We describe how we use the Ising model paradigm, in conjunction with kinetic Monte Carlo simulations, for generating dynamical network configurations that are consistent with Kawasaki lattice dynamics (i.e. constant conducting-site density). At any point during the simulations conducting-site pathways (with density) are taken to be given by the network of up spins, using the Ising terminology, with the non-conducting-sites represented by the network of down spins.
In addition to the simulation results we provide a theoretical analysis of the problem by firstly providing a rationale as to why we partition the net displacement of the RWs throughout the network into two terms representing: (1) the contribution to transport by ‘hopping' through nearest neighbor conducting sites (the so-called ‘percolation' mechanism) and (2) the self-diffusion of the site itself on which the RW finds itself at any given point in time, respectively. The ‘percolation- diffusion' component exhibits non-trivial scaling behavior, with a new scaling exponent that describes the cage trapping time of the RWs in conducting site clusters. We show how the value of this exponent can be found from computer simulation results and compare our results to conductance measurements in supercritical microemulsions and recently published diffusion data taken in dense colloidal suspensions.