## Self-Organized Criticality

*27 Feb 2017 16:30*

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?

- Recommended (big picture):
- Ronald Dickman, Miguel A. Munoz, Alessandro Vespignani and Stefano
Zapperi, "Paths to Self-Organized Criticality",
Brazilian Journal of Physics
**30**(2000): 27--41, cond-mat/9910454 - Alessandro Vespignani and Stefano Zapperi, "How self-organized criticality works: A unified mean-field picture", cond-mat/9709192

- Recommended (close ups):
- 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.] - D. A. Head, "Implicit Parameter Tuning in the Bak-Sneppen Model",
European Physical Journal B
**17**(2000): 289--294, arxiv:cond-mat/9910207

- To Read:
- 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]

- "Neuronal avalanches in cultured slices of
neocortex", J Neurosci
- 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
- 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 - Ole Peters and Michelle Girvan, "Universality Under Conditions
of Self-tuning", Journal of Statistical
Physics
**141**(2010): 53--59, arxiv:0902.1956 - Gunnar Pruessner, Self-Organised Criticality: Theory, Models and Characterisation
- 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.]

- To write:
- "Is Self-Organized Criticality Actually Self-Organizing?"