Simple Lessons from Complexity
Nigel Goldenfeldand Leo P. Kadanoff
87-89 (Apr 2, 1999).
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The complexity of the world is contrasted with the simplicity of the basic laws of physics. In recent years, considerable study has been devoted to systems which exhibit complex outcomes. This experience has not given us any new laws of physics, but instead has given us a set of lessons about appropriate ways of approaching complex systems.
One of the most striking aspects of physics is the simplicity of its laws. Maxwell's equations, Schrödinger's equation, and Hamiltonian Mechanics can each be expressed in a few lines. The ideas which form the foundation of our world-view are also very simple indeed: the world is lawful and the same basic laws hold everywhere. Everything is simple, neat, and expressible in terms of everyday mathematics, either partial differential equations, or ordinary differential equations.
Everything is simple and neat -- except, of course, the world.
Every place one looks--outside the physics classroom-- one sees a world of amazing complexity. The world contains complex ecologies at all levels, from huge mountain ranges, to the delicate ridge on the surface of a sand dune, to the salt spray coming off a wave, to the interdependencies of financial markets, to the true ecologies formed by living things. Each situation is highly organized and distinctive, with biological systems forming a limiting case of exceptional complexity. So why, if the laws are so simple, is the world so complicated? Here, we try in part to answer this question and to summarize what general lessons can be drawn from recent work on complexity in physical systems.
To us, Complexity means that we have structure with variations. Thus, a living organism is complex because it has many different working parts, each formed by variations in the working out of the same genetic coding. One look at ocean or sky gives the conviction that there is some natural tendency toward the formation of structure in the physical world. Chaos is also found very, very frequently. Chaos is the sensitive dependence of a final result upon the initial conditions that bring it about. In a chaotic world, it is hard to predict which variation will arise in a given place and time. In fact, errors and uncertainties often grow exponentially with time. A complex world is interesting, because it is highly structured. A chaotic world is interesting because we do not know what is coming next. But the world contains regularities as well. For example, climate is very complex, but Winter follows Summer in a predictable pattern. Our world is both complex and chaotic. An elementary lesson follows:
Nature can produce complex structures, even in simple situations, and some simple laws, even in complex situations.
Creating Complexity: Fluids frequently produce complex behavior, which can be either highly organized (think of a tornado) or chaotic (like. a highly turbulent flow). What is seen often depends on the size of the observer. A fly caught in a tornado would be surprised to learn that it is participating in a highly structured flow.
The equations which describe how the fluid velocity at one point in space affects the velocity at other points in space are derived from three basic ideas. First: Locality. A fluid contains many particles in motion. A particle is influenced only by other particles in its immediate neighborhood. Second: Conservation. Some things are never lost, only moved around, such as particles and momentum. Third: Symmetry. A fluid is isotropic and rotationally invariant.
To make a computer fluid, construct a kind of square dance in which particles move around, obeying the three basic ideas. In the simplest case, the dance is done on a regular hexagonal lattice. See the top panel of the figure. Each particle is characterized by a lattice position and by one of six directions of motion. These arrows are momentum vectors. The square dance starts out by the caller saying PROMENADE, which means step one step in the direction of your arrow. The result is shown in the middle panel. And then the caller calls SWING YOUR PARTNER, which instructs you to rotate all the arrows on a given site through sixty degrees, if they happen to add up to zero total momentum, with the result shown in the bottom panel. Both particle number and momentum are conserved in each step. Take thousands of particles and thousands of steps, and average a bit to smooth out the data. We find a pattern of motion identical to fluid motion. The square dance behaves like a fluid simply because its steps obey the three fundamental laws of fluid motion.
Gradually, through examples like this, it has dawned upon us that very simple ingredients can produce very beautiful, rich, and patterned outputs. Thus, our square dancers, through their simple hops and swings, produce the entire beautiful world of fluids in motion. For simple elementary actors to produce patterned and complex output, we require many events. Our example included many events, because it had many actors and much time.
For physicists, it is delightful, but not surprising that the computer generates realistic fluid behavior, regardless of the precise details of how we do the coding. For if this were not the case, then we would have extreme sensitivity to the microscopic modeling -- what one might loosely call model chaos -- and physics as a science could not exist: in order to model a bulldozer, we would need to be careful to model its constituent quarks! Nature has been kind enough to have provided us with a convenient separation of length, energy and time scales, allowing us to excavate physical laws from well-defined strata, even though the consequences of these laws are very complex. But we might not be so lucky with complexity in biological or economic situations.
To extract physical knowledge from a complex system, one must focus on the right level of description. There are three modes of investigation of systems like this: experimental, computational, and theoretical. Experiment is best for exploration, since experimental techniques, combined with the human eye, can scan large ranges of data very efficiently.
Computer simulations are often used to check our understanding of a particular physical process or situation. In our fluid dynamics example, the large-scale structure is independent of detailed description of the motion on the small scales. We can exploit this kind of universality, by designing the most convenient minimal model. For example, most fluid flow programs should not be modeled by Molecular Dynamics simulations. These simulations are so slow that they may not be able to reach a regime which will enable us to safely extrapolate to large systems. So we are likely to get the wrong answer. Instead, we should model at the macro level, use large time steps, large systems. For example, some computational biologists try to simulate protein dynamics by following each and every small part of the molecule. Result: most of the computer cycles are spent watching little OH groups wiggling back and forth. Nothing biologically significant occurs in the time they can afford.
Use the right level of description to catch the phenomena of interest. Don't model bulldozers with quarks.
This lesson applies with equal strength to theoretical work aimed at understanding complex systems. Modeling complex systems by tractable closure schemes or complicated free-field theories in disguise does not work. These may yield a successful description of the small-scale structure, but this description is likely to be irrelevant for the large-scale features. To get these gross features, one should most often use a more phenomenological and aggregated description, aimed specifically at the higher level. Thus, financial markets should not be modeled by simple geometric Brownian motion based models, all of which form the basis for modern treatments of derivative markets. These models were created to be analytically tractable and derive from very crude phenomenological modeling. They cannot reproduce the observed strongly non-Gaussian probability distributions in many markets, which exhibit a feature so generic that it even has a whimsical name, fat tails. Instead, the modeling should be driven by asking what are the simplest non-linearities or non-localities that should be present, trying to separate universal scaling features from market specific features. The inclusion of too many processes and parameters will obscure the desired qualitative understanding.
So every good model starts from a question. The modeler should aways pick the right level of detail to answer the question.
Complexity and Statistics:
As a fluid moves around it may carry with it some passive elements, that do not themselves influence the flow. Both energy and the density of impurities undergo this kind of motion, in which they convect (go with the flow) and diffuse (move randomly). The convective motion tends to move initially distant regions of the fluid close to one another, and thereby produce enhanced gradients. The diffusion tends to smooth out the gradients.
In many situations, these passive scalars are carried along by a rapid and turbulent flow, so that the convective mixing tends to dominate the diffusion. Computer simulations and experiments demonstrate that the density of the scalar soon develops a profile in which there are many flat regions surrounded by abrupt jumps. The flat regions are produced by the combined effects of convection and diffusion, in well-mixed regions of the sample. However, because the density must, in the large, follow the initial gradient, mixed regions must be separated by jumps.
This behavior, in which the the system is dominated by really big events is called intermittency. Intermittency seems to be a ubiquitous feature of dynamical systems. The weather turns stormy suddenly. There are ice ages. The stock market crashes. A plague takes hold. An airplane runs into turbulence. In every case, there is a big jump in the behavior of a dynamical system, and that big jump can have big human consequences.
In quantifying these ubiquitous jumps, one finds that they come in all sizes with the big jumps being less likely. Empirically, the size of the jumps is often given by a probability distribution which, for large jumps, takes the form:
P(jump)= exp(-|jump|/s) (1)
wheres is the standard deviation. Contrast this with the usual Gaussian form
P(jump) = exp[-|jump|2 /2s2 ] /Ö (2p s2) (2)
which has been the usual guess in statistical problems since the time of Galton. Chaotic and turbulent systems often show exponential behaviors, like (1). Improbable (very bad!!) events are much more likely with the exponential form than with the Gaussian form (2). For example, a 6-s event has a chance of 10-9 of occurring in the Gaussian case , while with the exponential the chance is 0.0025. Estimates, particularly Gaussian estimates, formed by short time series will give an entirely incorrect picture of large-scale fluctuations. These considerations have important consequences in financial markets, as emphasized recently by Mandelbrot. Thus we come to the lesson:
Complex systems form structures and these structures vary widely in size and duration. Their probability distributions are rarely normal, so that exceptional events are not that rare.
The development of complexity:
Long ago, Katchalsky and Prigogine described the formation of complex structures in non-equilibrium systems. Their `dissipative structures could have a degree of complication which could grow rapidly in time. It is believed that comparably complex structures do not exist in equilibrium. A. Turing described a mechanism, involving reaction diffusion equations, for the development of organization in living things. As we have seen from the examples quoted here and many others, in non-equilibrium situations, many-particle systems can get very compliced indeed. It is likely that this tendency is the basis of life. A restricted version of this idea is given in Bak, Tang, and Wiesenfeld's self-organized criticality. In an essay entitled `More is different, P.W. Anderson has described how features of organization may arise as an emergent property of systems. An example of this point of view is given by work on complexity phase transitions and accompanying speculations that various aspects of biological systems sit on a critical point, between order and complexity.
The next few years are likely to lead to an increasing study of complexity in the context of statistical dynamics, with a view to better understanding physical, economic, social, and especially biological systems. It will be an exciting time. As science turns to complexity, one must realize that complexity demands quite different attitudes from those heretofore common in physics. Up to now, physicists looked for fundamental laws true for all times and all places. But, each complex system is different. Apparently there are no general laws for complexity. Instead one has to reach for lessons which might, with insight and understanding, be learned in one system and applied to another. Maybe physics studies will become more like human experience.
Acknowledgments: NG acknowledges partial support of the National Science Foundation through grant NSF-DMR-93-14938. LPK acknowledges partial support by the ASCI Flash Center at the University of Chicago under DOE contract B341495.
Fig. 1. Three stages in the update algorithm of a lattice gas. Between the top panel and the middle panel each particle moves in the direction of its arrow, to arrive at a nearest neighboring site. Next, particles 'collide' whenever the total momentum on a site is zero. These collisions occur between the middle and bottom panels.
REFERENCES AND NOTES
Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801-3080. E-mail: firstname.lastname@example.org
Departments of Physics and Mathematics, University of Chicago, James Franck Institute, 5640 S. Ellis Ave, Chicago, IL 60615. E-mail: LeoP@UChicago.EDU
U. Frisch, B. Hasslacher, Y. Pomeau, Phys. Rev. Lett. 56, 1505 (1986).; J. Hardy, O. de Pazzis, U. Frisch, J. Math. Phys. 14, 1746 (1973) ; Phys. Rev. A13 1949 (1976).
Early work on the derivation of hydrodynamics from conservation laws can be found in S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, 3rd edition, 1970).
A.R. Kerstein, J. Fluid Mech. 291, 261 (1997). Scott Wunsch, Ph. D. thesis, University of Chicago (1998). For experiments see, for example, B. Castaing, et al. J. Fluid Mech. 204, 1-30 (1989). For theory see E. Siggia and Boris Shraiman, Phys. Rev. E49, 2912 (1994).
B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (Springer-Verlag, New York, 1997).
A. Katchalsky and P.F. Curan, Nonequilibrium Processes in Biophysics, (Harvard University Press, Cambridge, Mass., 1967).
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (John Wiley, New York, 1977).
A. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. B237 (1952).
For example, Leo Kadanoff, A. Libchaber, E. Moses, G. Zocchi, La Recherche, 22, 629 (22 Mai 1991) discuss the development of interlinked structures in a Rayleigh Benard flow.
Per Bak, Chao Tang, and Kurt Wiesenfeld,. Phys. Rev. Lett. 59, 381-384 (1987).; J. M. Carlson, J.T, Chayes, E.R. Grannan, G.H. Swindle, Phys. Rev. Lett. 65, 2547-2550 (1990).
P.W. Anderson , Science 177, 393-396 (1972).
Stuart A. Kauffman, The Origin of Order (Oxford University Press, Oxford, 1993). Stuart A. Kauffman, At Home in the Universe (Oxford University Press, Oxford, 1995).