We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numeri
cs to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z = 3/2 in one dimension, the mean-field value z = 2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.
The totally asymmetric simple exclusion process (TASEP), a well-known model in its strictly one-dimensional (chain) version, is generalized to cylinder (nanotube) and ribbon (nanoribbon) geometries. A mean-field theoretical description is given for v
ery narrow ribbons (necklaces), and nanotubes. For specific configurations of bond transmissivity rates, and for a variety of boundary conditions, theory predicts equivalent steady state behavior between (sublattices on) these structures and chains. This is verified by numerical simulations, to excellent accuracy, by evaluating steady-state currents. We also numerically treat ribbons of general width. We examine the adequacy of this model to the description of electronic transport in carbon nanotubes and nanoribbons, or specifically-designed quantum dot arrays.
We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process (TASEP). With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic con
cepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functions.
Properties of the one-dimensional totally asymmetric simple exclusion process (TASEP), and their connection with the dynamical scaling of moving interfaces described by a Kardar-Parisi-Zhang (KPZ) equation are investigated. With periodic boundary con
ditions, scaling of interface widths (the latter defined via a discrete occupation-number-to-height mapping), gives the exponents $alpha=0.500(5)$, $z=1.52(3)$, $beta=0.33(1)$. With open boundaries, results are as follows: (i) in the maximal-current phase, the exponents are the same as for the periodic case, and in agreement with recent Bethe ansatz results; (ii) in the low-density phase, curve collapse can be found to a rather good extent, with $alpha=0.497(3)$, $z=1.20(5)$, $beta=0.41(2)$, which is apparently at variance with the Bethe ansatz prediction $z=0$; (iii) on the coexistence line between low- and high- density phases, $alpha=0.99(1)$, $z=2.10(5)$, $beta=0.47(2)$, in relatively good agreement with the Bethe ansatz prediction $z=2$. From a mean-field continuum formulation, a characteristic relaxation time, related to kinematic-wave propagation and having an effective exponent $z^prime=1$, is shown to be the limiting slow process for the low density phase, which accounts for the above-mentioned discrepancy with Bethe ansatz results. For TASEP with quenched bond disorder, interface width scaling gives $alpha=1.05(5)$, $z=1.7(1)$, $beta=0.62(7)$. From a direct analytic approach to steady-state properties of TASEP with quenched disorder, closed-form expressions for the piecewise shape of averaged density profiles are given, as well as rather restrictive bounds on currents. All these are substantiated in numerical simulations.
