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, 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, (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, S.L.Ginzburg, N.E.Savitskaya, Acta Physica Slovaca, 52 (6),597, 2002

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.

BTW-model and sandpile-like model Ji-Ji- Ji+Ji+ J i-1 + J i+1 - Z i-1 zizi Z i+1 Perturbation rules Toppling rules BTW-model 11 Z i-1 Z i Z i+1

2.Model with spatial-temporal stochastity ( S.S.Manna, J. of Phys. A, 24, L363 (1992). ) 0 2 Z i-1 Z i Z i Z i-1 Z i Z i+1 1 Z i-1 Z i Z i+1

Model with intrinsic spatial randomness Ji-Ji- Ji+Ji+ J i-1 + J i+1 - Z i-1 zizi Z i+1 Perturbation rules Toppling rules Boundary conditions: Closed boundary: Open boundary:

SYSTEM SYMMETRY Ji-Ji- Ji+Ji+ J i-1 + J i+1 - Z i-1 zizi Z i+1 1.The system is called site-symmetrical if 2. The system is called cell-symmetrical if: JiJi JiJi J i-1 J i+1 Z i-1 zizi Z i+1 J i-1 JiJi JiJi Z i-1 zizi Z i+1

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.

METHODS OF PERTURBATION 1. The deterministic perturbation means that every time we increase z 1 at the closed boundary only. 2. The random perturbation implies that we perturb the system by variation of z i 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 W n is a measure of the total number of topplings (an avalanche size in the sandpile model): where k bn and k en 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.

POTENTIAL SYSTEMS J i-1 JiJi JiJi Z i-1 zizi Z i+1 J i =1 : BTW model, no self-organized criticality. J i-1 =1; J i =2 : 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 J i. The particular case of such a system is the multijunction SQUID we studied earlier. NON-POTENTIAL SYSTEM Ji-Ji- Ji+Ji+ J i-1 + J i+1 - Z i-1 zizi Z i+1

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.