Extensions of Effective-Medium Theory of Transport in Disordered Systems
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Abstract
The effective-medium theory of transport in disordered systems, whose basis is the replacement of spatial disorder by temporal memory, is extended in several practical directions. Restricting attention to a one-dimensional system with bond disorder for specificity, a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder. It is shown how to use the memory functions in the Laplace domain forms in which they first appear, and in the time domain forms which are obtained via numerical inversion algorithms, to address time evolution of the system beyond the asymptotic domain of large times normally treated. An analytic but approximate procedure is provided to obtain the memories, in addition to the inversion algorithm. Good agreement of effective-medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit, as expected. The use of ensemble averages is studied for normal as well as correlation observables. The effect of size on effective-medium theory is explored and it is shown that, even in the asymptotic limit, finite-size corrections develop to the well-known harmonic mean prescription for finding the effective rate. A percolation threshold is shown to arise even in one dimension for finite (but not infinite) systems at a concentration of broken bonds related to the system size. Spatially long-range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories, in spite of the fact that the original rates possess nearest neighbor character. Pausing time distributions in continuous-time random walks corresponding to the effective-medium memories are calculated.
