There is a great need for accurate and efficient computational approaches that can account for both the discrete and stochastic nature of chemical interactions as well as spatial inhomogeneities and diffusion. This is particularly true in biology and
nanoscale materials science, where the common assumptions of deterministic dynamics and well-mixed reaction volumes often break down. In this article, we present a spatial version of the partitioned-leaping algorithm (PLA), a multiscale accelerated-stochastic simulation approach built upon the tau-leaping framework of Gillespie. We pay special attention to the details of the implementation, particularly as it pertains to the time step calculation procedure. We point out conceptual errors that have been made in this regard in prior implementations of spatial tau-leaping and illustrate the manifestation of these errors through practical examples. Finally, we discuss the fundamental difficulties associated with incorporating efficient exact-stochastic techniques, such as the next-subvolume method, into a spatial-leaping framework and suggest possible solutions.
We investigate integer and half-integer filling states (uniform and unidimensional stripe states respectively) for graphene using the Hartree-Fock approximation. For fixed filling factor, the ratio between the scales of the Coulomb interaction and La
ndau level spacing $g=(e^2/epsilon ell)/(hbar v_F/ell)$, with $ell$ the magnetic length, is a field-independent constant. However, when $B$ decreases, the number of filled negative Landau levels increases, which surprisingly turns out to decrease the amount of Landau level mixing. The resulting states at fixed filling factor $ u$ (for $ u$ not too big) have very little Landau level mixing even at arbitrarily weak magnetic fields. Thus in the density-field phase diagram, many different phases may persist down to the origin, in contrast to the more standard two dimensional electron gas, in which the origin is surrounded by Wigner crystal states. We demonstrate that the stripe amplitudes scale roughly as $B$, so that the density waves ``evaporate continuously as $Bto 0$. Tight-binding calculations give the same scaling for stripe amplitude and demonstrate that the effect is not an artifact of the cutoff procedure used in the continuum calculations.
We investigate the transmission of electrons between conducting nanoribbon leads oriented at multiples of 60 degrees with respect to one another, connected either directly or through graphene polygons. A mode-matching analysis suggests that the trans
mission at low-energies is sensitive to the precise way in which the ribbons are joined. Most strikingly, we find that armchair leads forming 120-degree angles can support either a large transmission or a highly suppressed transmission, depending on the specific geometry. Tight-binding calculations demonstrate the effects in detail, and are also used to study transmission at higher energies as well as for zigzag ribbon leads.
