Dynamics of Sand
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BULLETIN/JUNE1991
of ail sizes (limited only by the size of the system) following a power-law distribution. The criticality can also be thought of as a critical chain reaction: the instability caused by a single falli n g g r a i n of s a n d p r o p a g a t e s by a branching process where the branching probability is precisely balanced by the probability that the activity dies. S t a r t i n g from an a r b i t r a r y configuration of sand, the chain reaction is generally supercritical or suberitical, but eventually the process of adding sand modifies the médium to the point where it is critical.
Figure 1. Evolution to the critical attractor in the space of metastable states.
Figure 1 schematically illustrâtes the configuration space of the sand pile and other self-organized critical Systems. Some states (the dots) represent stable configurations of not-too-steep piles. The states outside the surface represent unstable states of steep piles. Starting anywhere, the dynamics will eventually carry the System to the attractor inside the stable volume. The stable states outside the attractor can be reached only by carefully placing the individual grains (like cards in a card house), but will catastrophically
collapse urider small p e r t u r b a t i o n s . The attractor is high dimensional in the sensé that the motion is described by m a n y v a r i a b l e s . T h e n u m b e r of states belonging to the attractor grows exponentially with the size of the System. The complex dynamics cannot be thought of as low-dimensional chaotic, or low dimensional anything. The most important property of the self-organized critical state is its resiliency with respect to modifications of the system. Suppose that at some point, one starts to use wet sand instead of dry. For a t r a n s i e n t p e r i o d the avalanches will be smaller, but eventually the pile organizes itself into a steeper state, where again there will be collapses of ail sizes. Also, if one builds snow screens locally to prevent slidi n g , the pile will a g a i n r e s p o n d by b u i l d i n g u p s t e e p e r s t a t e s , a n d the large avalanches will résume. The sand pile picture provides an intuitive picture of the consistency of a stable dynamical attractor, and the existence of large fluctuations. Without this resiliency one would not expect the concept to apply to real Systems in nature. Simulations The convergence to the self-organized critical state can be demonstrated by computer simulations on toy sand models. In the simplest model 1 one defines a " h e i g h t " variable Z(i,j) on a t w o - d i m e n s i o n a l lattice. The pile is grown by adding sand, Z —» Z + 1 at random position, one grain at a time. When the height Z at some point exceeds a critical value Z cr , the pile relaxes by sending one grain of sand to e a c h of t h e four n e i g h b o r s of t h a t point, i.e., Z —-* Z - 4, Z n n —* Zn„ + 1. (Hère, Z n n dénotes the nearest neighbors of Z.) At the edges or corners, only three or two neighbors, respectively, are affected, an
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