Criticality, self-organized

Transcription

1 Criticality, self-organized Bai-Lian Li Volume 1, pp in Encyclopedia of Environmetrics (ISBN ) Edited by Abdel H. El-Shaarawi and Walter W. Piegorsch John Wiley & Sons, Ltd, Chichester, 2002

2 Criticality, self-organized The causes and possible relations between the abundance of fractal structures (see Fractal dimensions) and 1/f signals found in nature have puzzled scientists for years. Based on observations of computer simulations of the cellular automata sandpile model, the term self-organized criticality (SOC) was invented by theoretical physicists [3] and was used to explain a universal dynamic organizing process governing various fractal growths in space and time. SOC describes that systems organize themselves so their internal dynamics produce critical behavior over a very wide range of length scales. The term consists of two parts. Self-organization has been used to describe the ability of certain nonequilibrium systems to develop structures and patterns often without explicit pressure or involvement from outside the systems [23]. In other words, the constraints on form (i.e. organization) of interest to us are internal to the system, resulting from the interactions among the components and usually independent of the physical nature of those components. The organization can evolve in either time or space, maintain a stable form, or show transient phenomena. Examples include lasers, Bernard cells, Belousov Zhabotinski and Brusselator reactions, cellular autocatalysis, organism structures, immune systems, the brain, ecosystems, economies, etc. Criticality is a technical word used in connection with phase transitions (strictly speaking, continuous or second-order transitions), and has a very precise meaning, in which the system becomes critical when all members of the system influence each other and events of all sizes occur. The key point of view in [3] is that there are important analogies between the marginal stability reached at the critical point of a phase transition and the complex self-organized features that arise from the marginal stability of a dynamic system. The aim of the study of SOC is to yield insight into the fundamental question of why nature is complex, not simple, as the laws of physics imply. The physicist is largely concerned with abstracting simple things from a complex world. In physics, a simple system might be defined as one that obeys simple laws. But simplicity of the rules of the game does not necessarily imply triviality of outcome. On the contrary, the action of elementary laws on many particles over long periods of time will often give rise to interesting structures and events. All the richness of structures or patterns observed in nature may not be a consequence of the complexity of physical laws, but instead arises from many repeated applications of quite simple laws. SOC is a typical example of such a derived complexity. SOC combines two fascinating concepts self-organization and critical behavior to explain a third, no less fascinating and fashionable notion: complexity [11]. This has gained intense interest in physics and a variety of other disciplines, including ecology and evolutionary biology (e.g. [9], [14], [16], [20] and [27]). It has also been considered as a new way of viewing nature and a new paradigm in the physics of large, complex, nonequilibrium, dynamical systems. Bak [2] ambitiously claims that SOC is so far the only known general mechanism to generate complexity of the systems. The systems are complex in the same sense as fractals in that no single characteristic event size exists; there is no single temporal and spatial scale that controls the evolution of these systems. SOC is hypothesized to link the multitude of complex phenomena observed in nature to simplistic physical laws and/or one underlying process. It reflects a universal tendency of the internal interactions of large dynamic systems. Specifically, it states that large interactive systems will self-organize into a critical state that is governed by a power law f x / x a. This differs from systems in thermodynamic equilibrium, for which external tuning (such as temperature) is essential. Once in this state, small perturbations result in chain reactions that can affect any number of elements within the system. This stationary state is characterized by statistical fluctuations, which are referred to generically as avalanches. A separation of time-scales is required for the system evolving into an SOC dynamical state, since such separation has intimate connection with the existence of threshold and metastability. The process connected with external driving of the system needs to be much slower than the internal relaxation or dissipation processes. The existence of local thresholds is a necessary (although certainly not sufficient) condition for self-organization to criticality. The nature of the critical state is characterized by the response of an system to external disturbance. The crucial features of the response distributions of an SOC system are fractal structure in space, 1/f noise in time, and the power-law distribution of event sizes.

3 2 Criticality, self-organized The idea of SOC is conceptually illustrated with the intuitive example of avalanches in a pile of sand grains. Imagine sand being added grain by grain on the top of a sand pile. At first, the grains land harmlessly on the stable slope of a proto-sand pile. As more grains are added the slope of the pile increases. Eventually, the slope locally reaches a critical value such that the addition of one more grain results in an avalanche: a response to a too large local slope of the pile. These avalanche events can be small or they can cover the entire system many times over. The avalanche fills in empty areas of the sandbox. With the addition of still more grains the sandbox will overflow. Sand is thus added and lost from the system. When the count of grains added equals the count of grains lost (on average) then, according to the theory, the sand pile has self-organized to a critical state. The underlying algorithm for SOC models is relatively simple to understand. In essence, the algorithm keeps track of numbers associated with points on a grid. Numbers on the grid can increase, decrease, or stay the same. If a number on the grid gets too large, then the algorithm decreases that number and subsequently increases numbers elsewhere. For example, consider a one-dimensional version of the avalanche model with square sand stacked in a region of size L. The slope (height difference) will be measured by z i D h ic1 h i, where the integer h i denotes the number of grains in the sand column above position i. The dynamics of the model are defined in terms of the following two steps: (a) adding a grain to the pile, and (b) relaxing the slope of the pile whenever the local gradient exceeds the stability threshold z c ;see Figure 1. Computer programs of such models can be found in appendices of the book by Jensen [11]. The concept of SOC has been applied in fields spanning statistical mechanics, condensed matter theory, geophysics, economy, biology, and ecology. But the basic picture remains the same: many slowly driven nonequilibrium systems organize in a poised state the critical state where anything can happen within well-defined statistical laws. In ecology, for example, Li and Forsythe [17] use a cellular automata-based simulation model of a multipatch landscape subjected to different intensities and scales of disturbances to study criticality of spatially heterogeneous vegetation landscape responses to disturbances. Their results indicate that disturbanceinfluenced vegetation systems will exhibit SOC states and that ecosystems may operate persistently out of equilibrium at or near a threshold of instability and coevolve to the edge of chaos [16]. Sole and Manrubia [26] examine whether a rainforest exhibits SOC by using both empirical data and a simulation of forest gap distributions, and claim that the rainforest also evolves to SOC. Ito and Gunji [10] argue that during evolutionary processes self-organization of living systems moves toward a critical state. Jørgensen et al. [12] regard ecosystems as far-from-equilibrium, complex adaptive systems expanding into the adjacent possible under SOC near the edge of order and chaos. Malamud et al. [19] demonstrate from empirical data and a computer simulation model that forest fires are an example of SOC (see Forest fire models). Many other ecological case studies involving SOC can be found in [6] [9], [15], [16], [20], [24] and [27] among others. Many different views have been expressed regarding SOC. There is neither general consensus nor agreement about ingredients necessary to create an SOC state because the lack of a fundamental understanding prevents the construction of a unifying framework. The term SOC has not always been used with the same meaning, and in some cases it has A. Add a grain at a random site (Avalanche begins) B. If the slope is greater than z c = 2, then two grains from a stack fall over. At the right-hand end, grains fall off. Continue until no more stacks are unstable (Avalanche ends) C. Return to A Figure 1 An avalanche model

4 Criticality, self-organized 3 even been misused. One has to recognize that to be useful this concept must be specified accurately and related to a well-defined situation based on physical mechanisms rather than on simple observations [28]. As Kadanoff [13] put forward, SOC was a suggestion, not a fact. It is very likely that some systems will indeed show SOC; however, we still do not know how robust that behavior will be. And we certainly do not know if SOC is closely connected with the mechanisms by which nature produces complexity. According to numerous published studies, it is becoming evident that SOC can produce power-law or fractal-like phenomena; but power-law or fractallike patterns are not necessarily generated by SOC alone. Many different statistical, mathematical, and physical mechanisms can also generate similar phenomena [1, 7, 18, 21, 22, 25, 28, 29]. One alternative theory to SOC recently attracted much attention: highly optimized tolerance (HOT) proposed by Carlson and Doyle [4, 5] is a new mechanism for generating power-law distributions. These systems are optimized either through natural selection or engineering design. They suggest that power laws in these systems are due to tradeoffs among yield, cost of resources, and tolerance to risks. The tradeoffs lead to highly optimized designs that allow for occasional large events. The characteristic features of HOT systems include: (a) high efficiency, performance, and robustness to designed-for uncertainties; (b) hypersensitivity to design flaws and unanticipated perturbations; (c) nongeneric, specialized, structured configurations; and (d) power laws. In summary, the theory of SOC seeks to explain how the multitude of large interactive systems observed in nature develops power-law relationships from simple rules of interaction. Since some SOC models generate power-law distributions, proponents of the theory claim that many natural phenomena can eventually be understood via SOC. Since the mechanism by which an open system self-organizes into a state with no characteristic scales is not unique, we may need to ask how SOC works before we question how nature works by using SOC. SOC as a subfield of nonequilibrium statistical mechanics has played a key role in understanding self-organized, complex, many-body systems. Despite the problems mentioned above, its contribution to complexity research should be acknowledged. In his book, Bak [2] presents the historical development and general ideas behind the theory of SOC in a form easily absorbed by the nonmathematically inclined reader. However, to get a better understanding with a certain mathematical rigor, and comprehensive, balanced views about SOC, see the books by Jensen [11] and Sornette [28], as well as others referenced in this entry. References [1] Allen, A.P., Li, B.L. & Charnov, E.L. (2001). Population fluctuations, power laws and mixtures of lognormal distributions, Ecology Letters 4, 1 3. [2] Bak, P. (1996). How Nature Works: The Science of Selforganized Criticality, Springer-Verlag, New York. [3] Bak, P., Tang, C. & Wiesenfeld, K. (1987). Self-organized criticality: an explanation of 1/f noise, Physical Review Letters 59, [4] Carlson, J.M. & Doyle, J. (1999). Highly optimized tolerance: a mechanism for power laws in designed systems, Physical Review E 60, [5] Carlson, J.M. & Doyle, J. (2000). Highly optimized tolerance: robustness and design in complex systems, Physical Review Letters 84, [6] Choi, J.S., Mazumder, A. & Hansell, R.I.C. (1999). Measuring perturbation in a complicated, thermodynamic world, Ecological Modelling 117, [7] Fernandez, J. & Plastino, A. (2000). Structural stability can shape biological evolution, Physica A 276, [8] Gleiser, P.M., Tamarit, F.A. & Cannas, S.A. (2000). Self-organized criticality in a model of biological evolution with long-range interactions, Physica A 275, [9] Halley, J.M. (1996). Ecology, evolution and 1/f noise, Trends in Ecology and Evolution 11, [10] Ito, K. & Gunji, Y. (1994). Self-organisation of living systems towards criticality at the edge of chaos, BioSystems 33, [11] Jensen, H.J. (1998). Self-organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press, Cambridge. [12] Jørgensen, S.E., Mejer, H.F. & Nielsen, S.N. (1995). Ecosystem as self-organized critical systems, Ecological Modelling 111, [13] Kadanoff, L.P. (2000). Statistical Physics: Statics, Dynamics and Renormalization, World Scientific, Singapore. [14] Kauffman, S.A. & Johnsen, S. (1991). Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches, Journal of Theoretical Biology 149, [15] Keitt, T.H. & Stanley, H.E. (1998). Dynamics of North American breeding bird populations, Nature 393, [16] Li, B.L. (2000). Fractal geometry applications in description and analysis of patch patterns and patch dynamics, Ecological Modelling 132,

5 4 Criticality, self-organized [17] Li, B.L. & Forsythe, W.C. (1992). Spatio-temporal characteristics of vegetation response to disturbances: analysis of a cellular automata model, Bulletin of the Ecological Society of America 73, 249. [18] Li, B.L., Wu, H. & Zou, G. (2000). Self-thinning rule: a causal interpretation from ecological field theory, Ecological Modelling 132, [19] Malamud, B., Morein, G. & Turcotte, D.L. (1998). Forest fires: an example of self-organized critical behavior, Science 281, [20] Milne, B.T. (1998). Motivation and benefits of complex systems approaches in ecology, Ecosystems 1, [21] Newman, M. (2000). The power of design, Nature 405, [22] Podobnik, B., Ivanov, P.C., Lee, Y., Chessa, A. & Stanley, H.E. (2000). Systems with correlations in the variance: generating power law tails in probability distributions, Europhysics Letters 50, [23] Prigogine, I. (1997). The End of Certainty: Time, Chaos, and the New Laws of Nature, Free Press, New York. [24] Rodriguez-Iturbe, I. & Rinaldo, A. (1997). Fractal River Basins: Chance and Self-organization, Cambridge University Press, Cambridge. [25] Shiner, J.S. (2000). Self-organized criticality: self-organized complexity?, Open Systems and Information 7, [26] Sole, R.V. & Manrubia, S.C. (1995). Are rainforests self-organized in a critical state?, Journal of Theoretical Biology 173, [27] Sole, R.V., Manrubia, S.C., Benton, M., Kauffman, S. & Bak, P. (1999). Criticality and scaling in evolutionary ecology, Trends in Ecology and Evolution 14, [28] Sornette, D. (2000). Critical Phenomena in Natural Sciences, Springer-Verlag, Berlin. [29] Zolotarev, V.M. (1986). One-dimensional Stable Distributions, American Mathematical Society, Providence. (See also Population dynamics) BAI-LIAN LI