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 τ and an average number of topplings Δ spread over a radial distance ξ. We find that these quantities all depend on the average height hₐᵥ of the pile and they all diverge as hₐᵥ approaches the critical height hc from below: ${Δ}{~}{(h}c{-}{h}ₐᵥ{)}^{{-}{δ}},{τ}{~}{(h}c{-}{h}ₐᵥ{)}^{{-}{γ}},$ and ${ξ}{~}{(h}c{-}{h}ₐᵥ{)}^{{-}{ν}}.$ Numerically, we find ${δ}{}2.0,{γ}{}1.2,$ and ${ν}{}1.0$ for both BTW and Manna model in two dimensions. In the strained RFB model, we find that the breakdown susceptibility ${χ}$ (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time ${τ},$ both diverge as the applied load or stress ${σ}$ approaches the network failure threshold ${{σ}}c$ from below: ${χ}{~}({{σ}}c{-}{σ}{)}^{{-}1/2}$ and ${τ}{~}({{σ}}c{-}{σ}{)}^{{-}1/2}.$ 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.