  In several experiments for measuring various classes of responses, performed
at least some four decades ago, on driven physical systems in a
far-from-equilibrium (or, from a steady-state) situation, early stage
inverse-power-law relaxation dynamics had been observed. Since then, this
intriguing behavior raised its head off and on until it regained its central
role in the mainstream physical sciences about a decade ago with a breakdown
and/or avalanche type (also called self-organized critical) behavior of the
sand-pile model and a host of other similar problems. In this communication, we
report on the non-equilibrium dynamics in our Random Resistor cum
Tunneling-bond Network (RRTN) model. Previously, this semi-classical, or
semi-quantum percolative model has been highly successful in explaining the
static behavior for various random composite systems. In our dynamic studies
for the last several years, we observe two initial power-laws (more than a
decade each) and then an exponential relaxation for asymptotically large time
scales. Efforts were made to interpret our results with various existing
theoretical wisdom/s (which give, only one power-law relaxation for each such
system near its breakdown or run-away type state). Obviously, our results (with
two different power-laws) are richer than those particular cases. Further, a
complete theory is still lacking probably due to a much deeper issue of entropy
at stake. The appearance of two power-laws seems to be connected to some
non-extensive information-loss / entropy (the experimental systems being mostly
athermal) for such systems near their brinks (catastrophic failure not
necessarily due to criticality).
