Analytic results for scaling function and moments for a different type of avalanche in the Bak-Sneppen evolution model

Abstract

Starting from the master equation for the hierarchical structure of avalanches of a different kind within the frame of the Bak-Sneppen evolution model, we derive the exact formula of the scaling function describing the probability distribution of avalanches. The scaling function displays features required by the scaling ansatz and verified by simulations. Using the scaling function we investigate the avalanche moment, denoted by $〈S^k{〉}_{\Delta \overline{f}}.$ It is found that for any non-negative integer k, $〈S^k{〉}_{\Delta \overline{f}}$ diverges as $\Delta f^{-k},$ which gives an infinite group of exact critical exponents. Simulation outcomes of avalanche moments with $k=1,2,3,$ are found to be consistent with the corresponding analytical results.