BTW Model Research Articles

Avalanche dynamics in nonconservative water droplet

The self-organized criticality (SOC) exhibits excellent performance in describing the dynamic features of multi-body systems in nature. It is usually considered that conservative law is a necessary condition for the existence of SOC. However, mass or energy dissipation typically exists in the process from self-organization to a critical state in natural systems such as earthquakes and air pollution, namely, non-conservation. At present, few physical experiments have systematically explored the SOC behavior in a non-conservative system, compared to numerical simulations. This study is the first to design a non-conservative sandpile experiment using water droplets. Based on this experiment, the dynamic evolution process of avalanches in a non-conservative system has been analyzed. The results show that the avalanche size of water droplets follows a power-law distribution, as the water mass decays with time. This phenomenon obeys the rules of scale-free distribution, which is consistent with SOC behavior. Moreover, the waiting time of water droplets avalanche follows a stretched exponential distribution. These results suggest that SOC behavior still exists in non-conservative multi-body systems, which is affected by the decay coefficient. We then explore the mechanism by which the decay coefficient affects the transition of the system to a critical state in the non-conservative sandpile model (NBTW). We find that only when the decay coefficient is small do the inequality measures, Gini coefficient g and Kolkata index k, exhibit nearly universal values, similar to the mechanism of the system transitioning to a critical state in the traditional BTW model. When the decay coefficient is large, the NBTW model shows scaling-limited effects and the scaling exponent of avalanche size increases with the decay coefficient. Furthermore, we use the NBTW model to study how the decay coefficient influences the scaling exponent of avalanche size. We find that the NBTW model can effectively explain the variability of scaling exponents observed in different SOC systems and establish a quantitative relationship between the mass decay coefficient and the avalanche size scaling exponent. This study provides new insight into SOC behavior in non-conservative systems in nature.

Eigen microstates in self-organized criticality.

We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents, i.e., sqrt[σ_{0}w_{1}]/v[over ̃]_{1}∝L^{D} and v[over ̃]_{1}∝L^{D(1-τ_{s})/2}. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

1/x power-law in a close proximity of the Bak\u2013Tang\u2013Wiesenfeld 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.

Controlling cost in sandpile models through local adjustment of drive

In this paper we consider sandpile models and modify the drive mechanisms to control the size of avalanches. The modification to the drive mechanism is local. We have studied the scaling behavior of the BTW and Manna models. We have found that the BTW model is more sensitive to the modification than the Manna model. Furthermore we have assigned a cost function to each avalanche and have found an optimum value for the modification to arrive at the lowest cost

Power spectrum of rare events in a two-dimensional BTW model

One of the primary aims of the two-dimensional BTW model had been to explain the 1/fα noise which is widely seen in the natural systems. In this paper we study some time signals, namely the activity inside an avalanche (x(t)), the avalanches sizes (s(T)) and the rare events waiting time (τ(n) as a new type of noise). The latter is expected to be important in predicting the period and also the vastness of the upcoming large scale events in a sequence of self-organized natural events. Especially we report some exponential anti-correlation behaviors for s(T) and τ(n) which are finite size effects. Two characteristic time scales δTs and δTτ emerge in our analysis, and the power spectrum of s(T) and τ(n) behave like , in which bs and bτ are some L-dependent parameters and ω is the angular frequency. The 1/f2 noise is therefore obtained in the limit . bs and bτ decrease also in a power-law fashion with the system size L, which signals the fact that in the thermodynamic limit the power spectrum tends to the Dirac delta function.

Statistical investigation of avalanches of three-dimensional small-world networks and their boundary and bulk cross-sections.

In many situations we are interested in the propagation of energy in some portions of a three-dimensional system with dilute long-range links. In this paper, a sandpile model is defined on the three-dimensional small-world network with real dissipative boundaries and the energy propagation is studied in three dimensions as well as the two-dimensional cross-sections. Two types of cross-sections are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross-section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long-range links (α) increases. This trend is numerically shown to be a power law in a manner described in the paper. Two regimes of α are considered in this work. For sufficiently small αs the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is nontrivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with α just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) m^{2}∼α for the three-dimensional system as well as its two-dimensional cross-sections.

Sandpile on uncorrelated site-diluted percolation lattice; from three to two dimensions

The BTW sandpile model is considered on the three dimensional percolation lattice which is tuned by the occupation parameter p. Along with the three-dimensional avalanches, we study the avalanches in two-dimensional cross-sections. We use the moment analysis (along with some other methods) to extract the exponents for two separate cases: the lattice at critical percolation () and the supercritical one (). Our numerical data is consistent with the conjecture that the three-dimensional avalanches at p = pc have nearly the same exponents as the regular 2D BTW model. The moment analysis shows that finite size scaling theory is fulfilled, and some hyper-scaling relations hold. The main finding of the paper is the logarithmic dependence of the exponents on p − pc, for which the cut-off exponents ν change discontinuously from p = pc to the values for the supercritical case. Moreover we show that there is a singular point ( and being three- and two-dimensional percolation thresholds) for 2D cross-sections, which separate the behaviors to two distinct intervals: which, due to the lack of 2D percolation cluster, has no thermodynamic limit, and which involves the percolated clusters.

Generalisation of BTW Model with Next Nearest Neighbour

Generalisation of BTW Model with Next Nearest Neighbour

Mapping of the Bak, Tang, and Wiesenfeld sandpile model on a two-dimensional Ising-correlated percolation lattice to the two-dimensional self-avoiding random walk.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

BTW Model with Probabilistically Nonuniform Distribution of Particles Coming from the Unstable Sites

BTW Model with Probabilistically Nonuniform Distribution of Particles Coming from the Unstable Sites

Annealed and quenched disorder in sand-pile models with local violation of conservation

In this paper we consider the Bak, Tang, and Wiesenfeld (BTW) sand-pile model with local violation of conservation through annealed and quenched disorder. We have observed that the probability distribution functions of avalanches have two distinct exponents, one of which is associated with the usual BTW model and another one which we propose to belong to a new fixed point; that is, a crossover from the original BTW fixed point to a new fixed point is observed. Through field theoretic calculations, we show that such a perturbation is relevant and takes the system to a new fixed point.

Continuous transforming the BTW to the Manna model

Abstract In this paper we define some stochastic perturbations of the BTW model which make it into Manna model. These models have a continuous parameter p , where p = 0 and 1 correspond to the BTW and Manna models respectively. We have investigated the properties of the statistical observables of the waves of avalanches for various values of p . Our data supports the expectation of a crossover in such systems; at large scales Manna model is dominant. Therefore we find strong evidence in favor of the universality classes being distinct. Also it is observed that the BTW fixed point is unstable when Manna-type perturbation is added to the model.

Bak–Tang–Wiesenfeld model in the finite range random link lattice

We consider the BTW model in random link lattices with finite range interaction (RLFRI). The degree distribution for nodes is considered to be uniform in the interval $(0,n_0)$. We numerically calculate the exponents of the distribution functions in terms of $(n_0,R)$ in which $R$ is the range of interactions. Dijkstra radius is utilized to calculate the fractal dimension of the avalanches. Our analysis shows that there is, at least one length scale ($r_0(n_0,R)$) in which the fractal dimension is changed. We find that for the scales smaller than $r_0(n_0,R)$, which is typically one decade, the fractal dimension is nearly independent of $n_0$ and $R$ and is equal to $1.4$, i.e. close to that of the BTW in the regular lattice ($1.25$). Using this fact and other analysis, we conclude that the BTW-type behaviors are dominant for small values of $n_0$ and $R$, whereas for large values of these parameters a new regime is seen in which the exponent of distribution function of avalanche masses is nearly $1.4$. We also numerically calculate the explicit form of the \textit{number of unstable nodes} (NUN) as a time dependent process and show that for regular lattice it is (up to a normalization) proportional to a one dimensional Weiner process and for RLFRI it acquires a drift term. Using this dynamical variable it is numerically shown that we can not continuously approach the regular lattice limit by decreasing $R$.

Self Organised Criticality in Dilute Lattice

The variation of z in BTW model in presence of holes (dissipative sites) has been studied. The value of z decreases as the fraction of number of holes increases. Interstingly, it is observed that the variation of the rate of change of z with the fraction of number of holes is different for the two different types of distribution of holes over the lattice. When the holes are randomly distributed over the lattice then the dissipation is more compared to that of the case when the holes are present in the form of a single compact cluster with same fraction. The value of z is less in the first case than that observed in the second case.

Critical behavior of a small-world sandpile model

We have considered the Abelian sandpile model on a small-world network. To obtain a small-world network, we have added a few long-ranged bonds to an ordinary square lattice. It is observed that the probability distribution of the avalanches has two distinct power-law regimes: one corresponding to the ordinary BTW model and another that emerges because of the small-world structure of the network.

A dissipative deterministic BTW model with an activation scenario of strong events

An increase in seismic activity is usually a precursor of a strong earthquake. Grid models of earthquake formation related to the BTW sandpile model [Bak et al., 1987] reflect regularities close to those observed in reality; however, they demonstrate “incorrect” prognostic properties. Strong model earthquakes are preceded by a certain quiescence [Pepke and Carlson, 1994]. Weak dissipation of the BTW model is evidently responsible for its inversion scenario. We will construct a generalization of the BTW model with nontrivial dissipation. In the new model, a strong earthquake is preceded by an outburst of activity, which agrees with the dynamics of real seismicity. Strong events can be predicted with definite efficiency. The prediction efficiency is nonuniform in time.

Finite-size effects on the statistics of extreme events in the BTW model

The BTW Abelian sandpile model is a prominent example of systems showing self-organised criticality (SOC) in the infinite size limit. We study finite-size effects with special focus on the statistics of extreme events, i.e., of particularly large avalanches. Not only the avalanche size probability distribution, but also the mutual independence of large avalanches in the critical state is affected by finite-size effects. Instead of a Poissonian recurrencetime distribution, in the finite system we find a repulsion of extreme events that depends on the avalanche size and not on the respective probability. The dependence of these effects on the system size is investigated and some data collapse is found. Our results imply that SOC is an unsuitable mechanism for the explanation of extreme events which occur in clusters.

Avalanches in networks of weakly coupled phase-shifting rotators

The networks of phase-shifting rotators interacting through exchange of weak ±-kicks are considered. Such a rotator consists of a particle rotating on a circle which at some discrete mo- ments receives some ±-kicks. We assume that the kicks are not of mechanical character: they change a particle's position but not the rate. A comparison of the rotator networks with the BTW model of self-organized criticality, Burridge-Knopoff's model in seismicity, Herz-Hopfield neural networks, and the Turing-Smale system in biological cells is presented. This work studies the avalanches in rotator networks — the events when a number of rotators almost simultaneously hit some threshold levels. The asymptotic relations linking distribution of avalanches with the architecture of a network are proved. The equivalence of two well-known power-law conjectures, in lattice models of statistical physics and in interacting threshold systems, is established.

UNIVERSALITY ASPECTS OF SELF-ORGANIZATION MODELS

A sandpile model on the square lattice with periodic boundary conditions is studied as a problem of variable density of grains per site. We define an order parameter and we show the existence of a critical density. This model is the closed version of the deterministic BTW model and belongs to the "Fixed Energy Sandpiles" class. An algorithm identifying the periodic states of the system permits us to have a good estimation of the order parameter, which is free from transient effects. The universality of the model with respect to two different forms of oriented flows is studied and compared with that of the isotropic flow. We find two universality classes characterized by different critical exponents. The Manhattan flow appears to belong to the same universality class as the isotropic flow. However, the directed flow gives a completely different critical behavior. The complexity of the model for densities larger than the critical is also studied. It is observed that, both the Manhattan model and the isotropic model have an interesting structure characterized by several plateaus. The right edges of these plateaus are characterized by a sudden increase of the order parameter, giving rise to new critical points. The directed flow model has a simple structure without plateaus.

Sensitivity to Initial Conditions in Self-Organized Critical Systems

We discuss sensitivity to initial conditions in a model for avalanches in granular media displaying self-organized criticality. We show that damage, due to a small perturbation in initial conditions, does not spread. The damage persists in a statistically time-invariant and scale-free form. We argue that the origin of this behavior is the Abelian nature of the model, which generalizes our results to all Abelian models, including the BTW model and the Manna model. An ensemble average of the damage leads to seemingly time dependent damage spreading. Scaling arguments show that this numerical result is due to the time lag before avalanches reach the initial perturbation.