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.
Subject(s)
Models, Chemical , Motion , DiffusionABSTRACT
The relaxation time approximation (RTA) is commonly employed in nonequilibrium statistical mechanics to approximate solutions to the Boltzmann equation in terms of an exponential relaxation to equilibrium. Despite its common use, the RTA suffers from the drawback that relaxation times commonly employed are independent of initial conditions. We derive a variational principle for solutions to the Boltzmann equation, which allows us to extend the standard RTA using relaxation times that depend on the initial distribution. Tests of the approach on a calculation of the mobility for a one-dimensional (1D) tight-binding band indicate that our analysis typically provides a better approximation than the standard RTA.
ABSTRACT
We study static annihilation on complex networks, in which pairs of connected particles annihilate at a constant rate during time. Through a mean-field formalism, we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections. This general formalism, which is exact for disordered networks, is applied to Kronecker, Erdös-Rényi (i.e., Poisson), and scale-free networks. We compare our theoretical results with extensive numerical simulations obtaining excellent agreement. Although the mean-field approach applies in an exact way neither to ordered lattices nor to small-world networks, it qualitatively describes the annihilation dynamics in such structures. Our results indicate that the higher the connectivity of a given network element, the faster it annihilates. This fact has dramatic consequences in scale-free networks, for which, once the "hubs" have been annihilated, the network disintegrates and only isolated sites are left.
ABSTRACT
Polaronic theories for charge transport in disordered organic solids, particularly molecularly doped polymers, have been plagued by issues of internal consistency related to the magnitude of physical parameters. We present a natural resolution of the problem by showing that, in the presence of correlated disorder, polaronic carriers with binding energies Delta approximately 50-500 meV and transfer integrals J approximately 1-20 meV are completely consistent with the magnitudes of field and temperature dependent mobilities observed.
