Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

Dependence of the transportation time on the sequence in which particles with different hopping probabilities enter a lattice

Smooth transportation has drawn the attention of many researchers and practitioners in several fields. In the present study, we propose a modified model of a totally asymmetric simple exclusion process (TASEP), which includes multispecies of particles and takes into account the sequence in which the particles enter a lattice. We investigate the dependence of the transportation time on this "entering sequence" and show that, for a given collection of particles, group sequence in some cases minimizes the transportation time better than a random sequence. We also introduce the "sorting cost" necessary to transform a random sequence into a group sequence and show that when this is included a random sequence can become advantageous in some conditions. We obtain these results not only from numerical simulations but also by theoretical analyses that generalize the simulation results for some special cases.

Velocity correlations of a discrete-time totally asymmetric simple-exclusion process in stationary state on a circle

The discrete-time version of totally asymmetric simple-exclusion process (TASEP) on a finite one-dimensional lattice is studied with the periodic boundary condition. Each particle at a site hops to the next site with probability 0≤p≤1 if the next site is empty. This condition can be rephrased by the condition that the number n of vacant sites between the particle and the next particle is positive. Then the average velocity is given by a product of the hopping probability p and the probability that n≥1. By mapping the TASEP to another driven diffusive system called the zero-range process, it is proved that the distribution function of vacant sites in the stationary state is exactly given by a factorized form. We define k-particle velocity correlation function as the expectation value of a product of velocities of k particles in the stationary distribution. It is shown that it does not depend on positions of k particles on a circle but depends only on the number k. We give explicit expressions for all velocity correlation functions using the Gauss hypergeometric functions. Covariance of velocities of two particles is studied in detail, and we show that velocities become independent asymptotically in the thermodynamic limit.

Mean-field analysis for parallel asymmetric exclusion process with anticipation effect

This paper studies an extended parallel asymmetric exclusion process, in which the anticipation effect is taken into account. The fundamental diagram of the model has been investigated via cluster mean field analysis. Different from previous mean field analysis, in which the n -cluster probabilities P(σ{i},…,σ{i+n-1}) involve the (n+2) -cluster probabilities P(τ{i-1},…,τ{i+n}) , our mean-field analysis is asymmetric because the three-cluster probabilities P(σ{i},σ{i+1},σ{i+2}) involve the six-cluster probabilities P(τ{i-1},…,τ{i+4}) . We find an excellent agreement between Monte Carlo simulations and cluster mean field analysis, which indicates that the mean field analysis might give the exact expression.

Velocity and cluster distributions in a bottleneck system

Velocity and cluster distributions for particles with unidirectional motion in one dimension are studied. The particles never pass each other, like cars on a narrow road that does not allow overtaking. As a result, particles cluster behind slow particles (queues are formed behind slow cars). Thus, the actual velocity of each particle is to a large extent determined by slow particles further ahead. Considering all possible permutations of N particles with initial velocities {vi}, the average number of particles with actual velocity vi is (N+1)/[i(i+1)] (in the sequence {vi}, the initial velocities are listed with monotonically increasing values). For i large and vi proportional, variant i the average number of actual velocities is thus a power law in vi, even though the average cluster density is found to be independent of cluster size, L. On the other hand, the cluster density varies significantly with cluster velocity; we obtain [(N-i)!(N-L)!]/[NN!(N-L-i+1)!]. The average velocity at a given position in the sequence of N particles, and the average global velocity are determined. Explicit results for several distributions of the initial velocities show that the global velocity depends sensitively on the form of this distribution.

Queueing phase transition: theory of translation

We study the current of particles on a lattice, where to each site a different hopping probability has been associated and the particles can move only in one direction. We show that the queueing of the particles behind a slow site can lead to a first-order phase transition, and derive analytical expressions for the configuration of slow sites for this to happen. We apply this stochastic model to describe the translation of mRNAs. We show that the first-order phase transition, uncovered in this work, is the process responsible for the classification of the proteins having different biological functions.

Asymmetric exclusion processes with disorder: effect of correlations

Multiparticle dynamics in one-dimensional asymmetric exclusion processes with disorder is investigated theoretically by computational and analytical methods. It is argued that the general phase diagram consists of three nonequilibrium phases that are determined by the dynamic behavior at the entrance, at the exit and at the slowest defect bond in the bulk of the system. Specifically, we consider dynamics of asymmetric exclusion process with two identical defect bonds as a function of distance between them. Two approximate theoretical methods that treat the system as a sequence of segments with exact description of dynamics inside the segments and neglect correlations between them, are presented. In addition, a numerical iterative procedure for calculating dynamic properties of asymmetric exclusion systems is developed. Our theoretical predictions are compared with extensive Monte Carlo computer simulations. It is shown that correlations play an important role in the particle dynamics. When two defect bonds are far away from each other the strongest correlations are found at these bonds. However, bringing defect bonds closer leads to the shift of correlations to the region between them. Our analysis indicates that it is possible to develop a successful theoretical description of asymmetric exclusion processes with disorder by properly taking into account the correlations.

Dynamics of condensation in growing complex networks

A condensation transition was predicted for growing technological networks evolving by preferential attachment and competing quality of their nodes, as described by the fitness model. When this condensation occurs, a node acquires a finite fraction of all the links of the network. Earlier studies based on steady-state degree distribution and on the mapping to Bose-Einstein condensation were able to identify the critical point. Here we characterize the dynamics of condensation and we present evidence that below the condensation temperature there is a slow down of the dynamics and that a single node (not necessarily the best node in the network) emerges as the winner for very long times. The characteristic time t;{*} at which this phenomenon occurs diverges both at the critical point and at T-->0 when new links are attached deterministically to the highest quality node of the network.

Characteristics of the asymmetric simple exclusion process in the presence of quenched spatial disorder

We investigate the effect of quenched spatial disordered hopping rates on the characteristics of the asymmetric simple exclusion process with open boundaries both numerically and by extensive simulations. Disorder averages of the bulk density and current are obtained in terms of various input and output rates. We study the binary and uniform distributions of disorder. It is verified that the effect of spatial inhomogeneity is generically to enlarge the size of the maximal-current phase. This is in accordance with the mean-field results obtained by Harris and Stinchcombe [Phys. Rev. E 70, 016108 (2004)]. Furthermore, we obtain the dependence of the current and the bulk density on the characteristics of the disorder distribution function. It is shown that the impact of disorder crucially depends on the particle input and out rates. In some situations, disorder can constructively enhance the current.

Bottleneck-induced transitions in a minimal model for intracellular transport

We consider the influence of disorder on the nonequilibrium steady state of a minimal model for intracellular transport. In this model particles move unidirectionally according to the totally asymmetric exclusion process (TASEP) and are coupled to a bulk reservoir by Langmuir kinetics. Our discussion focuses on localized point defects acting as a bottleneck for the particle transport. Combining analytic methods and numerical simulations, we identify a rich phase behavior as a function of the defect strength. Our analytical approach relies on an effective mean-field theory obtained by splitting the lattice into two subsystems, which are effectively connected exploiting the local current conservation. Introducing the key concept of a carrying capacity, the maximal current that can flow through the bulk of the system (including the defect), we discriminate between the cases where the defect is irrelevant and those where it acts as a bottleneck and induces various novel phases (called bottleneck phases). Contrary to the simple TASEP in the presence of inhomogeneities, many scenarios emerge and translate into rich underlying phase diagrams, the topological properties of which are discussed.

Stochastic optimal velocity model and its long-lived metastability

In this paper, we propose a stochastic cellular automaton model of traffic flow extending two exactly solvable stochastic models, i.e., the asymmetric simple exclusion process and the zero range process. Moreover, it is regarded as a stochastic extension of the optimal velocity model. In the fundamental diagram (flux-density diagram), our model exhibits several regions of density where more than one stable state coexists at the same density in spite of the stochastic nature of its dynamical rule. Moreover, we observe that two long-lived metastable states appear for a transitional period, and that the dynamical phase transition from a metastable state to another metastable/stable state occurs sharply and spontaneously.

Partially asymmetric exclusion models with quenched disorder

We consider the one-dimensional partially asymmetric exclusion process with random hopping rates, in which a fraction of particles (or sites) have a preferential jumping direction against the global drift. In this case, the accumulated distance traveled by the particles, x, scales with the time, t, as x approximately t(1/z), with a dynamical exponent z>0. Using extreme value statistics and an asymptotically exact strong disorder renormalization group method, we exactly calculate z(PW) for particlewise disorder, which is argued to be related as z(SW)=z(PW)/2 for sitewise disorder. In the symmetric case with zero mean drift, the particle diffusion is ultraslow, logarithmic in time.

Phases of a conserved mass model of aggregation with fragmentation at fixed sites

To study the effect of quenched disorder in a class of reaction-diffusion systems, we introduce a conserved mass model of diffusion and aggregation in which fragmentation occurs only at certain fixed sites. On most sites, the mass moves as a whole to a nearest neighbor while it leaves the fixed sites only as a single monomer (i.e., chips off). Once the mass leaves any site, it coalesces with the mass present on its neighbor. We study in detail the effect of a single chipping site on the steady state in arbitrary dimensions, with and without bias. In the thermodynamic limit, the system can exist in one of the following three phases. (a) Pinned aggregate (PA) phase in which an infinite aggregate (with mass proportional to the volume of the system) appears at the chipping site with probability one but not in the bulk. (b) Unpinned aggregate (UA) phase in which the infinite aggregate occurs at the chipping site with a probability strictly less than one and can coexist with infinite aggregates in the bulk. (c) Nonaggregate (NA) phase in which there is no infinite cluster. The steady state of the system depends on the dimension and drive. A sitewise inhomogeneous mean field theory predicts that the system exists in the UA phase in all cases. Monte Carlo simulations in one and two dimensions support this prediction in all but one-dimensional, biased case. In the latter case, there is a phase transition from the NA phase to the PA phase as the density is increased. We identify the critical point exactly and calculate the mass distribution in the PA phase. The NA phase and the critical point are studied by Monte Carlo simulations and using scaling arguments. A variant of the above aggregation model is also considered in which total particle number is conserved and chipping occurs at a fixed site, but the particles do not interact with each other at other sites. This model is solved exactly by mapping it to a zero range process. With increasing density, it exhibits a phase transition from the NA phase to the PA phase in all dimensions, irrespective of bias. The free-particle model is also solved with an extensive number of chipping sites with random chipping rates and we argue that it qualitatively describes the behavior of the aggregation model with extensive disorder.

Minimal current phase and universal boundary layers in driven diffusive systems

We investigate boundary-driven phase transitions in open driven diffusive systems. The generic phase diagram for systems with short-ranged interactions is governed by a simple extremal principle for the macroscopic current, which results from an interplay of density fluctuations with the motion of shocks. In systems with more than one extremum in the current-density relation, one finds a minimal current phase even though the boundaries support a higher current. The boundary layers of the critical minimal current and maximal current phases are argued to be of a universal form. The predictions of the theory are confirmed by Monte Carlo simulations of the two-parameter family of stochastic particle hopping models of Katz, Lebowitz, and Spohn and by analytical results for a related cellular automaton with deterministic bulk dynamics. The effect of disorder in the particle jump rates on the boundary layer profile is also discussed.

Phase transition in a traffic model with passing

We investigate a traffic model in which cars either move freely with quenched intrinsic velocities or belong to clusters formed behind slower cars. In each cluster, the next-to-leading car is allowed to pass and resume free motion. The model undergoes a phase transition from a disordered phase for the high passing rate to a jammed phase for the low rate. In the disordered phase, the cluster size distribution decays exponentially in the large size limit. In the jammed phase, the distribution of finite clusters is independent on the passing rate, but it accounts only for a fraction of all cars; the "excessive" cars form an infinite cluster moving with the smallest velocity. Mean-field equations, describing the model in the framework of Maxwell approximation, correctly predict the existence of phase transition and adequately describe the disordered phase; properties of the jammed phase are studied numerically.

Exactly solvable two-way traffic model with ordered sequential update

Within the formalism of the matrix product ansatz, we study a two-species asymmetric exclusion process with backward and forward site-ordered sequential updates. This model, which was originally introduced with the random sequential update [J. Phys. A 30, 8497 (1997)], describes a two-way traffic flow with a dynamic impurity and shows a phase transition between the free flow and the traffic jam. We investigate characteristics of this jamming and examine similarities and differences between our results and those with a random sequential update.