Self-organized Critical Phenomena Research Articles

Generalized sandpile model and the characterization of the existence of self-organized criticality.

We consider a generalized sandpile model where the particle addition and toppling are formulated in terms of mappings. In this way, the toppling rules and toppling conditions of the system can also be completely general. Even under these extremely relaxed conditions, we can still find an if and only if condition for the existence of an absolute steady state. Moreover, such a kind of absolute steady state often exhibits self-organized criticality. Our model is a superset of both the original Bak-Tang-Wiesenfeld sandpile and the Abelian sandpile models. Finally, we shall demonstrate the importance of both the particle-addition methods and the boundary conditions to the self-organized critical phenomena of a physical system.

Long-range spatial correlations for anisotropic zero-range processes

The spatial correlations are investigated for a homogeneous system of indistinguishable particles undergoing stochastic anisotropic hopping dynamics on the d-dimensional lattice, d>or=2. The interaction is zero range, i.e. the rate at which particles leave a given site only depends on the occupation number at that site. A series expansion around the independent particle system is given for the equal time correlations and is shown to converge for small times t. The formal t to infinity limiting expansion is analysed termwise from which a quadrupole type decay ( approximately r-d) is derived for the stationary two-points function ( eta (0) eta (r))-( eta (0))2. This phenomenon of self-organized criticality is a direct consequence of the anisotropy causing the system to violate the condition of detailed balance, combined with the conservation law forcing a diffusive decay ( approximately t-d2/) of the temporal correlations.

Theoretical studies of self-organized criticality

We study various aspects of the self-organized critical phenomena. A continuous-energy model is analyzed using a self-consistent effective medium method. We find that energy is homogeneously and isotropically distributed in space. We have calculated analytically the critical energy, which is very close to that of numerical simulations. We found that the critical energy is quite universal whereas the energy distribution per site depends on particular models. Using scaling arguments we have estimated the various exponents. We have also performed a detailed study of the noise spectrum. The result shows a 1/ f 2 behavior in the frequency region that we have probed.

Abelian sandpiles

A class of models for self-organized critical phenomena possesses an isomorphism between the recursive states under addition and the Abelian operator algebra on them. Several exact results follow, including the existence of a unique identity state, which when added to any configuration C in the recursive set relaxes back to that configuration. In this relaxation process, the number of topplings at any lattice site is independent of the configuration C.

Critical phenomena in fluid invasion of porous media.

We present a phase diagram for fluid invasion of porous media as a function of pressure P and the contact angle \ensuremath{\theta} of the invading fluid. Increasing P leads to perculation above a critical ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$, and depinning below ${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$. Depinning is characterized by a diverging coherence length and the power-law distribution of events typical of self-organized critical phenomena. At the transition from percolation to depinning another correlation length diverges and an order parameter, an effective macroscopic surface tension, becomes nonzero. The fluid interface changes from self-similar to self-affine. Results are compared to experiments.

Kinetic roughening of interfaces in driven systems.

We study the dynamics of an interface driven far from equilibrium in three dimensions. First we derive the Kardar-Parisi-Zhang equation from the Langevin equation for a system with a nonconserved scalar order parameter, for the cases where an external field is present, and where an asymmetric coupling to a conserved variable exists. The relationship of the phenomena to self-organized critical phenomena is discussed. Numerical results are then obtained for three models that simulate the growth of an interface: the Kardar-Parisi-Zhang equation, a discrete version of that model, and a solid-on-solid model with asymmetric rates of evaporation and condensation. We first make a study of crossover effects. In particular, we propose a crossover scaling ansatz and verify it numerically. We then estimate the dynamical scaling exponents. Within the precision of our study, the Kardar-Parisi-Zhang equation and the solid-on-solid model have the same asymptotic behavior, indicating that the models share a dynamical universality class. Furthermore, the discrete models exhibit a kinetic roughening transition. We study this by monitoring the surface step energy, which shows a dramatic jump at a finite temperature for a given driving force. At the same temperature, a finite-size-scaling analysis of the bond-energy fluctuation shows a diverging peak.

地震現象の新しい見方

The modern seismology took over the classical seismology in 1950s. The former is based on the analysis of individual earthquakes, while the latter took the statistical approach as a whole in understanding earthquakes. The earthquake phenomenon is a good example of fractal, in which the part is as complex as the whole. For understainding this type of complex phenomena, a new view is required other than the reductionist's (modern) view or the wholist's (classical) one. The whole is not a mere sum of the part, due to the nonlinear interaction among the parts. Recent advances in the nonlinear dynamical system theory are clarifying the various effects of nonlinear interactions. Earthquakes are re-examined on this line. In order to explain power-law relations known in seismology, earthquakes are viewed as a self-organized critical phenomenon (SOC). Earthquakes occur as an energy dissipation process in the earth's crust to which the energy is continuously input due to plate tectonics. The crust self-organizes into the critical state and the temporal and spatial fractal structure emerges naturally. Power-law relations are the expression of the critical state of the crust. The SOC model for earthquakes explains the Gutenberg-Richter relation, the Omori's formula of aftershocks and the fractal distribution of hypocenters.

Kinetic Roughening of Interfaces in Driven Systems

We study the dynamics of an interface driven far from equilibrium in three dimensions. The relationship of the phenomena to self-organized critical phenomena is discussed. Numerical results are obtained for three models which simulate the growth of an interface: the Kardar-Parisi-Zhang equation, a discrete version of that model, and a solid-on-solid model with asymmetric rates of evaporation and condensation. We show that the three models belong to the same dynamical universality class by estimating the dynamical scaling exponents and the scaling functions. We confirm the results by a careful study of the crossover effects. In particular, we propose a crossover scaling ansatz and verify it numerically. Furthermore, the discrete models exhibit a kinetic roughening transition. We study this phenomenon by monitoring the surface step energy which shows a drastic jump at a finite temperature for a given driving force. At the same temperature, a finite size scaling analysis on the bond energy fluctuation shows a diverging peak [1].

Exactly solved model of self-organized critical phenomena.

We define a variant of the model of Bak, Tang, and Wiesenfeld of self-organized critial behavior by introducing a preferred direction. We characterize the critical state and, by establishing equivalence to a voter model, determine the critical exponents exactly in arbitrary dimension d. The upper critical dimension for this model is three. In two dimensions the model is equivalent to an earlier solved special case of directed percolation.

Scaling theory of self-organized criticality.

We study the phenomena of self-organized criticality originally proposed by Bak, Tang, and Wiesenfeld. A continuous-energy model is introduced. Using numerical simulations, we find that energy is homogeneously and isotropically distributed in space, and that it is concentrated around discrete values. We propose a scaling theory to estimate the various exponents. The activation cluster size distribution is found to be D(s)\ensuremath{\sim}1/${s}^{\ensuremath{\tau}}$, \ensuremath{\tau}=2-2/d; and the dispersion relation t\ensuremath{\sim}${r}^{z}$, z=(d+2)/3.

Mean-field exponents for self-organized critical phenomena.

The mean-field critical exponents, \ensuremath{\tau}=(5/2, \ensuremath{\sigma}=1/2, \ensuremath{\gamma}=\ensuremath{\beta}=1, etc., for the self-organized critical state are derived viewing the self-organized critical state as a critical branching process.

Critical exponents and scaling relations for self-organized critical phenomena.

Critical indices beta, gamma delta, nv, etc. are defined and calculated for self-organized critical phenomena. Scaling relations are derived and checked numerically. The order-parameter exponent beta describes the spontaneous current and the relaxation to the criticl point. The power spectrum has 'l/f' behavior with the exponent phi = nv x z, where z is the dynamical critical exponent.

Mean field theory of self-organized critical phenomena

A mean field theory is presented for the recently discovered self-organized critical phenomena. The critical exponents are calculated and found to be the same as the mean field values for percolation. The power spectrum has “1/f” behavior with exponentg4=1.