leading to generic scale invariance.
Figure 5 Flows in the space of earthquake models. Full legend
High resolution image and legend (36k)
Figure 6 Attracting fixed point. Full legend
High resolution image and legend (33k)
Sometimes, even when the system space has an unstable direction as in Fig. 4, the observed
behaviour always has avalanches of all scales. This can occur simply because the physical system averages over a range of model parameters (that is, averaging over a range of R 85including R in Fig. 4). For example, this can occur by the sweeping of a parameter slowly c
in time, or varying it gradually in space — either deliberately or through large-scale
36, 37Self-organized criticality can also occur, where the system is controlled so that it sits naturally on the critical surface. Self- organization to the critical point can occur through many mechanisms. In some models of earthquake faults (Fig. 5b), the external force 20naturally stays near the rupture point because the plates move at a fixed, but very small,
velocity with respect to one another (Fig. 5b). (This probably does not occur during large 28, 86earthquakes, where inertial effects lead to temporary strain relief.) Sandpile models
self-organize (if sand is added to the system at an infinitesimal rate) when open boundary 87conditions are used (which allows sand to leave until the sandpile slope falls to the critical 88-90value). Long-range interactions between domains can act as a negative feedback in
some models, yielding a net external field that remains at the critical point. For each of these cases, once the critical point is understood, adding the mechanism for self-organization is relatively easy.
15, 73-7688-The case shown in Fig. 4 of 'plain old criticality' is what is seen in some but not all924228 models of magnetic materials, in foams, and in some models of earthquakes.
Beyond power laws
The renormalization group is the theoretical basis for understanding why universality and self-similarity occur. Once we accept that different systems should sometimes share long-distance properties, though, we can quite easily derive some powerful predictions. To take a tangible example, consider the relation between the duration of an avalanche and its size. In paper crumpling, this is not interesting: all the avalanches seem to be without 11internal temporal structure. But in magnets, large events take longer to finish, and have an interesting internal statistical self-similarity (Fig. 7a). If we look at all avalanches of a
certain duration T in an experiment, they will have a distribution of sizes S around some
average S(T). If we look at a theoretical model, it will have a corresponding experiment
average size S(T). If our model describes the experiment, these functions must be theory
essentially the same at large S and large T. We must allow for the fact that the experimental