Self-Organized Criticality

One of many, many ways of generating power law distributions; not "how nature works".

Conjecture: Reed and Hughes (see here) have shown that you generically get power-law size distributions if you observe things which grow exponentially (in the mean) at random times. Could one show that this accounts for the avalanche size distributions, by tracking the size of individual avalanches over time?

Ronald Dickman, Miguel A. Munoz, Alessandro Vespignani and Stefano Zapperi, "Paths to Self-Organized Criticality", cond-mat/9910454
Alessandro Vespignani and Stefano Zapperi, "How self-organized criticality works: A unified mean-field picture", cond-mat/9709192

S. C. Chapman, R. O. Dendy and N. W. Watkins, "Robustness and scaling: key observables in the complex dynamic magnetosphere", Plasma Physics and Controlled Fusion 46 (2004): B157--B166 [How to get sandpile-like event-size distributions from a linear model. Thanks to Prof. Chapman for a reprint.]

A. Arenas, A. Diaz-Guilera, C.J. Perez and F. Vega-Redondo, "Self-organized criticality in evolutionary systems with local interaction," cond-mat/0103496
Siva R. Athreya and Antal A. Járai, "Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models", Communications in Mathematical Physics 249 (2004): 197--213
Marco Baiesi and Christian Maes, "Realistic time correlations in self-organized criticality from a random walk driving", cond-mat/0505274 [Contradicting the arguments of X. Yang et al., below; also, this seems like a cool model in its own right.]
Viorel Barbu, Philippe Blanchard, Giuseppe Da Prato, Michael Röckner, "Self-organized criticality via stochastic partial differential equations", arxiv:0811.2093
Claude Bedard, Helmut Kroeger and Alain Destexhe, "Does the 1/f frequency-scaling of brain signals reflect self-organized critical states?", q-bio.NC/0608026
J. M. Beggs and D. Plenz
- "Neuronal avalanches in cultured slices of neocortex", J Neurosci 23 (2003): 11167--77 [Link to abstract, PDF]
- "Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures", Journal of Neuroscience 24 (2004): 5216--5229 [Link to abstract, PDF]
Ginestra Bianconi and Matteo Marsili, "Clogging and self-organized criticality in complex networks", Physical Review E 70 (2004): 035105(R) = cond-mat/0312537
Xavier Bressaud and Nicolas Fournier, "On the invariant distribution of a one-dimensional avalanche process", math.PR/0703750
J. M. Carlson and G. H. Swindle, "Self-organized criticality: Sandpiles, singularities, and scaling", Proceedings of the National Academy of Sciences (USA) 92 (1995): 6712--6719
B. Cessac, Ph. Blanchard, T. Krüger and J. L. Meunier, "Self-Organized Criticality and Thermodynamic Formalism", Journal of Statistical Physics 115 (2004): 1283--1326 = nlin.CD/0209038
Alvaro Corral, "Comment on 'Do Earthquakes Exhibit Self-Organized Criticality?'", cond-mat/0412441
Michael Creutz, "Playing with Sandpiles", cond-mat/0401302
J. Davidsen and N. Luethje, "Comment on "1/f noise in the Bak-Sneppen model"," cond-mat/0201201
Roman Frigg, "Self-organised criticality --- what it is and what it isn't", Studies in History and Philosophy of Science A 34 (2003): 613--632
Guilherme J. M. Garcia and Ronald Dickman, "On the thresholds, probability densities, and critical exponents of Bak-Sneppen-like models", cond-mat/0403036
C. Haldeman and J. M. Beggs, "Critical branching captures activity in living neural networks and maximizes the number of metastable states", Physical Review Letters 94 (2005): 058101 [Link to abstract, PDF]
Julianne D. Halley, Andrew C. Warden, Suzanne Sadedin and Wentian Li, "Rapid self-organized criticality: Fractal evolution in extreme environments", Physical Review E 70 (036118)
Henrik Jeldt Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems [Blurb]
Boris S. Kruglikov and Martin Rypdal, "Dynamics and entropy in the Zhang model of Self-Organized Criticality", math.DS/0411590
Lasse Laurson, Mikko J. Alava and Stefano Zapperi, "Power spectra of self-organized critical sandpiles", cond-mat/0509401
D.-S. Lee, K.-I. Goh, B. Kahng and D. Kim, "Sandpile avalanche dynamics on scale-free networks", cond-mat/0401531
C. Maes, F. Redig and E. Saada, "The Infinite Volume Limit of Dissipative Abelian Sandpiles", Communications in Mathematical Physics 244 (2004): 395--417
Ronald Meester and Dmitri Znamenski, "Limit Behavior of the Bak-Sneppen Evolution Model", The Annals of Probability 31 (2003): 1986--2002
Gunnar Pruessner and Ole Peters, "Self-organizied Criticality and Absorbing States: Lessons from the Ising Model", cond-mat/0411709
Matthew Stapleton, Kim Christensen, "Mean-Field Theory and Sandpile Models", cond-mat/0510626
Matthew Stapleton, Martin Dingler and Kim Christensen, "Sensitivity to Initial Conditions in Self-Organized Critical Systems", Journal of Statistical Physics 117 (2004): 891--900
Ronaldo Vidigal and Ronald Dickman, "Asymptotic Behavior of the Order Parameter in a Stochastic Sandpile", Journal of Statistical Physics 118 (2005): 1--25
Felix Wissel and Barbara Drossel, "Transient and stationary behavior of the Olami-Feder-Christensen earthquake model", cond-mat/0608324 ["the transient time diverges nonanalytically when $\alpha$ approaches zero, and that the avalanche-size distribution will not approach a power law with increasing system size."]
C. B. Yang and Rudolph C. Hwa, "The origin of power-law distributions in self-organized criticality", cond-mat/0403304
Xiaosong Yang, Shuming Du and Jin Ma, "Do Earthquakes Exhibit Self-Organized Criticality?", Physical Review Letters 92 (2004): 228501 [Ans.: No.]

"Is Self-Organized Criticality Actually Self-Organizing?"

Notebooks

Self-Organized Criticality