  We have studied precursors of the global failure in some self-organised
critical models of sand-pile (in 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) (\sim (h_{c}-h_{av}))(^{-\delta}), (\tau \sim
(h_{c}-h_{av})^{-\gamma}) and (\xi) (\sim) ((h_{c}-h_{av})^{-\nu}). Numerically
we find (\delta \simeq 2.0), (\gamma \simeq 1.2) and (\nu \simeq 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: (\chi) (\sim) ((\sigma_{c})
(-)(\sigma)^{-1/2}) and (\tau) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}). These
self-organised dynamical models of failure therefore show some definite
precursors with robust power laws long before the failure point. Such
well-characterised precursors should help predicting the global failure point
of the systems in advance.

Statistical Mechanics