Precursors of catastrophe in the Bak-Tang-Wiesenfeld, Manna, and random-fiber-bundle models of failure

We have studied precursors of the global failure in some self-organized critical models of sandpile [in Bak-Tang-Wiesenfeld (BTW) and Manna models] and in the random-fiber-bundle model (RFB), In both BTW and Manna model, as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time tau and an average number of topplings Delta spread over a radial distance xi. We find that these quantities all depend on the average height h(av) of the pile and they all diverge as h(av) approaches the critical height h(c) from below: Delta similar to (h(c) - h(av))(-delta), tau similar to (h(c) - h(av))(-gamma), and xi similar to (h(c) - h(av))(-v). Numerically, we find delta similar or equal to 2.0, gamma similar or equal to 1.2, and v similar or equal to 1.0 for both BTW and Manna model in two dimensions. In the strained RFB model, we find that the breakdown susceptibility chi (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time tau, both diverge as the applied load or stress sigma approaches the network failure threshold sigma(c) from below: and chi similar to (sigma(c)-sigma)(-1/2) and tau similar to (sigma(c)-sigma)(-1/2). These self-organized dynamical models of failure, therefore, show some definite precursors with robust power laws long before the failure point, Such well-characterized precursors should help predicting the global failure point of the systems in advance.