Abstract
During this book we have encountered many examples of critical phenomenon and have highlighted their common characteristics of divergence of the range of correlations, absence of characteristic scales, fluctuations of all sizes and anomalous response (at all scales in amplitude, spatial extent and duration) to even a tiny perturbation. These characteristics are reflected in many scaling laws, expressing quantitatively the scale invariance of the phenomena. Typically, criticality occurs for a particular value of the control parameter which is adjusted externally, for example the critical temperature in second order phase transitions, percolation threshold p c or bifurcation point in a dynamic system. However, it turns out that certain systems, maintained in a nonequilibrium state by a continuous supply of material or energy, can evolve spontaneously to a critical state, without external regulation. This is the concept of self-organised criticality, which we will present in this chapter, discussing various examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Other than the “trivial” scales of system size, duration of observation and, at the other extreme, the scales of the constitutive elements.
- 2.
In this preliminary statement, we have deliberately omitted to take into account the spatial extension of the system. The quantity X is actually a function \(X(\vec{r})\) and will therefore evolve differently at each spatial point \(\vec{r}\). Self-organised criticality appears when, in addition, spatial correlations develop as local values of X approach the threshold X c . We will return to this point in Sect. 10.1.3.
- 3.
We call hard core interaction a short ranged quasi-infinite repulsion, modelled by a hard core of radius equal to the range of this repulsive interaction.
- 4.
In three dimensions a percolating path no longer makes a border and the front will extend over a whole range of concentrations. However, its foremost part will still be spontaneously localised in the region where p = p c .
- 5.
A cellular automaton is a model in which time, space but also state variables take only discrete values. Such a model is therefore particularly suitable for numerical studies. Typically, particles are moved from site to site according to very simple probabilistic rules. Cellular automata are used to study many transport phenomena, for example reaction-diffusion phenomena or those encountered in hydrodynamics and population dynamics [8]. In the context of self-organised criticality, sandpiles, forest fires and traffic flow are numerically studied using this type of simulation [20].
- 6.
We do not discuss here the reliability of observations or how real the phenomenon the model intends to reproduce and explain actually is.
References
P. Bak, How Nature Work: The Science of Self-Organized Criticality (Springer, Berlin, 1996)
P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of 1 ∕ f noise. Phys. Rev. Lett. 59, 381 (1987)
P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality. Phys. Rev. A 38, 364 (1988)
P. Bak, K. Chen, C. Tang, A forest-fire model and some thoughts on turbulence. 147, 297 (1990)
P. Bak, K. Sneppen, Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71, 4083 (1993)
A.L. Barabasi, S.V. Buldyrev, H.E. Stanley, B. Suki, Avalanches in the lung: a statistical mechanical model. Phys. Rev. Lett. 76, 2192 (1996)
K. Chen, P. Bak, Self-organized criticality. Sci. Am. 46 (1991)
B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems. Collection “Aléa Saclay” (Cambridge University Press, Cambridge, 1998)
K. Christensen, L. Danon, T. Scanlon, P. Bak, Unified scaling law for earthquakes. Proc. Natl. Acad. Sci USA 99, 2509 (2002)
B. Drossel, F. Schwabl, Self-organized critical forest-fire model. Phys. Rev. Lett. 69, 1629 (1992)
N. Fraysse, A. Sornette, D. Sornette, Critical phase transitions made self-organized: proposed experiments. J. Phys. I France 3, 1377 (1993)
L. Gil, D. Sornette, Landau–Ginzburg theory of self-organized criticality. Phys. Rev. Lett. 76, 3991 (1996)
J.F. Gouyet, Physics and Fractal Structures (Springer, New York, 1996)
B. Gutenberg, C.F. Richter, Bull. Seismol. Soc. Am. 64, 185 (1944)
B. Gutenberg, C.F. Richter, Seismicity of the Earth and Associated Phenomenon (Princeton University Press, Princeton, 1954)
D. Helbing, M. Treiber, Jams, waves and clusters. Science, 282, 2001 (1998)
H.J. Jensen, Self-Organized Criticality (Cambridge University Press, Cambridge, 1998)
S.A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution (Oxford University Press, 1993)
H.Y. Lee, H.W. Lee, D. Kim, Origin of synchronized traffic flow on highways and its dynamic phase transitions. Phys. Rev. Lett. 81, 1130 (1998)
K. Nagel, M. Schreckenberg, A cellular automaton model for freeway traffic. J. Phys. I France 2, 2221 (1992)
S.R. Nagel, Instabilities in a sandpile. Rev. Mod. Phys. 64, 321 (1992)
M.E.J. Newman, Evidence for self-organized criticality in evolution. Physica D 107, 293 (1997)
M.E.J. Newman, Simple models of evolution and extinction. Comput. Sci. Eng. 2, 80 (2000)
M.E.J. Newman, G.J. Eble, Power spectra of extinction in the fossil record. Proc. R. Soc. London B 266, 1267 (1999)
M. Paczuski, K. Nagel, in Traffic and Granular Flow, ed. by D.E. Wolf, M. Schreckenberg, A. Bachem. Self-organized criticality and 1/f noise in traffic (World Scientific, Singapore, 1996) pp. 73–85
L.F. Richardson, Frequency of occurrence of wars and other fatal quarrels. Nature 148, 598 (1941)
D. Sornette, Critical phase transitions made self-organized: a dynamical system feedback mechanism for self-organized criticality. J. Phys. I France 2, 2065 (1992)
D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Self-Organization and Disorder: Concepts and Tools (Springer, Berlin, 2000)
B. Suki, A.L. Barabasi, Z. Hantos, F. Peták, H.E. Stanley, Avalanches and power-law behaviour in lung inflation. Nature 368, 615 (1994)
D.L. Turcotte, Self-organized criticality. Rep. Prog. Phys. 62, 1377 (1999)
D.E. Wolf, M. Schrenckenberg, A. Bachem (eds.), Traffic and Granular Flow (World Scientific, Singapore, 1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lesne, A., Laguës, M. (2012). Self-Organised Criticality. In: Scale Invariance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15123-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-15123-1_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15122-4
Online ISBN: 978-3-642-15123-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)