Self-organized criticality.

This paper concerns the behavior of spatially extended dynamical systems —that is, systems with both temporal and spatial degrees of freedom. Such systems are common in physics, biology, and even social sciences such as economics. Despite their abundance, there is little understanding of the spatiotemporal evolution of these complex systems. ' Seemingly disconnected from this problem are two widely occurring phenomena whose very generality require some unifying underlying explanation. The first is a temporal effect known as 1/f noise or flicker noise; the second concerns the evolution of a spatial structure with scale-invariant, self-similar (fractal) properties. Here we report the discovery of a general organizing principle governing a class of dissipative coupled systems. Remarkably, the systems evolve naturally toward a critical state, with no intrinsic time or length scale. The emergence of the self-organized critical state provides a connection between nonlinear dynamics, the appearance of spatial self-similarity, and 1/f noise in a natural and robust way. A short account of some of these results has been published previously. The usual strategy in physics is to reduce a given problem to one or a few important degrees of freedom. The effect of coupling between the individual degrees of freedom is usually dealt with in a perturbative manner —or in a "mean-field manner" where the surroundings act on a given degree of freedom as an external field —thus again reducing the problem to a one-body one. In dynamics theory one sometimes finds that complicated systems reduce to a few collective degrees of freedom. This "dimensional reduction'* has been termed "selforganization, " or the so-called "slaving principle, " and much insight into the behavior of dynamical systems has been achieved by studying the behavior of lowdimensional at tractors. On the other hand, it is well known that some dynamical systems act in a more concerted way, where the individual degrees of freedom keep each other in a more or less stab1e balance, which cannot be described as a "perturbation" of some decoupled state, nor in terms of a few collective degrees of freedom. For instance, ecological systems are organized such that the different species "support" each other in a way which cannot be understood by studying the individual constituents in isolation. The same interdependence of species also makes the ecosystem very susceptible to small changes or "noise." However, the system cannot be too sensitive since then it could not have evolved into its present state in the first place. Owing to this balance we may say that such a system is "critical. " We shall see that this qualitative concept of criticality can be put on a firm quantitative basis. Such critical systems are abundant in nature. We shaB see that the dynamics of a critical state has a specific ternporal fingerprint, namely "flicker noise, " in which the power spectrum S(f) scales as 1/f at low frequencies. Flicker noise is characterized by correlations extended over a wide range of time scales, a clear indication of some sort of cooperative effect. Flicker noise has been observed, for example, in the light from quasars, the intensity of sunspots, the current through resistors, the sand flow in an hour glass, the flow of rivers such as the Nile, and even stock exchange price indices. ' All of these may be considered to be extended dynamical systems. Despite the ubiquity of flicker noise, its origin is not well understood. Indeed, one may say that because of its ubiquity, no proposed mechanism to data can lay claim as the single general underlying root of 1/f noise. We shall argue that flicker noise is in fact not noise but reflects the intrinsic dynamics of self-organized critical systems. Another signature of criticality is spatial selfsimilarity. It has been pointed out that nature is full of self-similar "fractal" structures, though the physical reason for this is not understood. " Most notably, the whole universe is an extended dynamical system where a self-similar cosmic string structure has been claimed. Turbulence is a phenomenon where self-similarity is believed to occur in both space and time. Cooperative critical phenomena are well known in the context of phase transitions in equilibrium statistical mechanics. ' At the transition point, spatial selfsirnilarity occurs, and the dynamical response function has a characteristic power-law "1/f" behavior. (We use quotes because often flicker noise involves frequency spectra with dependence f ~ with P only roughly equal to 1.0.) Low-dimensional nonequilibrium dynamical systems also undergo phase transitions (bifurcations, mode locking, intermittency, etc.) where the properties of the attractors change. However, the critical point can be reached only by fine tuning a parameter (e.g. , temperature), and so may occur only accidentally in nature: It