Book contents
2 - Scaling
from Part I - Introduction
Published online by Cambridge University Press: 05 September 2012
Summary
In broad terms, the aim of the analysis of a supposed self-organised critical system is to determine whether the phenomenon is merely the sum of independent local events, or is caused by interactions on a global scale, i.e. cooperation, which is signalled by algebraic correlations and non-Gaussian event distributions. Self-organised criticality therefore revolves around scaling and scale invariance, as it describes the asymptotic behaviour of large, complex systems and hints at their universality (Kadanoff, 1990). Numerical and analytical work generally concentrates on the scaling features of a model. Understanding their origin and consequences is fundamental to the analysis as well as to the interpretation of SOC models, beginning at the very motivation of a particular model and permeating down to the level of the presentation of data.
During the last fifteen years or so, the understanding of scaling in complex systems has greatly improved and some standard numerical techniques have been established, which allow the comparison of different models, assumptions and approaches. Yet, there is still noticeable confusion regarding the implications of scaling as well as its quantification.
Most concepts, such as universality and generalised homogeneous functions, are taken from or are motivated by the equilibrium statistical mechanics of phase transitions (Stanley, 1971; Privman et al., 1991), and were first applied to SOC in a systematic manner by Kadanoff et al. (1989). Yet, what appears to be rather natural in the context of equilibrium statistical mechanics, might not be so for complex systems.
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- Self-Organised CriticalityTheory, Models and Characterisation, pp. 25 - 51Publisher: Cambridge University PressPrint publication year: 2012