50 research outputs found
Metastability, Criticality and Phase Transitions in brain and its Models
This essay extends the previously deposited paper "Oscillations, Metastability and Phase Transitions" to incorporate the theory of Self-organizing Criticality. The twin concepts of Scaling and Universality of the theory of nonequilibrium phase transitions is applied to the role of reentrant activity in neural circuits of cerebral cortex and subcortical neural structures
The abelian sandpile and related models
The Abelian sandpile model is the simplest analytically tractable model of
self-organized criticality. This paper presents a brief review of known results
about the model. The abelian group structure allows an exact calculation of
many of its properties. In particular, one can calculate all the critical
exponents for the directed model in all dimensions. For the undirected case,
the model is related to q= 0 Potts model. This enables exact calculation of
some exponents in two dimensions, and there are some conjectures about others.
We also discuss a generalization of the model to a network of communicating
reactive processors. This includes sandpile models with stochastic toppling
rules as a special case. We also consider a non-abelian stochastic variant,
which lies in a different universality class, related to directed percolation.Comment: Typos and minor errors fixed and some references adde
Avalanches in the Weakly Driven Frenkel-Kontorova Model
A damped chain of particles with harmonic nearest-neighbor interactions in a
spatially periodic, piecewise harmonic potential (Frenkel-Kontorova model) is
studied numerically. One end of the chain is pulled slowly which acts as a weak
driving mechanism. The numerical study was performed in the limit of infinitely
weak driving. The model exhibits avalanches starting at the pulled end of the
chain. The dynamics of the avalanches and their size and strength distributions
are studied in detail. The behavior depends on the value of the damping
constant. For moderate values a erratic sequence of avalanches of all sizes
occurs. The avalanche distributions are power-laws which is a key feature of
self-organized criticality (SOC). It will be shown that the system selects a
state where perturbations are just able to propagate through the whole system.
For strong damping a regular behavior occurs where a sequence of states
reappears periodically but shifted by an integer multiple of the period of the
external potential. There is a broad transition regime between regular and
irregular behavior, which is characterized by multistability between regular
and irregular behavior. The avalanches are build up by sound waves and shock
waves. Shock waves can turn their direction of propagation, or they can split
into two pulses propagating in opposite directions leading to transient
spatio-temporal chaos. PACS numbers: 05.70.Ln,05.50.+q,46.10.+zComment: 33 pages (RevTex), 15 Figures (available on request), appears in
Phys. Rev.
25 Years of Self-Organized Criticality: Numerical Detection Methods
The detection and characterization of self-organized criticality (SOC), in
both real and simulated data, has undergone many significant revisions over the
past 25 years. The explosive advances in the many numerical methods available
for detecting, discriminating, and ultimately testing, SOC have played a
critical role in developing our understanding of how systems experience and
exhibit SOC. In this article, methods of detecting SOC are reviewed; from
correlations to complexity to critical quantities. A description of the basic
autocorrelation method leads into a detailed analysis of application-oriented
methods developed in the last 25 years. In the second half of this manuscript
space-based, time-based and spatial-temporal methods are reviewed and the
prevalence of power laws in nature is described, with an emphasis on event
detection and characterization. The search for numerical methods to clearly and
unambiguously detect SOC in data often leads us outside the comfort zone of our
own disciplines - the answers to these questions are often obtained by studying
the advances made in other fields of study. In addition, numerical detection
methods often provide the optimum link between simulations and experiments in
scientific research. We seek to explore this boundary where the rubber meets
the road, to review this expanding field of research of numerical detection of
SOC systems over the past 25 years, and to iterate forwards so as to provide
some foresight and guidance into developing breakthroughs in this subject over
the next quarter of a century.Comment: Space Science Review series on SO
Introduction to the Sandpile Model
This article is based on a talk given by one of us (EVI) at the conference
``StatPhys-Taipei-1997''. It overviews the exact results in the theory of the
sandpile model and discusses shortly yet unsolved problem of calculation of
avalanche distribution exponents. The key ingredients include the analogy with
the critical reaction-diffusion system, the spanning tree representation of
height configurations and the decomposition of the avalanche process into waves
of topplings
Experiments in vortex avalanches
Avalanche dynamics is found in many phenomena spanning from earthquakes to
the evolution of species. It can be also found in vortex matter when a type II
superconductor is externally driven, for example, by increasing the magnetic
field. Vortex avalanches associated with thermal instabilities can be an
undesirable effect for applications, but "dynamically driven" avalanches
emerging from the competition between intervortex interactions and quenched
disorder constitute an interesting scenario to test theoretical ideas related
with non-equilibrium dynamics. However, differently from the equilibrium phases
of vortex matter in type II superconductors, the study of the corresponding
dynamical phases - in which avalanches can play a role - is still in its
infancy. In this paper we critically review relevant experiments performed in
the last decade or so, emphasizing the ability of different experimental
techniques to establish the nature and statistical properties of the observed
avalanche behavior.Comment: To be published in Reviews of Modern Physics April 2004. 17 page