Self-organization of the critical state in one-dimensional systems with intrinsic spatial randomness

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Self-organization of the critical state in
one-dimensional systems with intrinsic spatial
randomness S.L.Ginzburg, N.E.Savitskaya Petersburg
Nuclear Physics Institute, Gatchina, Leningrad
district, 188300 Russia
1. Concept of self-organized criticality
(SOC) 2. BTW-model and sandpile-like model 3.
Model with intrinsic spatial randomness 4.Computer
simulation results 5.Conclusions
REFERENCE 1. P.Bak, C.Tang, K.Wiesenfeld, Phys.
Rev. Lett., 59, 381 (1987). 2.S.S.Manna, J. of
Phys. A, 24, L363 (1992). 3. S.L.Ginzburg,
N.E.Savitskaya, Phys. Rev. E, 66, 026128
(2002). 4. S.L.Ginzburg, N.E.Savitskaya, Jornal
of Low Temperature Physics, 130, N 3/4, 333
(2003). 5.A.Ali, D.Dhar, Phys.Rev.E, 51, R2705,
1995. 6. S.L.Ginzburg, N.E.Savitskaya, Acta
Physica Slovaca, 52 (6),597, 2002  
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Concept of self-organized criticality (SOC) (
P.Bak, C.Tang, K.Wiesenfeld, Phys. Rev. Lett.,
59, 381 (1987))   1.Giant dissipative dynamical
systems naturally evolve into self-reproducing
critical state without fine tuning of the
external parameters.   2.This critical state is
an ensemble of various metastable
states.   3.During its evolution, the critical
system never comes to a stable state but
migrates from one metastable state to another by
means of dynamical processes, so called
"avalanches".   4. The avalanches are initiated
by small external perturbations. Avalanche sizes
for se lf-organized system demonstrate a
power-law distribution.
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BTW-model and sandpile-like model  
Perturbation rules

Toppling rules
 
1 1  
BTW-model
Zi-1 Zi
Zi1
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2.Model with spatial-temporal stochastity
(S.S.Manna, J. of Phys. A, 24, L363 (1992).)



 

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Model with intrinsic spatial randomness
Perturbation rules

Toppling rules
Boundary conditions
Closed boundary
Open boundary
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SYSTEM SYMMETRY
1.The system is called site-symmetrical if
Zi1
2. The system is called cell-symmetrical if
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The cell-symmetry is an essential physical
characteristic of the system. If the system is
potential, it is cell-symmetrical. Site-symmetry
is not that important for the system dynamics and
its critical behavior. Why? Earlier we studied
the multijunction SQUID that can serve as
physical example of system with intrinsic spatial
randomness. Then the general view for equations,
describing a system with intrinsic spatial
randomness, may by written as
These equations can be rewritten as
The system is potential if
it means that the system is
cell-symmetrical one.
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METHODS OF PERTURBATION  1. The deterministic
perturbation means that every time we increase z1
at the closed boundary only. 2. The random
perturbation implies that we perturb the system
by variation of zi in a randomly chosen site.
The latter is usually used in classical sandpile
model.  AVALANCHE SIZE For each avalanche in the
critical state we can calculate its size Wn is
a measure of the total number of topplings (an
avalanche size in the sandpile model)
where kbn and ken are the initial and the
final moment of the n-th avalanche,
respectively.  Then we calculate the probability
densities for the avalanche sizes for potential
(cell-symmetrical) and non-potential
(cell-asymmetrical) systems.
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POTENTIAL SYSTEMS
Ji1 BTW model, no self-organized
criticality. Ji-11 Ji2 model from A.Ali,
D.Dhar, Phys.Rev.E, 51, R2705, 1995. The system
demonstrates self-organized-like behavior. We
consider the stochastical system with random Ji.
The particular case of such a system is the
multijunction SQUID we studied earlier.
NON-POTENTIAL SYSTEM
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CONCLUSIONS   1. A class of self-organized
systems is presented. We show that intrinsic
spatial randomness which we introduced in our
system effectively substitutes temporal
stochasticity that is necessary for realization
of SOC in the classical models of
self-organization. Systems under consideration
can be divided into two subclasses --- potential
and non-potential ones.     2. It is shown that
for some degree of intrinsic randomness the
critical state of the system is self-organized
even for deterministic perturbation. The
critical state of non-potential systems becomes
self-organized at lesser degree of randomness
than it is required for potential systems.