Critical Behavior of Sandpile Models with Sticky Grains

We revisit the question whether the critical behavior of sandpile models with sticky grains is in the directed percolation universality class. Our earlier theoretical arguments in favor, supported by evidence from numerical simulations [ Phys. Rev. Lett., 89 (2002) 104303], have been disputed by Bonachela et al. [Phys. Rev. E 74 (2004) 050102] for sandpiles with no preferred direction. We discuss possible reasons for the discrepancy. Our new results of longer simulations of the one-dimensional undirected model fully support our earlier conclusions. After the pioneering work of Bak, Tang and Wiesenfeld[1] in 1987, sandpile models have been studied extensively in the past two decades, both as paradigms of self-organized critical systems in general[2], and also as models of real granular matter [3]. Many different types of sandpile models with different toppling rules have been studied [4] : deterministic and stochastic, with or without preferred direction, different instability criteria [5], or particle distribution rules [6], with fixed energy [7] etc.. Most of these models could only be studied numerically, and for a while it seemed that each new variation studied belonged to a new universality class of critical behavior. Though not complete, a broad picture of the different universality classes of self-organized critical behavior has emerged in recent years [8,9]. In an earlier paper [10], we have argued that the generic behavior of sandpile models is in the universality class of directed percolation (DP), and models with deterministic toppling rules like the original BTW model, and models with stochastic toppling rules like the Manna models, are unstable to a Preprint submitted to Elsevier 1 February 2008 perturbation of introduction of “stickiness” in the toppling rules, and under renormalization, the flows are directed towards the DP-fixed point. These arguments are reasonable, but not rigorous, and we presented a detailed study of a specific model, where some of the steps in the arguments could be shown to be valid, and we used detailed Monte Carlo simulations to check our conclusions. Some of the conclusions of this paper have recently been disputed by Bonachela et al. [11]. These authors contend that while the directed sandpiles with sticky grains show DP behavior in the SOC limit, our arguments do not apply to undirected sandpiles, where, even with stickiness, the critical behavior continues be the same as that of the Manna model, i. e., in the universality class of the directed percolation with a conservation law (hereafter referred to as the Manna/C-DP universality class). In this paper, we will try discussing these conflicting claims, and also present some data from more recent extensive simulations, which supports our original conclusions. We start by defining the model precisely, and then summarize the arguments of [10]. We then discuss the simulations of [11], and finally present the results of our new more extensive simulations. First the precise definition of the model. We consider the directed model on a (1 + 1)-dimensional square lattice, for definiteness. Generalizations to higher dimensions are straight forward. The sites on an L×M torus are labelled by euclidean coordinates (i, j) with (i + j) even and j increasing downward. At each site (i, j), there is a non-negative integer hi,j to be called the height of the pile at that site. Initially all hi,j are zero. The system is driven by choosing a site at random and increasing the height at that site by one. The ‘stickiness’ of the grains is characterized by a parameter p, and its role in the dynamics of sandpiles is defined as follows: A site is said to become unstable at time t, if at least one particle is added to it at time t, and its height becomes greater than 1. A site (i, j) made unstable at time t relaxes at the time (t+ 1) stochastically: With probability (1− p), it becomes stable without losing any grains, and the added particle(s) sticks to the existing grains. Otherwise (with probability p), the site topples, and the height at the site decreases by two, and the site becomes stable. We introduce bulk dissipation: at each toppling, with probability δ both grains from the toppling are lost, other wise (with probability 1− δ), the two grains are transferred to the two downward neighbors (i± 1, j + 1). We relax all the unstable sites by parallel dynamics. An unstable site is relaxed in one step, independent of whether it received one or more grains at the previous time step. Once a site has relaxed, it remains stable until perturbed again by new grains coming to the site. This relaxation process is repeated