Two early-stage inverse power law dynamics in nonlinear complex systems far from equilibrium

We consider the dynamics of the charge carriers in a tunneling-enhanced percolation network, named as a Random Resistor cum Tunneling-bond Network (RRTN), where we allow tunneling in the gap between two randomly thrown nearest-neighbour metallic bonds only. Our earlier studies involve the dc and ac nonlinear response, the percolative aspects, dielectric breakdown, the finite-temperature variable-range hopping conductivity, etc. in the RRTN. Here we study the non-equilibrium dynamics of the carriers. With two far-from-equilibrium, initial inverse power law relaxations extending over several decades, the dynamics has a lot of similarities with many avalanche-like, run-away phenomena occurring in a wide variety of naturally occuring driven, disordered systems with statistically correlated randomness. In this regime, the RRTN violates the Boltzmann's (or, Debye) relaxation time approximation strongly. Beyond this regime, the response decays exponentially fast (acquires a time-scale) to a steady state.