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Register a new userDynamical properties of the Zhang model of Self-Organized Criticality
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Added by
Albert Diaz-Guilera
Publication date
1998
fields
Physics
and research's language is
English
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Critical exponents of the infinitely slowly driven Zhang model of self-organized criticality are computed for $d=2,3$ with particular emphasis devoted to the various roughening exponents. Besides confirming recent estimates of some exponents, new quantities are monitored and their critical exponents computed. Among other results, it is shown that the three dimensional exponents do not coincide with the Bak, Tang, and Wiesenfeld (abelian) model and that the dynamical exponent as computed from the correlation length and from the roughness of the energy profile do not necessarily coincide as it is usually implicitly assumed. An explanation for this is provided. The possibility of comparing these results with those obtained from Renormalization Group arguments is also briefly addressed.
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The concept of percolation is combined with a self-consistent treatment of the interaction between the dynamics on a lattice and the external drive. Such a treatment can provide a mechanism by which the system evolves to criticality without fine tuning, thus offering a route to self-organized criticality (SOC) which in many cases is more natural than the weak random drive combined with boundary loss/dissipation as used in standard sand-pile formulations. We introduce a new metaphor, the e-pile model, and a formalism for electric conduction in random media to compute critical exponents for such a system. Variations of the model apply to a number of other physical problems, such as electric plasma discharges, dielectric relaxation, and the dynamics of the Earths magnetotail.
A system is in a self-organized critical state if the distribution of some measured events (avalanche sizes, for instance) obeys a power law for as many decades as it is possible to calculate or measure. The finite-size scaling of this distribution function with the lattice size is usually enough to assume that any cut off will disappear as the lattice size goes to infinity. This approach, however, can lead to misleading conclusions. In this work we analyze the behavior of the branching rate sigma of the events to establish whether a system is in a critical state. We apply this method to the Olami-Feder-Christensen model to obtain evidences that, in contrast to previous results, the model is critical in the conservative regime only.
The shape of clouds has proven to be essential for classifying them. Our analysis of images from fair weather cumulus clouds reveals that, besides by turbulence they are driven by self-organized criticality (SOC). Our observations yield exponents that support the fact the clouds, when projected to two dimensions (2D), exhibit conformal symmetry compatible with $c=-2$ conformal field theory (CFT), in contrast to 2D turbulence which has $c=0$ CFT. By using a combination of the Navier-Stokes equation, diffusion equations and a coupled map lattice (CML) we successfully simulated cloud formation, and obtained the same exponents.
The well known Sandpile model of self-organized criticality generates avalanches of all length and time scales, without tuning any parameters. In the original models the external drive is randomly selected. Here we investigate a drive which depends on the present state of the system, namely the effect of favoring sites with a certain height in the deposition process. If sites with height three are favored, the system stays in a critical state. Our numerical results indicate the same universality class as the original model with random depositition, although the stationary state is approached very differently. In constrast, when favoring sites with height two, only avalanches which cover the entire system occur. Furthermore, we investigate the distributions of sites with a certain height, as well as the transient processes of the different variants of the external drive.
Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attacted much comment, ranging from the very positive to the polemical. The other papers in this special issue (Aschwanden et al, 2014; McAteer et al, 2014; Sharma et al, 2015) showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. In Baks own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner, 2012). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfelds original papers.
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