We introduce a model of a totally asymmetric simple exclusion process (TASEP) on a tree network where the aggregate hopping rate is constant from level to level. With this choice for hopping rates the model shows the same phase diagram as the one-dimensional case. The potential applications of our model are in the area of distribution networks; where a single large source supplies material to a large number of small sinks via a hierarchical network. We show that mean field theory (MFT) for our model is identical to that of the one-dimensional TASEP and that this mean field theory is exact for the TASEP on a tree in the limit of large branching ratio, $b$(or equivalently large coordination number). We then present an exact solution for the two level tree (or star network) that allows the computation of any correlation function and confirm how mean field results are recovered as $b\rightarrow\infty$. As an example we compute the steady-state current as a function of branching ratio. We present simulation results that confirm these results and indicate that the convergence to MFT with large branching ratio is quite rapid.
We consider the one-dimensional totally asymmetric simple exclusion process (TASEP) with position-dependent hopping rates. The problem is solved,in a mean field/adiabatic approximation, for a general (smooth) form of spatial rate variation. Numerical simulations of systems with hopping rates varying linearly against position (constant rate gradient), for both periodic and open boundary conditions, provide detailed confirmation of theoretical predictions, concerning steady-state average density profiles and currents, as well as open-system phase boundaries, to excellent numerical accuracy.
We study a one-dimensional totally asymmetric simple exclusion process with one special site from which particles fly to any empty site (not just to the neighboring site). The system attains a non-trivial stationary state with density profile varying over the spatial extent of the system. The density profile undergoes a non-equilibrium phase transition when the average density passes through the critical value 1-1/[4(1-ln 2)]=0.185277..., viz. in addition to the discontinuity in the vicinity of the special site, a shock wave is formed in the bulk of the system when the density exceeds the crit…
We study steady state of the totally asymmetric simple exclusion process with inhomogeneous hopping rates associated with sites (site-wise disorder). Using the fact that the non-normalized steady-state weights which solve the master equation are polynomials in all the hopping rates, we propose a general method for calculating their first few lowest coefficients exactly. In case of binary disorder where all slow sites share the same hopping rate r<1, we apply this method to calculate steady-state current up to the quadratic term in r for some particular disorder configurations. For the most gen…
We investigate particle condensation in a driven pair exclusion process on one- and two- dimensional lattices under the periodic boundary condition. The model describes a biased hopping of particles subject to a pair exclusion constraint that each particle cannot stay at a same site with its pre-assigned partner. The pair exclusion causes a mesoscopic condensation characterized by the scaling of the condensate size $m_{\rm con}\sim N^β$ and the number of condensates $N_{\rm con}\sim N^α$ with the total number of sites $N$. Those condensates are distributed randomly without hopping bias. We fin…
We study the nonequilibrium steady states in totally asymmetric exclusion processes (TASEP) with open boundary conditions having spatially inhomogeneous hopping rates. Considering smoothly varying hopping rates, we show that the steady states are in general classified by the steady state currents in direct analogy with open TASEPs having uniform hopping rates. We calculate the steady state bulk density profiles, which are now spatially nonuniform. We also obtain the phase diagrams in the plane of the control parameters, which though have phase boundaries that are in general curved lines, have …
We consider a one-dimensional totally asymmetric exclusion process on a ring with extended inhomogeneities, consisting of several segments with different hopping rates. Depending upon the underlying inhomogeneity configurations and for moderate densities, our model displays both localised (LDW) and delocalised (DDW) domain walls and delocalisation transitions of LDWs in the steady states. Our results allow us to construct the possible steady state density profiles for an arbitrary number of segments with unequal hopping rates. We explore the scaling properties of the fluctuations of LDWs and D…
We consider the totally asymmetric simple exclusion process (TASEP) on the periodic chain in the presence of a single impurity site that is inaccessible to other particles and therefore acts as a static defect. Particles are allowed to advance any distance l \geq 1 on the right with the probability that decays as l^-(1+sigma), where sigma > 1. Despite the long range of hopping, we find the same type of phase transition that occurs in the standard short-range TASEP with a defect site where defect induces a macroscopic shock in the stationary state. In particular, our model displays two main fea…
Recently, it was shown that spatial correlations may have a drastic effect on the dynamics of real-space condensates in driven mass-transport systems: in models with a spatially correlated steady state, the condensate is quite generically found to drift with a non-vanishing velocity. Here we examine the condensate dynamics in the accelerate exclusion process (AEP), where spatial correlations are present. This model is a "facilitated" generalization of the totally asymmetric simple exclusion process (TASEP) where each hopping particle may trigger another hopping event. Within a mean-field appro…
We present a one-dimensional nonlocal hopping model with exclusion on a ring. The model is related to the Raise and Peel growth model. A nonnegative parameter $u$ controls the ratio of the local backwards and nonlocal forwards hopping rates. The phase diagram and consequently the values of the current, depend on $u$ and the density of particles. In the special case of half-filling and $u = 1$ the system is conformal invariant and an exact value of the current for any size $L$ of the system is conjectured and checked for large lattice sizes in Monte Carlo simulations. For $u > 1$ the current ha…
We construct a one-dimensional totally asymmetric simple exclusion process (TASEP) on a ring with two segments having unequal hopping rates, coupled to particle non-conserving Langmuir kinetics (LK) characterized by equal attachment and detachment rates. In the steady state, in the limit of competing LK and TASEP, the model is always found in states of phase coexistence. We uncover a nonequilibrium phase transition between a three-phase and a two-phase coexistence in the faster segment, controlled by the underlying inhomogeneity configurations and LK. The model is always found to be half-fille…
We reconsider the long-standing question of the critical defect hopping rate $r_c$ in the one-dimensional totally asymmetric exclusion process (TASEP) with a slow bond (defect). For $r< r_c$ a phase separated state is observed due to queuing at the defect site whereas for $r\geq r_c$ the defect site has only local effects on the stationary state of the homogeneous system. Mean-field theory predicts $r_c=1$ (when hopping rates outside the defect bond are equal to 1) but numerical investigations seem to indicate $r_c \approx 0.80(2)$. Here we improve the numerics to show that $r_c > 0.99$ and gi…
The TASEP is a paradigmatic model from non-equilibrium statistical physics, which describes particles hopping along a lattice of discrete sites. The TASEP is applicable to a broad range of different transport systems, but does not consider the fact that in many such systems the availability of resources required for the transport is limited. In this paper we extend the TASEP to include the effect of a limited number of two different fundamental transport resources: the hopping particles, and the "fuel carriers", which provide the energy required to drive the system away from equilibrium. As as…
Finite Dimensional Representations of the Quadratic Algebra: Applications to the Exclusion Process
本文研究了一维开边界部分非对称简单排斥过程（ASEP），它描述了一个硬核粒子在两端耦合到储层的链上跳跃随机过程的系统.德里达、埃文斯、哈基姆和帕斯奎尔[J. Phys. A 26，1493（1993）]已经证明，这个模型的平稳概率分布可以表示为二次代数上的迹，与变形代数密切相关。我们构造了这个代数的所有有限维不可约表示。这使我们能够计算静态堆积密度以及所有的相关长度的ASEP上的一组特殊的曲线的相图。
We study the evolution of a system of free fermions in one dimension under the simultaneous effects of coherent tunneling and stochastic Markovian noise. We identify a class of noise terms where a hierarchy of decoupled equations for the correlation functions emerges. In the special case of incoherent, nearest-neighbour hopping the equation for the two-point functions is solved explicitly. The Green's function for the particle density is obtained analytically and a timescale is identified where a crossover from ballistic to diffusive behaviour takes place. The result can be interpreted as a co…
We investigate a non-Poissonian version of the asymmetric simple exclusion process, motivated by the observation that coarse-graining the interactions between particles in complex systems generically leads to a stochastic process with a non-Markovian (history-dependent) character. We characterize a large family of one-dimensional hopping processes using a waiting-time distribution for individual particle hops. We find that when its variance is infinite, a real-space condensate forms that is complete in space (involves all particles) and time (exists at almost any given instant) in the thermody…
We report here our preliminary results in the study of a new version of the generalized Totally Asymmetric Simple Exclusion process (gTASEP) on open tracks. In the gTASEP an additional interaction between the particles is considered, besides the hard-core exclusion, which exists in the standard TASEP. It is modeled by the introduction of a second hopping probability $p_m$ for the particles, belonging to the same cluster, in addition to the standard hopping probability p, which now applies only for single particles and the head (rightmost) particle of a cluster. We briefly describe how one can …
By considering the master equation of the totally asymmetric exclusion process on a one-dimensional lattice and using two types of boundary conditions (i.e. interactions), two new families of the multi-species reaction-diffusion processes, with particle-dependent hopping rates, are investigated. In these models (i.e. reaction-diffusion and drop-push systems), we have the case of distinct particles where each particle $A_α$ has its own intrinsic hopping rate $v_α$. They also contain the parameters that control the annihilation-diffusion rates (including pair-annihilation and coagulation to the …
We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective long-range potential which in the limit of very large flux takes the simple form $U= -2\sum_{i\neq j}\log|\sinπ(n_{i}/L-n_{j}/L)|$, where $n_{1}% n_{2},\ldots n_{N}$ are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and det…
Adding quenched disorder to the one-dimensional asymmetric exclusion process is known to always induce phase separation. To test the robustness of this result, we introduce two modifications of the process that allow particles to bypass defect sites. In the first case, particles are allowed to jump l sites ahead with the probability p_l ~ l^-(1+sigma), where sigma>1. By using Monte Carlo simulations and the mean-field approach, we show that phase coexistence may be absent up to enormously large system sizes, e.g. lnL~50, but is present in the thermodynamic limit, as in the short-range case. In…
We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1-|t|/tau with a finite support [-tau,tau]. This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact tau=L/c, and in the special case of maximum velocity 1 (corresponding to the totally asymmetric simple exclusion process) we can ide…
Modelling the propagation of a pulse in a dense {\em milieu} poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the concept of persistent random walk to account for spatial exclusion effects. This is achieved by introducing an explicit constraint in the hopping rates, that weights the occupancy of the target sites. We derive the mean-field equations, which display nonlinear terms that are important at high density. We compute the evolution of the mean square displacement (MSD) …
The exclusive queueing process (EQP) has recently been introduced as a model for the dynamics of queues which takes into account the spatial structure of the queue. It can be interpreted as a totally asymmetric exclusion process of varying length. Here we investigate the case of deterministic bulk hopping p=1 which turns out to be one of the rare cases where exact nontrivial results for the dynamical properties can be obtained. Using a time-dependent matrix product form we calculate several dynamical properties, e.g. the density profile of the system.
We consider the asymmetric simple exclusion process on a ring, with an arbitrary asymmetry between the hopping rates of the particles. Using a functional formulation of the Bethe equations of the model, we derive exact expressions for all the cumulants of the current in the stationary state. These expressions involve tree structures with composite nodes. In the thermodynamic limit, three regimes can be observed for the current fluctuations depending on how the asymmetry scales with the size of the system.
We study asymmetric exclusion processes (TASEP) on a nonuniform one-dimensional ring consisting of two segments having unequal hopping rates, or {\em defects}. We allow weak particle nonconservation via Langmuir kinetics (LK), that are parameterised by generic unequal attachment and detachment rates. For an extended defect, in the thermodynamic limit the system generically displays inhomogeneous density profiles in the steady state - the faster segment is either in a phase with spatially varying density having no density discontinuity, or a phase with a discontinuous density changes. Nonequili…
We introduce a new update algorithm for exclusion processes, more suitable for the modeling of pedestrian traffic. Pedestrians are modeled as hard-core particles hopping on a discrete lattice, and are updated in a fixed order, determined by a phase attached to each pedestrian. While the case of periodic boundary conditions was studied in a companion paper, we consider here the case of open boundary conditions. The full phase diagram is predicted analytically and exhibits a transition between a free flow phase and a jammed phase. The density profile is predicted in the frame of a domain wall th…
Invariant measures of exclusion processes with a look-ahead rule
Faculty of Information Technology, Ho Chi Minh City University of Economics and Finance（胡志明市经济与财经大学信息技术学院）；Institute of Research and Development, Duy Tan University（大慈大学研发研究所）；Faculty of Natural Sciences, Duy Tan University（大慈大学自然科学学院）
Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the prob…
We obtain an exact matrix product steady state for a class of multi species asymmetric simple exclusion process with impurities, under periodic boundary condition. Alongside the usual hopping dynamics, an additional flip dynamics is activated only in the presence of impurities. Although the microscopic dynamics renders the system to be non-ergodic, exact analytical results for observables are obtained in steady states for a specific class of initial configurations. Interesting physical features including negative differential mobility and transition of correlations from negative to positive wi…
We consider the nonabelian sandpile model defined on directed trees by Ayyer, Schilling, Steinberg and Thiéry (Commun. Math. Phys, 2013) and restrict it to the special case of a one-dimensional lattice of $n$ sites which has open boundaries and disordered hopping rates. We focus on the joint distribution of the integrated currents across each bond simultaneously, and calculate its cumulant generating function exactly. Surprisingly, the process conditioned on seeing specified currents across each bond turns out to be a renormalised version of the same process. We also remark on a duality proper…
We consider current statistics for a two species exclusion process of particles hopping in opposite directions on a one-dimensional lattice. We derive an exact formula for the Green's function as well as for a joint current distribution of the model, and study its long time behavior. For a step type initial condition, we show that the limiting distribution is a product of the Gaussian and the GUE Tracy-Widom distribution. This is the first analytic confirmation for a multi-component system of a prediction from the recently proposed non-linear fluctuating hydrodynamics for one dimensional syste…
We study an exclusion process with 4 segments, which was recently introduced by T Banerjee, N Sarkar and A Basu [J. Stat. Mech. (2015) P01024]. The segments have hopping rates 1, r(<1), 1 and r, respectively. In a certain parameter region, two shocks appear, which are not static but synchronized. We explore dynamical properties of each shock and correlation of shocks, by means of the so-called second-class particle. The mean-squared displacement of shocks has three diffusive regimes, and the asymptotic diffusion coefficient is different from the known formula. In some time interval, it also ex…
We introduce an extension of the M/M/1 queueing process with a spatial structure and excluded- volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary-state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We …
We introduce and solve a model of fermions hopping between neighbouring sites on a line with random Brownian amplitudes and open boundary conditions driving the system out of equilibrium. The average dynamics reduces to that of the symmetric simple exclusion process. However, the full distribution encodes for a richer behaviour entailing fluctuating quantum coherences which survive in the steady limit. We determine exactly the system state steady distribution. We show that these out of equilibrium quantum fluctuations fulfil a large deviation principle and we present a method to recursively co…
We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-μ/\sqrt{L}$ respectively in the forward and in the backward direction. Using Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $μ=0$ by solving order by order a simple functional equation. A key point is that when $μ\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum, but converge instead at large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a re…
Stochastic processes of interacting particles with varying length are relevant e.g. for several biological applications. We try to explore what kind of new physical effects one can expect in such systems. As an example, we extend the exclusive queueing process that can be viewed as a one-dimensional exclusion process with varying length, by introducing Langmuir kinetics. This process can be interpreted as an effective model for a queue that interacts with other queues by allowing incoming and leaving of customers in the bulk. We find surprising indications for breaking of ergodicity in a certa…
