Self-Organized Criticality in Nonconserved Systems.

The origin of self-organized criticality in a model without conservation law (Olami, Feder, and Christensen, Phys. Rev. Lett. 68, 1244 (1992)) is studied. The homogeneous system with periodic boundary condition is found to be periodic and neutrally stable. A change to open boundaries results in the invasion of the interior by a “self-organized” region. The mechanism for the self-organization is closely related to the synchronization or phase-locking of the individual elements with each other. A simplified model of marginal oscillator locking on a directed lattice is used to explain many of the features in the non-conserved model: in particular, the dependence of the avalanchedistribution exponent on the conservation parameter α is examined. 05.70.Jk,05.45.+b,91.30.-f Typeset using REVTEX 1 The phenomenon of the self-organized criticality (SOC) [1] is characterized by spontaneous and dynamical generation of scale-invariance in an extended non-equilibrium system. One of the key issues in this field is to identify the mechanism(s) of SOC. It has been conjectured that conservation laws or special symmetries are necessary [2]. Conservation laws certainly are of great importance in “sandpile” models [1,3,4], where the scale invariance can be shown to follow from a local conservation law (sand grains are conserved except at the boundaries of the pile) [5]. In this sense, the origins of long range correlations in SOC systems with conservation are well understood, though not all exponents have been calculated analytically. However, models have been constructed [6–9] that have no apparent conservation law, and yet display a power-law distribution of avalanche sizes. Of particular interest is the model proposed by Olami, Feder, and Christensen (OFC) [9–11] which, they argue, models earthquake dynamics. The OFC model is very similar to the conserved sandpile model [1], but it has a parameter which defines the degree of conservation. In this paper, we study in detail the OFC model and we find that the self-organization is due to synchronization or “phase-locking” – a mechanism very different from that in the conserved models. In the OFC model, dynamical “height” variables hi are defined on sites i of a square lattice. The hi increase at unit rate until h = 1 at some location. The site j where hj = 1 is considered to be unstable and will “topple”. The rule of toppling is that when hj ≥ 1, then hj → 0 and hk → hk + αhj, for all k neighboring j. The toppling on site j may cause its neighbors to become unstable (hk ≥ 1) and to topple. This procedure is repeated until all sites are stable (hi < 1 everywhere). The magnitude of the avalanche is given by the total “energy” dissipated in the process, i.e., the total change in ∑ i hi. The avalanche, which happens instantly on the time scale of driving [13], is then followed by growth. The parameter α is the measure of conservation (of h’s). When α = 1/4, the model is conserved and it is in the same universality class as for the BTW model [1,9,12]. We study here the non-conservative case 0 < α < 1/4. We note that there is an ambiguity in this model. After some number of avalanches, it 2 will occur that more than one site will have exactly the same height, as more than one site may topple to a height of exactly zero in a single avalanche. This can lead to two neighboring sites toppling simultaneously; the above procedure does not define the result of such events. We have modified the rules in several ways, e.g. by adding a very small amount of random noise to ensure that no two sites have the same height, and they all give the similar results. As most avalanches are small, and do not change the hi at many sites, we do not need to scan the whole lattice after each avalanche in order to determine the most unstable site. We use a tree structure to keep track of the highest values of hi in order to determine avalanche trigger sites. Using this technique, we have simulated up to more than 10 avalanches for single systems. We find that the system with periodic boundary conditions quickly reaches an exactly periodic state [10], with a unique period in the slow time. It has been noted [9,11] that in the case of periodic boundary conditions, the avalanche size distribution function drops very quickly with size. In the case of our modified model, which prevents two sites from having the same value of h, all avalanches consist of the toppling of exactly one lattice site, after a brief transient time. In a periodic state, the hi’s take turns to topple, one by one. The height h decreases by one each time a site topples and increases by 4α due to the toppling of the four neighbors when they reach h = 1. Thus the period of all these periodic states is 1 − 4α in the slow time variable, so that the slow “growth” is balanced exactly by the dissipation due to toppling. These periodic states are highly degenerate and neutrally stable in the sense that a typical small perturbation of the height at a single site in a periodic state is still a periodic state. They are similar to the neutrally stable periodic states in coupled oscillators [14]. There is a continuous set of periodic states in the attractor, with measure (1− 4α) in the initial phase space, where V is the system volume. Any inhomogeneity, such as a change in boundary conditions, destroys such simple periodic states. When the boundary conditions are open, the system can no longer have period 1− 4α, as the boundary sites have 3 neighbors (we study a system that is open on one axis, with the other directions periodic). Initially, the interior sites quickly converge to a nearly 3 periodic state and topple with period 1 − 4α, but the boundaries are aperiodic. At longer times, the aperiodic region invades the periodic interior, as shown in Fig. 1. This invasion, which destroys the periodicity in the interior and builds up long-range correlations, occurs by a mechanism similar to oscillator locking, as we describe below. The interface between the two regions is well defined on scales larger than one or two lattice constants. The invasion distance appears to have a power law dependence on time, y(t) ∼ t as shown in Fig. 2, with β = 0.23 ± 0.08, 0.63 ± 0.08 for α = 0.07, 0.15, respectively. We see such invasion occurring even for values of α < 0.05, though β appears to be quite small, so that the invasion is extremely slow. This suggests that the transition to non-SOC behavior claimed by Olami, Feder, and Christensen [9] may only be apparent, due to the finite time of the simulations; we note that the time for complete invasion of a 128 system with α = 0.07 is greater than 10 avalanches. In the limit of long times, when the invasion crosses the whole sample, the distribution P (s;α) of avalanches of size s is a power-law, with