Formation of avalanches and critical exponents in an Abelian sandpile model

Signatures of self-organized dynamics in rapidly driven critical sandpiles

We study two prototypical models of self-organized criticality, namely sandpile automata with deterministic (Bak-Tang-Wiesenfeld) and probabilistic (Manna model) dynamical rules, focusing on the nature of stress fluctuations induced by driving-adding grains during avalanche propagation, and dissipation through avalanches that hit the system boundary. Our analysis of stress evolution time series reveals robust cyclical trends modulated by collective fluctuations with dissipative avalanches. These modulated cycles attain higher harmonics, characterized by multifractal measures within a broad range of timescales. The features of the associated singularity spectra capture the differences in the dynamic rules behind the self-organized critical states at adiabatic driving and their pertinent response to the increased driving rate, which alters the process of stochasticity and causes a loss of avalanche scaling. In sequences of outflow current carried by dissipative avalanches, the first return distributions follow the q-Gaussian law in the adiabatic limit. They appear to follow different laws at an intermediate scale with an increased driving rate, describing different pathways to the gradual loss of cooperative behavior in these two models. The robust appearance of cyclical trends and their multifractal modulation thus represents another remarkable feature of self-organized dynamics beyond the scaling of avalanches. It can also help identify the prominence of self-organizational phenomenology in an empirical time series when underlying interactions and driving modes remain hidden.

Finite-size scaling of critical avalanches

We examine probability distribution for avalanche sizes observed in self-organized critical systems. While a power-law distribution with a cutoff because of finite system size is typical behavior, a systematic investigation reveals that it may also decrease with increasing the system size at a fixed avalanche size. We implement the scaling method and identify scaling functions. The data collapse ensures a correct estimation of the critical exponents and distinguishes two exponents related to avalanche size and system size. Our simple analysis provides striking implications. While the exact value for avalanches size exponent remains elusive for the prototype sandpile on a square lattice, we suggest the exponent should be 1. The simulation results represent that the distribution shows a logarithmic system size dependence, consistent with the normalization condition. We also argue that for the train or Oslo sandpile model with bulk drive, the avalanche size exponent is slightly less than 1, which differs significantly from the previous estimate of 1.11.

1/x power-law in a close proximity of the Bak-Tang-Wiesenfeld sandpile

A cellular automaton constructed by Bak, Tang, and Wiesenfeld (BTW) in 1987 to explain the 1/f noise was recognized by the community for the theoretical foundations of self-organized criticality (SOC). Their conceptual work gave rise to various scientific areas in statistical physics, mathematics, and applied fields. The BTW core principles are based on steady slow loading and an instant huge stress-release. Advanced models, extensively developed far beyond the foundations for 34 years to successfully explain SOC in real-life processes, still failed to generate truncated 1/x probability distributions. This is done here through returning to the original BTW model and establishing its larger potential than the state-of-the-art expects. We establish that clustering of the events in space and time together with the core principles revealed by BTW lead to approximately 1/x power-law in the size-frequency distribution of model events.

Effects of inhibitory and excitatory neurons on the dynamics and control of avalanching neural networks

The statistical analysis of the collective neural activity known as avalanches provides insight into the proper behavior of brains across many species. We consider a neural network model based on the work of Lombardi, Herrmann, De Arcangelis et al. that captures the relevant dynamics of neural avalanches, and we show how tuning the fraction of inhibitory neurons in this model alters the connectivity of the network over time, removes exponential cut-offs present in the distributions of avalanche size and duration, and transitions the power spectral density of the network into an "epileptic" regime. We propose that the brain operates away from this power-law regime of low inhibitory fraction to protect itself from the dominating avalanches present in these extended distributions. We present control strategies that curtail these power-law distributions through either random or, more effectively, targeted disabling of excitatory neurons.

Hydrodynamics, density fluctuations, and universality in conserved stochastic sandpiles

We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density ρ. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation σ^{2}(ρ)=χ(ρ)/D(ρ), which connects bulk-diffusion coefficient D(ρ), conductivity χ(ρ), and mass fluctuation, or scaled variance of subsystem mass, σ^{2}(ρ). Consequently, density large-deviations are governed by an equilibrium-like chemical potential μ(ρ)∼lna(ρ), where a(ρ) is the activity in the system. By using the above hydrodynamics, we derive two scaling relations: As Δ=(ρ-ρ_{c})→0^{+}, ρ_{c} being the critical density, (i) the mass fluctuation σ^{2}(ρ)∼Δ^{1-δ} with δ=0 and (ii) the dynamical exponent z=2+(β-1)/ν_{⊥}, expressed in terms of two static exponents β and ν_{⊥} for activity a(ρ)∼Δ^{β} and correlation length ξ∼Δ^{-ν_{⊥}}, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality-not that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).

Critical properties of a dissipative sandpile model on small-world networks

A dissipative sandpile model is constructed and studied on small-world networks (SWNs). SWNs are generated by adding extra links between two arbitrary sites of a two-dimensional square lattice with different shortcut densities ϕ. Three regimes are identified: regular lattice (RL) for ϕ≲2(-12), SWN for 2(-12)<ϕ<0.1, and random network (RN) for ϕ≥0.1. In the RL regime, the sandpile dynamics is characterized by the usual Bak, Tang, and Weisenfeld (BTW)-type correlated scaling, whereas in the RN regime it is characterized by mean-field scaling. On SWNs, both scaling behaviors are found to coexist. Small compact avalanches below a certain characteristic size s(c) are found to belong to the BTW universality class, whereas large, sparse avalanches above s(c) are found to belong to the mean-field universality class. A scaling theory for the coexistence of two scaling forms on a SWN is developed and numerically verified. Though finite-size scaling is not valid for the dissipative sandpile model on RLs or on SWNs, it is found to be valid on RNs for the same model. Finite-size scaling on RNs appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves.

Statistical properties of directed avalanches

A two-dimensional directed stochastic sandpile model is studied both numerically and analytically. One of the known analytical approaches is extended by considering general stochastic toppling rules. The probability density distribution for the first-passage time of stochastic process described by a nonlinear Langevin equation with power-law dependence of the diffusion coefficient is obtained. Large-scale Monte Carlo simulations are performed with the aim to analyze statistical properties of the avalanches, such as the asymmetry between the initial and final stages, scaling of voids and the width of the thickest branch. Comparison with random walks description is drawn and different plausible scenarios for the avalanche evolution and the scaling exponents are suggested.

Patterned and disordered continuous Abelian sandpile model

We study critical properties of the continuous Abelian sandpile model with anisotropies in toppling rules that produce ordered patterns on it. Also, we consider the continuous directed sandpile model perturbed by a weak quenched randomness, study critical behavior of the model using perturbative conformal field theory, and show that the model has a random fixed point.

Numerical renormalization-group approach to a sandpile

Multiple topplings and grain redistribution are two essential features in sandpile dynamics. A renormalization group (RG) approach incorporating these features is investigated. The full enumeration of all relaxations involving such an approach is difficult. Instead, we developed an efficient procedure to sample the relaxations. We applied this RG scheme to a square lattice and a triangular lattice. As shown by the fixed point analysis on a square lattice, the effect of multiple topplings leads the resultant height probabilities towards the exact solution while the effect of grain redistribution does not.

Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model

Rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in an out-of-equilibrium situation like self-organized criticality, a two state "quasideterministic" rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.

Effects of bulk dissipation on the critical exponents of a sandpile

Bulk dissipation of a sandpile on a square lattice with the periodic boundary condition is investigated through a dissipating probability f during each toppling process. We find that the power-law behavior is broken for f>10(-1) and not evident for 10(-1)}>f>10(-2). In the range 10(-2)>or=f>or=10(-5), numerical simulations for the toppling size exponents of all, dissipative, and last waves have been studied. Two kinds of definitions for exponents are considered: the exponents obtained from the direct fitting of data and the exponents defined by the simple scaling. Our result shows that the exponents from these two definitions may be different. Furthermore, we propose analytic expressions of the exponents for the direct fitting, and it is consistent with the numerical result. Finally, we point out that small dissipation drives the behavior of this model toward the simple scaling.

Exact multileg correlation functions for the dense phase of branching polymers in two dimensions

We consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k. We find that besides the standard power-law factor the leading term also has a logarithmic multiplier.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Evidence for universality within the classes of deterministic and stochastic sandpile models

Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

We show that in the loop-erased random-walk problem, the exponent characterizing the probability distribution of areas of erased loops is superuniversal. In d dimensions, the probability that the erased loop has an area A varies as A(-2) for large A, independent of d, for 2< or =d< or =4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d=2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5/4. In three dimensions, we get z=1.6183+/-0.0004. The exponent for the distribution of the durations of avalanches in the three-dimensional Abelian sandpile model is determined from this by using scaling relations.

From waves to avalanches: two different mechanisms of sandpile dynamics

Time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics. For the d=2 Bak-Tang-Wisenfeld model, long range correlations determine a modification of the wave size distribution under coarse graining in time, and multifractal scaling for avalanches. In the Manna model, the distribution of avalanche coincides with that of waves, which are uncorrelated and obey finite size scaling, a result expected also for the d=3 Bak-Tang-Wiesenfeld sandpile.

Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche <T> exactly, it is shown that for any alpha>1, <T><infinity even in the infinite-volume limit. The power-law divergence of <T> with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.

Scaling of waves in the bak-tang-wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D</=3. For D=2 they seem not easy to interpret.

Dynamical real space renormalization group applied to sandpile models

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.