Adaptation Noise And Self Organizing Systems

We consider simple examples of self-organized critical systems on one-dimensional superlattices without local particle conservation laws. The set of all recurrence states are also found in these examples using a method similar to the burning algorithm.

The train model which is a variant of the Burridge-Knopoff earthquake model is investigated for a velocity-strengthening friction law. It shows self-organized criticality with complex scaling exponents. That is, the probability density function of the avalanche strength is a power law times a log-periodic function. Exact results (scaling exponent: 3/2+2πi/ln4 ) are found for a nonlocal cellular automaton which approximates the overdamped train model. Further the influence of random static friction is discussed.

The disorder and a simple convex measure of complexity are studied for rank ordered power law distributions, indicative of criticality, in the case where the total number of ranks is large. It is found that a power law distribution may produce a high level of complexity only for a restricted range of system size (as measured by the total number of ranks), with the range depending on the exponent of the distribution. Similar results are found for disorder. Self-organized criticality thus does not guarantee a high level of complexity, and when complexity does arise, it is self-organized itself only if self-organized criticality is reached at an appropriate system size.

Analytical investigation is made on the two-dimensional traffic-flow model with alternative movement and exclude-volume effect between right and up arrows [Phys. Rev. {\bf A} 46 R6124 (1992)]. Several exact results are obtained, including the upper critical density above which there are only jamming configurations, and the lower critical density below which there are only moving configurations. The observed jamming transition takes place at another critical density p c (N) , which is in the intermidiate region between the lower and upper critical densities. It is derived that p c (N)=C N α , where C and α are determined to be respectively 0.76 and −0.14 from previous numerical simulation. This transition is suggested to be a second-order phase transition, the order parameter is found. The nature of self-organization, ergodicity breaking and synchronization are discussed, Comparison with the sandpile model is made.

We suggest that ensembles of self-replicating entities such as biological systems naturally evolve into a self-organized critical state in which fluctuations, as well as waiting-times between phase transitions are distributed according to a 1/f power law. We demonstrate these concepts by analyzing a population of self-replicating strings (segments of computer-code) subject to mutation and survival of the fittest.

We examine exhaustively the behavior of avalanches in critical height sandpile models based in two- and three-dimensional lattices of various topologies. We get that for two-dimensional lattices the spatial and temporal distributions characterizing bulk avalanches do not depend on the lattice topology. For the three-dimensional case, we detect a small dependence of the topology for the temporal distribution, while the spatial ones are independent. The two-dimensional lattices studied are: the plane ( R 2 ), the cylinder ( S 1 ×R ), and the Möbius-strip ( M ); and the three-dimensional are: R 3 , S 1 × R 2 , S 1 × S 1 ×R , M×R , S 2 ×R , K×R , and RP×R , where K and RP are respectively the Klein bottle and the real projective plane.

Statistical analysis indicates that the fossil extinction record is compatible with a distribution of extinction events whose frequency is related to their size by a power law with an exponent close to two. This result is in agreement with predictions based on self-organized critical models of extinction, and might well be taken as evidence of critical behaviour in terrestrial evolution. We argue however that there is a much simpler explanation for the appearance of a power law in terms of extinctions caused by stresses (either biotic or abiotic) to which species are subjected by their environment. We give an explicit model of this process and discuss its properties and implications for the interpretation of the fossil record.

In this paper we consider the question whether a distributed network of sensors and data processors can form "perceptions" based on the sensory data. Because sensory data can have exponentially many explanations, the use of a central data processor to analyze the outputs from a large ensemble of sensors will in general introduce unacceptable latencies for responding to dangerous situations. A better idea is to use a distributed "Helmholtz machine" architecture in which the collective state of the network as a whole provides an explanation for the sensory data.

Simple analytically solvable model of 1/f noise is proposed. The model consists of one or few particles moving in the closed contour. The drift period of the particle round the contour fluctuates about some average value, e.g. due to the external random perturbations of the system's parameters. The model contains only one relaxation rate, however, the power spectral density of the current of particles reveals an exact 1/f spectrum in any desirable wide range of frequency and can be expressed by the Hooge formula. It is likely that the analysis and the generalizations of the model can strongly influence on the understanding of the origin of 1/f noise.

This article gives a brief introduction to the mathematical modeling of large-scale biological evolution and extinction. We give three examples of simple models in this field: the coevolutionary avalanche model of Bak and Sneppen, the environmental stress model of Newman, and the increasing fitness model of Sibani, Schmidt, and Alstrom. We describe the features of real evolution which these models are intended to explain and compare the results of simulations against data drawn from the fossil record.