Self-organizing criticality

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

The mechanism of Self-organized criticality, a concept first introduced by Bak, Tang and Wiesenfeld [bak87a], may possibly provide a fundamental link between such temporal scale invariant phenomena and phenomena exhibiting a spatial scale invariance - familiar examples of which are given by fractal coastlines, mountain landscapes and cloud formations [mandel82],... 

The most important property of the self-organized critical state is the presence of locally connected domains of all sizes. Since a given perturbation of the state 77 can lead to anything from a trivial one-site shift to a lattice-wide avalanche, there are no characteristic length scales in the system. Bak, et al. [bak87] have, in fact, found that the distribution function D s) of domains of size s obeys the power law... 

In physics, the late Danish physicist. Per Bak, and his colleagues Tang and Wiesenfeld startled the field in the late 1980s by producing a widely quoted paper on self-organized criticality. Bak and others applied this model widely - to the size distribution of earthquakes and the distribution of clusters of matter in the universe, to the size distribution of extinction events in the biological record. Self-organized... 

C.J. Perez, A. Corral, A. Diaz Guilera, K. Christensen, and A. Arenas. On self-organized criticality and synchronization in lattice models of coupled dynamical systems, http //xxx.lanl.gov, paperno. cond-mat/9601102, 1996. 

Complex adaptative systems such as membranes are said to self-organize, but this must not be misunderstood. Indeed, self-organized criticality at each hierarchical level of biological systems is not reached independently of other levels, but in a fully coordinated and integrated manner. The readily recognizable levels are listed in Section 1.2.1, and those to be briefly considered here appear in the title of this section. 

P. Bak, How Nature Works. The Science of Self-Organized Criticality, Oxford University Press, Oxford, 1997. 

Tang, C. and Bak, P., Critical exponents and scaling relations for self organized critical phenomena, Physical Review Letters, Vol. 60, 1988, pp. 2347-2350. 

Seuront L, Spilmont N (2002) Self-organized criticality in intertidal microphytobenthos patch patterns. Physica A 313 513-539... 

There are also some nice computer simulations, such as those by Cieplak and Robbins ), demonstrating the influence of the contact angle on fluid penetration into a two-dimensional porous material and providing an example of self-organized criticality. 

R Jung, A. Cornell-Bell, K. S. Madden, and F. Moss. Noise-induced spiral waves in astrocyte syncytia show evidence of self-organized criticality. J. Neurophysiol., 79 1098, 1998. 

Fig. 7.7 The distribution of numbers of atoms in t rpical large avalanches of STs with the distribution being of power-law type of the form 7.64 (s = —1.96 0.09) exhibiting events of self-organized criticality (from Argon and Demkowicz (2008) courtesy of TMS).

Jensen, H. J. (1998) Self Organized Criticality, Cambridge Cambridge University Press. 

The spontaneous emergence of avalanches, droplets and rivulets is very difficult to simulate with classical fluid dynamical models, due to the critical nature (self-organized criticality) and threshold character of these nonlinear phenomena. Therefore, the role of statistical fluctuations in thin-film dynamics cannot be underestimated, especially in the mesoscale. Unlike the classical approaches, we need not introduce any external and artificial perturbations. All phenomena occur spontaneously due to thermal noise inherent in the nonlinearly interacting particle dynamics. 

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