A Physicist’s View of SOC Models Presentation by: Markus J. Aschwanden 2013 September 16-20 International Space Science Institute (ISSI) Hallerstrasse 6, Bern, Switzerland http://www.issibern.ch/teams/s-o-turbulence/index.html

Solar Physics Social Physics Astrophysics Financial Physics It’s all physics : Solar Physics Social Physics Astrophysics Financial Physics Self-Organized Criticality Systems Magnetospheric Physics Biophysics Geophysics

A SOC event is an instability in a nonlinear dissipative system

An instability has an initally exponential-growing behavior, with subsequent saturation or quenching of the instability (exponential-growth and/or logistic-growth models)

Frequency Distribution Frequency distribution of dissipated energies N(W) and fluxes N(F) are power-laws for exponential or logistic growth curves WS(tS)

Binomial Statistics For incoherent (linear) random events: Gaussian, Poisson, exponential function Scale-free size probability for coherent (nonlinear) avalanches: Powerlaw function

The statistical probability distribution function (PDF) of all possible avalanche sizes in a finite volume scales reciprocally to the avalanche volume, which is a powerlaw function. S=Euclidean dimension of space (S=1,2,3,…)

All size distributions N(x) can be derived from scaling laws of L(x)~xa Statistical probability for avalanches with Euclidean size L: Physical scaling laws (2-parameter correlations): Derived occurrence probability frequency disributions:

Common observables in astrophysics: F = flux (or intensity in a given wavelength) P = peak flux of time profile of an event E = total (time-integrated energy) or total flux (fluence) Derived occurrence frequency distributions:

Summary of powerlaw indices : Powerlaw slopes:

3-Parameter Scaling Laws 3-Parameter scaling laws x=LaHb require the knowledge of 2 distributions N(L) and N(H) in order to derive the size distribution of the 3rd variable, N(x). The scaling law can then be substituted and the integration over the other two variables has to be performed under consideration of the truncated distributions.

Universal Probability Statistics Hydrodynamic Physcial System Instumental & Observable Parameters

Using the observed statistical size distributions (in particular their powerlaw slopes) we can retrieve the scaling laws and correlations between the underlying physical parameters. The generic SOC models (sandpile avalanches and cellular automatons) mimic the evolution of instabilities with discretized mathematical redistribution rules for next-neighbor interactions on a microscopic level  toy models for physical instabilities observed on a macrosocpic level.

Metrics of Observables, Statistical Distributions and Physical Processes