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 1-dimensional system with bond
disorder for specificity, a transformation procedure is developed to deduce,
from given distribution functions characterizing the system disorder, explicit
expressions for the memory functions. It is shown how to use the memory
functions in the Lapace 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
mediumtheory 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 1-d 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.Comment: 15 pages, 11 figure
