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.

The electrical mnductivities of FeUl-doped polypyrrole samples with different dopant concentrations have been measured in the tempenture range 25-300 K. Thermoelectric power mea+urements have been performed on two polypyrmle samples at low doping levels between 50 and 300 K. The variation in DC conductivity can be explained by Monk vasiablerange hopping (VRH) model. Fitting with MOD'S YRH theory. the conductivity data provide the values of the density N(€F) of states at the Fermi level and the hopping distance R. The low value and the sublinear character of the themoelechic power is consistent with the VRH theory.

The Random Resistor cum Tunneling Network (RRTN) model was proposed from our group by considering an extra phenomenological (semi-classical) tunneling process into a classical RRN bond percolation model. We earlier reported about early-stage two inverse power-laws, followed by large time purely exponential tail in some of the RRTN macroscopic current relaxations. In this paper, we investigate on the broader perspective of current relaxation. We present here an analytical argument behind the strong convergence (irrespective of initial voltage configuration) of the bulk current towards its steady-state, mapping the problem into a special kind of Gauss-Seidel method. We find two phenomenological time-scales (referred as τ t and τ s ), those emerge from the variation of macroscopic quantities during current dynamics. We show that not both, only one of them is independent. Thus there exists a single scale in time which controls the entire dynamics.

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