Transcript Self-Organized Criticality and Solar Flares

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
It’s all physics :
Social
Physics
Financial
Physics
Solar Physics
Astrophysics
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
dt S
N (WS )dWS  N [t S (WS )]
dWS
dWS
Frequency distribution of dissipated energies N(W) and fluxes N(F) are power-laws
for exponential or logistic growth curves WS(tS)
 G 2WS
2n
1
N (WS ) 
( )(
 1)  ,
W1 [1  exp( t 2 / t Se )] t Se W1
  (1 
G
t Se
)
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.
N(L) ~ V
1
~L
S=Euclidean dimension
of space (S=1,2,3,…)

S
All size distributions N(x) can be derived from scaling laws of L(x)~xa
Statistical probability for avalanches
with Euclidean size L:
N(L)dL  LS
Physical scaling laws
(2-parameter correlations):

L  T1/ 2,P  T DS / 2, E  T1DS / 2
Derived occurrence probability
frequency disributions:

dL
dt  T (1S )/ 2 ,
dt
dT
N(P)dP  N(T[P])
dP  P (1(S 1)/ D ),
dP
dT
N(E)dE  N(T[E])
dE  E [1(S 1)/(2D )]
dE
N(T)dt  N(T[L])
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