On the Asymmetric Zero-Range in the Rarefaction Fan

We consider one-dimensional asymmetric zero-range processes starting from a step decreasing profile leading, in the hydrodynamic limit, to the rarefaction fan of the associated hydrodynamic equation. Under that initial condition, and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps, we derive the Law of Large Numbers for a second class particle, under the initial configuration in which all positive sites are empty, all negative sites are occupied with infinitely many first class particles and there is a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle it picks randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation, through some sort of renormalization function. By coupling the constant-rate totally asymmetric zero-range with the totally asymmetric simple exclusion, we derive limiting laws for more general initial conditions.     

Exact solution of the totally asymmetric simple exclusion process: Shock profiles

The microscopic structure of macroscopic shocks in the one-dimensional, totally asymmetric simple exclusion process is obtained exactly from the complete solution of the stationary state of a model system containing two types of particles-first and second class. This nonequilibrium steady state factorizes about any second-class particle, which implies factorization in the one-component system about the (random) shock position. It also exhibits several other interesting features, including long-range correlations in the limit of zero density of the second-class particles. The solution also shows that a finite number of second-class particles in a uniform background of first-class particles form a weakly bound state.     

An exclusion process for modelling fungal hyphal growth

A simple model for mass transport within a growing fungal filament is reviewed. Inspired by the role of microtubule-transported vesicles, we embody the dynamics of mass along a quasi-one-dimensional hypha with mutually excluding particles hopping on a growing one-dimensional lattice. The model is a generalisation of the totally asymmetric simple exclusion process (TASEP) to a dynamically extending lattice. We discuss mean-field and improved mean-field equations and present a phase diagram of the model's steady-state behaviour which generalises that of the TASEP. In particular we identify a region in which a shock in the density travels forward more slowly than the tip of the lattice and thus moves away from both the boundaries.     

Realization of the Open-Boundary Totally Asymmetric Simple Exclusion Process on a Ring

We propose a misanthrope process, defined on a ring, which realizes the totally asymmetric simple exclusion process with open boundaries. In the misanthrope process, particles have no exclusion interaction in contrast to those in the simple exclusion process, while the hop rates depend on both numbers of particles at departure and arrival sites. Arranging the hop rates, we can recover the simple exclusion property and moreover have condensation if the number of particles exceeds that of sites. One condensate grows at an arbitrary single site and then behaves as an external reservoir providing and absorbing particles. It is known that, under some condition, the misanthrope process has an exact solution for the steady-state probability distribution. We exploit this to investigate the present model in an analytical manner.     

Generalized Green Functions and Current Correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the Airy₂ process.     

Brownian Bridges for Late Time Asymptotics of KPZ Fluctuations in Finite Volume

Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium steady state depends on the initial condition. Using a reformulation of the matrix product representation for the dominant eigenstate, the statistics of the height at large scales is expressed, for arbitrary initial conditions, in terms of extremal values of independent standard Brownian bridges. Comparison with earlier exact Bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non-intersecting Brownian bridges with exponentially distributed distances between the endpoints.     

Matrix Representation of the Stationary Measure for the Multispecies TASEP

In this work we construct the stationary measure of the N species totally asymmetric simple exclusion process in a matrix product formulation. We make the connection between the matrix product formulation and the queueing theory picture of Ferrari and Martin. In particular, in the standard representation, the matrices act on the space of queue lengths. For N>2 the matrices in fact become tensor products of elements of quadratic algebras. This enables us to give a purely algebraic proof of the stationary measure which we present for N=3.     

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of the TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.     

Exact results for the 1D asymmetric exclusion process and KPZ fluctuations

We discuss several properties of the current of the one-dimensional asymmetric simple exclusion process (ASEP) through exact solutions. First we explain the stationary measure for the finite system with boundaries and its average current. Then we study the fluctuation properties of the current and the position of a tagged particle for the infinite system in the Kardar-Parisi-Zhang (KPZ) scaling region.     

Asymmetric exclusion processes with site sharing in a one-channel transport system

This Letter investigates two-species totally asymmetric simple exclusion process (TASEP) with site sharing in a one-channel transport system. In the model, different species of particles may share the same sites, while particles of the same species may not (hard-core exclusion). The site-sharing mechanism is applied to the bulk as well as the boundaries. Such sharing mechanism within the framework of the TASEP has been largely ignored so far. The steady-state phase diagrams, currents and bulk densities are obtained using a mean-field approximation and computer simulations. The presence of three stationary phases (low-density, high-density, and maximal current) are identified. A comparison on the stationary current with the Bridge model [M.R. Evans, et al., Phys. Rev. Lett. 74 (1995) 208] has shown that our model can enhance the current. The theoretical calculations are well supported by Monte Carlo simulations.     

Some Properties of k-Step Exclusion Processes

We introduce k-step exclusion processes as generalizations of the simple exclusion process. We state their main equilibrium properties when the underlying stochastic matrix corresponds to a random walk or is positive recurrent and reversible. Finally, we prove laws of large numbers for tagged and second-class particles     

Spectral Degeneracies in the Totally Asymmetric Exclusion Process

We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at arbitrary filling. Although the system does not possess obvious symmetries except translation invariance, the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules. This behaviour is explained by a hidden symmetry property of the Bethe Ansatz. Assuming a one-to-one correspondence between the solutions of the Bethe equations and the eigenmodes of the Markov matrix, we derive combinatorial formulae for the orders of degeneracy and the number of multiplets. These results are confirmed by exact diagonalisations of small size systems. This unexpected structure of the TASEP spectrum suggests the existence of an underlying large invariance group.     

Order of Current Variance and Diffusivity in the Rate One Totally Asymmetric Zero Range Process

We prove that the variance of the current across a characteristic is of order t ^2/3 in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t ^1/3. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed in Balázs and Seppäläinen (Order of current variance and diffusivity in the asymmetric simple exclusion process, 2006) for simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t ^2/3-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.     

Semi-infinite TASEP with a Complex Boundary Mechanism

We consider a totally asymmetric exclusion process on the positive half-line. When particles enter the system according to a Poisson source, Liggett has computed all the limit distributions when the initial distribution has an asymptotic density. In this paper we consider systems for which particles enter according to a complex mechanism depending on the current configuration in a finite neighborhood of the origin. For this kind of models, we prove a strong law of large numbers for the number of particles which have entered the system at a given time. Our main tool is a new representation of the model as a multi-type particle system with infinitely many particle types.     

The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

In this paper we find explicit formulas for: (1) Green’s function for a system of one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.     

A family of discrete-time exactly-solvable exclusion processes on a one-dimensional lattice

A two-parameter family of discrete-time exactly-solvable exclusion processes on a one-dimensional lattice is introduced, which contains the asymmetric simple exclusion process and the drop-push model as particular cases. The process is rewritten in terms of boundary conditions, and the conditional probabilities are calculated using the Bethe-ansatz. This is the discrete-time version of the continuous-time processes already investigated in [1-3]. The drift- and diffusion-rates of the particles are also calculated for the two-particle sector.     

Exclusive Queueing Process with Discrete Time

In a recent study (Arita in Phys. Rev. E 80(5):051119, 2009), an extension of the M/M/1 queueing process with the excluded-volume effect as in the totally asymmetric simple exclusion process (TASEP) was introduced. In this paper, we consider its discrete-time version. The update scheme we take is the parallel one. A stationary-state solution is obtained in a slightly arranged matrix product form of the discrete-time open TASEP with the parallel update. We find the phase diagram for the existence of the stationary state. The critical line which separates the parameter space into regions with and without the stationary state can be written in terms of the stationary current of the open TASEP. We calculate the average length of the system and the average number of particles.     

Determinant Representation for Some Transition Probabilities in the TASEP with Second Class Particles

We study the transition probabilities for the totally asymmetric simple exclusion process (TASEP) on the infinite integer lattice with a finite, but arbitrary number of first and second class particles. Using the Bethe ansatz we present an explicit expression of these quantities in terms of the Bethe wave function. In a next step it is proved rigorously that this expression can be written in a compact determinantal form for the case where the order of the first and second class particles does not change in time. An independent geometrical approach provides insight into these results and enables us to generalize the determinantal solution to the multi-class TASEP.     

Strong Asymmetric Limit of the Quasi-Potential of the Boundary Driven Weakly Asymmetric Exclusion Process

We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasi-potential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida [16]. We consider the strong asymmetric limit of the quasi-potential and recover the functional derived by Derrida, Lebowitz, and Speer [15] for the asymmetric exclusion process.     

Existence of spontaneous symmetry breaking in two-lane totally asymmetric simple exclusion processes with an intersection

We study two-lane totally asymmetric simple exclusion processes(TASEPs) with an intersection. Monte Carlo simulations show that only symmetric phases exist in the system. To verify the existence of asymmetric phases, we carry out a cluster mean-field analysis. Analytical results show that the densities of the two upstream segments of the intersection site are always equal, which indicates that the system is not in asymmetric phases. It demonstrates that the spontaneous symmetry breaking does not exist in the system. The density profiles and the boundaries of the symmetric phases are also investigated. We find that the cluster mean-field analysis shows better agreement with simulations than the simple mean-field analysis where the correlation of sites is ignored.