Mean-field theory of avalanches in self-organized critical states

A new mean-field approximation is proposed for the sandpile model of Bak, Tang and Wiesenfeld and the distribution of heights in the self-organized critical state is calculated. We treat several series of successive topples to approximate various avalanches. In contrast to the previous mean-field theory given by Tang and Bak, which assumes the stationary condition among topples in an avalanche, our theory takes into account long-term behavior of the system. The results are in good agreement with the values estimated by computer simulations even in low dimensions d = 2 and 3. Higher approximations obtained by including local correlations or by imposing a reducibility condition, which is used in the cluster variation method, are also shown.