We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left-hand boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along the lattice, particles can adsorb or desorb, and the right-hand boundary is defined by a wall particle. The confining wall particle has intrinsic forward and backward hopping, a net leftward drift, and cannot desorb. Performing Monte Carlo simulations and using a moving-frame finite segment approach coupled to mean field theory, we find the parameter regimes in which the wall acquires a steady-state position. In other regimes, the wall will either drift to the left and fall off the lattice at the injection site, or drift indefinitely to the right. Our results are discussed in the context of nonequilibrium phases of the system, fluctuating boundary layers, and particle densities in the laboratory frame versus the frame of the fluctuating wall.
We unify step bunching ͑SB͒ instabilities occurring under various conditions on crystal surfaces below roughening. We show that when attachment-detachment of atoms at step edges is the rate-limiting process, the SB of interacting, concentric circular steps is equivalent to the commonly observed SB of interacting straight steps under deposition, desorption, or drift. We derive a continuum Lagrangian partial differential equation, which is used to study the onset of instabilities for circular steps. These findings place on a common ground SB instabilities from numerical simulations for circular steps and experimental observations of straight steps. Recent advances in the fabrication of small devices such as quantum dots have motivated theoretical research into the fundamental properties of surfaces at the nanoscale. Below the roughening transition temperature, 1 the evolution of crystal surfaces is governed by the motion of steps. 2 This motion is important in understanding phenomena such as pattern formation 3 and the self-assembly of nanostructures. 4 One of the most commonly studied surface phenomena is step bunching ͑SB͒, where steps cluster together tightly into widely separated bunches. 5 This instability has been observed experimentally on many different systems, e.g., see Refs. 6-8 Most theoretical studies of SB have focused on the idealized situation with straight steps. 3,9,10 However, as surface features become smaller, the step curvature should play an increasingly important role. 11 Therefore, a realistic model of step bunching must include both the effects of step curvature and step interactions. 1 Experimentally, the most common way to induce SB is to heat the surface using a direct current. 12 The resulting instability in straight steps has been understood on the basis of an asymmetry in the adatom density caused by a preferential drift. 13 However, the equivalent phenomenon in circular steps has received much less attention. Experiments for circular steps are not uncommon 14,15 and yet, very few theories currently exist that predict the onset of SB in circular steps ͑see Ref. 16, however, for a quasi-steady-state analysis͒.In this Brief Report, we discuss how curvature differences from one step to another can also induce a drift and, thus, give rise to SB. The main result is the derivation of a reduced partial differential equation ͑PDE͒, Eq. ͑12͒ below, which captures this effect and is able to predict the onset of SB instabilities for relaxing circular steps. We therefore unify SB phenomena in straight and circular steps by showing that they have a common physical and mathematical basis. In particular, our approach demonstrates that the effect of step line tension on SB is equivalent to that of a drift or of desorption and/or material deposition in the presence of a difference in the kinetic rates at step edges. This equivalence is shown by treating the step index as a continuum variable. 17 In the resulting PDEs for the step positions, the step line tension and the physical effects described ...
Plaques are fatty deposits that grow mainly in arteries and develop as a result of a chronic inflammatory response. Plaques are characterized as 'vulnerable' when they have large internal regions of necrosis and are heavily infiltrated by macrophages. The particular composition of a vulnerable plaque renders it susceptible to rupture, which releases thrombogenic agents into the bloodstream and can result in myocardial infarction. In this paper, we propose a mathematical model to predict the development of a plaque's necrotic core. By solving coupled reaction-diffusion equations for macrophages and dead cells, we focus on the joint effects of hypoxic cell death and chemoattraction to oxidized low-density lipoprotein (Ox-LDL), a molecule that is strongly linked to atherosclerosis. We do not model the mechanical properties of the plaque, its growth or rupture. Our model predicts cores that have approximately the right size and shape when compared to ultrasound images. Because our model is linear and autonomous, normal mode analysis and subsequent calculation of the smallest eigenvalue allow us to compute the times taken for the necrotic core to form. We find that the spatial distribution of Ox-LDL within the plaque determines not only the placement and size of cores, but their time of formation. Although plaques are biochemically complex, our study shows that certain aspects of their composition can be predicted and are, in fact, governed by simple physical models.
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