Anomalous scaling in the Zhang model

We apply the moment analysis technique to analyze large scale simulations of the Zhang sandpile model. We find that this model shows different scaling behavior depending on the update mechanism used. With the standard parallel updating, the Zhang model violates the finite-size scaling hypothesis, and it also appears to be incompatible with the more general multifractal scaling form. This makes impossible its affiliation to any one of the known universality classes of sandpile models. With sequential updating, it shows scaling for the size and area distribution. The introduction of stochasticity into the toppling rules of the parallel Zhang model leads to a scaling behavior compatible with the Manna universality class. PACS. 05.70.Ln Nonequilibrium and irreversible thermodynamics – 05.65.+b Self-organized systems The identification of universality classes is one of the most important and still open problems in the field of self-organized criticality (SOC) [1]. In spite of the great relevance of this issue, however, it has not been possible until very recently to clearly discern differences in the critical behavior of the various SOC models proposed so far. The situation seems now to have been settled in the case of the Bak-Tang-Wiesenfeld (BTW) [2] and the Abelian Manna [3,4] models that represent, respectively, the prototypical examples of deterministic and stochastic sandpile automata. In this particular case, recent large scale simulations [5,6,7,8] clearly indicate that Manna and BTW models belong to different universality classes. The universality class asset remains still uncertain for the Zhang model [9], which is the archetype of all sandpile automata with continuous variables. This model has deterministic dynamics like the BTW model, and in contrast with most other cases, it is non-Abelian. This means that the stable configurations obtained at the end of an avalanche depend on the order in which the active sites are updated. In spite of this essential peculiarity, earlier simulations of the model [9] placed it in the same universality class as the BTW model, of which it was supposed to be the continuous counterpart. This conclusion was confirmed by the large scale simulations performed by Lübeck in d = 2 [10]. On the other hand, Milshtein et al. [5], analyzing different magnitudes than Lübeck, arrived at the opposite result in d = 2, namely, they observed noticeable differences in the critical exponents. Finally, the simulations of Giacometti and Dı́az-Guilera [11] provided evidence that, even though the exponents of both models are similar in d = 2, they do not coincide in d = 3. Recently, it has been shown [12,13] that the deterministic nature of the BTW model renders its scaling incompatible with the standard finite-size scaling (FSS) hypothesis, and induces moreover peculiar non-ergodic effects [8]. Thus, it would not be surprising to find similar anomalies in the Zhang model because of its deterministic nature. In this paper we characterize the Zhang universality class by applying the recently proposed moment analysis technique [12,13]. In the following we will show that the scaling of the Zhang model depends very strongly on the updating mechanism implemented in the simulations, either parallel or sequential. The Zhang model with parallel updating, as it has been customarily defined in the literature, displays a complex scaling behavior that is not compatible neither with the standard finite-size scaling (FSS) hypothesis nor with the multifractal picture [14] proposed for the avalanche distribution of the BTW model [13]. On the other hand, the Zhang model with sequential updating shows well defined size and area exponents. The origin of the complex behavior of the parallel Zhang model can be ascribed to the deterministic microscopic dynamical rules of the model. In order to prove this conjecture, we study a variation of the Zhang model, the stochastic parallel Zhang model, which exhibits a standard FSS behavior, compatible with the universality class of the Manna model. We consider the definition of the original Zhang model [9]. On each site of a d dimensional hypercubic lattice of size L we assign a continuous variable Ei, called “energy”. At each time step, an amount of energy δ is added to a randomly selected site j (Ej → Ej+δ). The quantity δ is a random variable uniformly distributed in [0, δmax] [15]. In our simulations we consider the fixed value δmax = 0.25. When a site acquires an energy larger than or equal to 1, (Ek ≥ 1), it becomes active and topples. An active site k relaxes losing all its energy, which is equally dis2 Romualdo Pastor-Satorras, Alessandro Vespignani: Anomalous scaling in the Zhang model tributed among its nearest neighbors: Ek → 0, and Ek′ → Ek′ + Ek/2d. Here the index k ′ runs over the set of all nearest neighbors of the site k. The transported energy can activate the nearest neighbor sites and thus create an avalanche. Energy can be lost only at the boundary of the system (open boundary conditions). The avalanche stops when all sites in the lattice are subcritical (Ei < 1). Given these dynamical rules, it is easy to see that the Abelian nature of the model depends on the type of updating implemented. With parallel updating—parallel Zhang (P-Z) model— at each time step t in the evolution, all active sites are toppled simultaneously, and time is incremented t → t + 1. Since all the energy of an active site is transferred to its nearest neighbors, we notice that in a bipartite lattice (such as the hypercubic lattice here considered) all active sites at a give time step t reside onto the same sublattice, and that activity alternates between the dual sublattices in consecutive time steps. Again, since all the energy is transferred in a toppling, the state of the active sites in a sublattice at time t is independent of the order in which the active sites in the dual sublattice were updated at time t − 1. On the other hand, with sequential updating—sequential Zhang (S-Z) model— at each time step t an active site is randomly chosen among all the Na(t) active sites present at that time. The chosen site is toppled and time is incremented t → t + 1/Na(t). In this case, activity is not restricted to alternate sublattices, but spreads all over the system. Depending on the order in which the intermediate list of over-critical sites is updated, any active site with at least one active nearest neighbor can end up with an energy Ei = 0 (if its nearest neighbors are updated before it) or with energy Ei > 0 (if it is updated before its nearest neighbors). In this case the model fully exploits its non-abelian characters. We have analyzed both parallel and sequential Zhang models by determining their critical exponents [1]. In the limit of an infinite slow driving, i.e. the energy addition is interrupted during the avalanche evolution, defining a complete time scale separation [16], the system reaches a critical stationary state with avalanches of activity distributed according to power laws. If we define the probability distributions P (x) of occurrence of an avalanche of a given size s, time t, and area a, the FSS hypothesis [17], usually assumed in SOC systems, states that