We introduce and study a dynamic transport model exhibiting Self-Organized Criticality. The novel concepts of our model are the probabilistic propagation of activity and unbiased random repartition of energy among the active site and its nearest neighbors. For space dimensionality d _> 2 we argue that the model is related to (d + 1)-dimensional directed percolation, with time interpreted as the preferred direction.
Directed Percolation (DP) is one of the simplest and most recurrent models in statistical mechanics. Under very general guide-lines (locality, scalar variable, etc.) Grassberger [ 1] and Janssen have proposed that a wide range of models would fall into the DP universality class. This conjecture has stood the test of time. An impressive parade of models under various disguises turned out to belong to the same DP class (see for a review). However, they all share one common feature: the activation probability p has to be defined in advance, and when it is properly fine-tuned, a phase transition takes place. Our aim here is to design a dynamical model in which critical directed percolation would occur via self-organization.
Consider the following activation-transport problem. A conserved quantity called energy E can be stored at each site of a d-dimensional square lattice with open boundaries. An input energy 8E < 1 is added to the fixed input site in the center of the system and this site is declared active. An active site can propagate activity under the following rules: (i) Random repartition of energy among the active site and its 2d neighbors i.e. r--.,2d+ 1 ~"~2d+ 1
Ei ~ xi2_,j=l Ej (i = 1 ..... 2d+ 1), where xi = ri/z_,j=l rj, and ri are uncorrelated random numbers between 0 and 1. (ii) After repartition each of the above 2d + 1 sites becomes active with the probability given by its energy content. If Ei > 1, activation happens with certainty. There is no spontaneous activation of sites not connected to the current active site. The above process is repeated until no active sites are left in the