Extensions of Effective Medium Theory of Transport in Disordered Systems

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

Random Walk Access Times on Partially-Disordered Complex Networks: an Effective Medium Theory

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are different for steps across lattice bonds from the rates $f$ across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially-disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.Comment: 12 pages, 8 figures; added new results and discussion; added appendix on numerical procedures. To appear in PR

Effects of disorder in location and size of fence barriers on molecular motion in cell membranes

The effect of disorder in the energetic heights and in the physical locations of fence barriers encountered by transmembrane molecules such as proteins and lipids in their motion in cell membranes is studied theoretically. The investigation takes as its starting point a recent analysis of a periodic system with constant distances between barriers and constant values of barrier heights, and employs effective medium theory to treat the disorder. The calculations make possible, in principle, the extraction of confinement parameters such as mean compartment sizes and mean intercompartmental transition rates from experimentally reported published observations. The analysis should be helpful both as an unusual application of effective medium theory and as an investigation of observed molecular movements in cell membranes.Comment: 9 pages, 5 figure

Static Pairwise Annihilation in Complex Networks

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, Erd\"os-R\'enyi (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.Comment: 7 Figures, 10 page

Traversal Times for Random Walks on Small-World Networks

We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate F that is different from the rate f for steps across small-world connections. When f >> F, the mean time to traverse the network exhibits a transition associated with percolation of the random graph (i.e., small-world) part of the network, and a collapse of the data onto a universal curve. This transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps. We develop a simple self-consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition, as is characteristic of most effective medium theories.Comment: 9 pages, 5 figure

Adiabatic-Nonadiabatic Transition in the Diffusive Hamiltonian Dynamics of a Classical Holstein Polaron

We study the Hamiltonian dynamics of a free particle injected onto a chain containing a periodic array of harmonic oscillators in thermal equilibrium. The particle interacts locally with each oscillator, with an interaction that is linear in the oscillator coordinate and independent of the particle's position when it is within a finite interaction range. At long times the particle exhibits diffusive motion, with an ensemble averaged mean-squared displacement that is linear in time. The diffusion constant at high temperatures follows a power law D ~ T^{5/2} for all parameter values studied. At low temperatures particle motion changes to a hopping process in which the particle is bound for considerable periods of time to a single oscillator before it is able to escape and explore the rest of the chain. A different power law, D ~ T^{3/4}, emerges in this limit. A thermal distribution of particles exhibits thermally activated diffusion at low temperatures as a result of classically self-trapped polaronic states.Comment: 15 pages, 4 figures Submitted to Physical Review

Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases

We prove approach to thermal equilibrium for the fully Hamiltonian dynamics of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a $d$-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We establish that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature $T$, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics--they need not be in a state of thermal equilibrium. The temperature $T$ to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy $k_{\mathrm B}T$ is equal to an appropriately defined average of the scatterers' kinetic energy. In the equilibrated state, particle motion is diffusive

Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks

We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds ($F$) and across network shortcuts ($f$). For fast shortcuts ($f/F\gg 1$) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.Comment: 8 pages, 6 figures; expanded discussions, added figures and references. To appear in J Stat Phy

Anderson localization of polaron states

Using the vanishing of the typical polaron tunneling rate as an indicator of the breakdown of itinerancy, we study the localization of polaron states in a generic model for a disordered polaronic material. We find that extremely small disorder causes an Anderson localization of small polaron states. However, the ratio between the critical disorder strength needed to localize all states in the polaron band and the renormalized bandwidth is not necessarily smaller than for a bare electron.Comment: 4 pages, 3 figure

Normal transport properties for a classical particle coupled to a non-Ohmic bath

We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility over macroscopic time scales of order exp(-c/F). The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties