In a recent study, (Jain et al 2007 Phys. Rev. Lett. 99 190601), a symmetric exclusion process with time-dependent hopping rates was introduced. Using simulations and a perturbation theory, it was shown that if the hopping rates at two neighboring sites of a closed ring vary periodically in time and have a relative phase difference, there is a net DC current which decreases inversely with the system size. In this work, we simplify and generalize our earlier treatment. We study a model where hopping rates at all sites vary periodically in time, and show that for certain choices of relative phases, a DC current of order unity can be obtained. Our results are obtained using a perturbation theory in the amplitude of the time-dependent part of the hopping rate. We also present results obtained in a sudden approximation that assumes large modulation frequency.
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 analyze the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean field limit. The mean field equations for particle densities are written in terms of Ricatti equations with the steady-state current $J$ as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents $J$ are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for $J$ from asymptot…
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…
The exclusion process in which particles may jump any distance l>=1 with the probability that decays as l^-(1+sigma) is studied from coarse-grained equation for density profile in the limit when the lattice spacing goes to zero. For 1<sigma<2, the usual diffusion term of this equation is replaced by the fractional one, which affects dynamical-scaling properties of the late-time approach to the stationary state. When applied to an open system with totally asymmetric hopping, this approach gives two results: first, it accounts for the sigma-dependent exponent that characterizes the algebraic dec…
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…
Generalization of the one-dimensional totally asymmetric exclusion process (TASEP) with open boundary conditions in which particles are allowed to jump $l$ sites ahead with the probability $p_l\sim 1/l^{σ+1}$ is studied by Monte Carlo simulations and the domain-wall approach. For $σ>1$ the standard TASEP phase diagram is recovered, but the density profiles near the transition lines display new features when $1<σ<2$. At the first-order transition line, the domain-wall is localized and phase separation is observed. In the maximum-current phase the profile has an algebraic decay with a $σ$-depend…
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 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 …
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…
We investigate the role of the boundary in the symmetric simple exclusion process with competing nonlocal and local hopping events. With open boundaries, the system undergoes a first order phase transition from a finite density phase to an empty road phase as the nonlocal hopping rate increases. Using a cluster stability analysis, we determine the location of such an abrupt nonequilibrium phase transition, which agrees well with numerical results. Our cluster analysis provides a physical insight into the mechanism behind this transition. We also explain why the transition becomes discontinuous…
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…
A multi-species generalization of the Asymmetric Simple Exclusion Process (ASEP) has been considered in the presence of a single impurity on a ring. The model describes particles hopping in one direction with stochastic dynamics and hard core exclusion condition. The ordinary particles hop forward with their characteristic hopping rates and fast particles can overtake slow ones with a relative rate. The impurity, which is the slowest particle in the ensemble of particles on the ring, hops in the same direction of the ordinary particles with its intrinsic hopping rate and can be overtaken by or…
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…
An asymmetric exclusion model on an open chain with random rates for hopping particles, where overtaking is also possible, is studied numerically and by computer simulation. The phase structure of the model and the density profiles near the high and low density coexistence line are obtained. The effect of a site impurity is also studied.
We consider a multi-species generalization of the Asymmetric Simple Exclusion Process on an open chain, in which particles hop with their characteristic hopping rates and fast particles can overtake slow ones. The number of species is arbitrary and the hopping rates can be selected from a discrete or continuous distribution. We determine exactly the phase structure of this model and show how the phase diagram of the 1-species ASEP is modified. Depending on the distribution of hopping rates, the system can exist in a three-phase regime or a two phase regime. In the three- phase regime the phase…
We verify the finite time fluctuation theorem for a linear Ising chain at its ends in contact with heat reservoirs. Analytic results are derived for a chain consisting of only two spins. The system can be mapped onto a model for particle transport, namely the symmetric exclusion process, in contact with thermal and particle reservoirs. We modify the symmetric exclusion process to represent a thermal engine and reproduce universal features of the efficiency at maximum power.
We study symmetric simple exclusion processes (SSEP) on a ring in the presence of uniformly moving multiple defects or disorders - a generalization of the model proposed earlier [Phys. Rev. E 89, 022138 (2014)]. The defects move with uniform velocity and change the particle hopping rates locally. We explore the collective effects of the defects on the spatial structure and transport properties of the system. We also introduce an SSEP with ordered sequential (sitewise) update and elucidate the close connection with our model.
The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.
We investigate the nature of the dynamically inactive phase of a simple symmetric exclusion process on a ring. We find that as the system's activity is tuned to a lower-than-average value the particles progressively lump into a single cluster, thereby forming a kink in the density profile. All dynamical regimes, and their finite size range of validity, are explicitly determined.
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…
For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current $Q_t$ during time $t$ through the origin when, in the initial condition, the sites are occupied with density $ρ_a$ on the negative axis and with density $ρ_b$ on the positive axis. All the cumulants of $Q_t$ grow like $\sqrt{t}$. In the range where $Q_t \sim \sqrt{t}$, the decay $\exp [-Q_t^3/t]$ of the distribution of $Q_t$ is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities recently derived by Tracy and Widom for exclusion process…
We consider single-file diffusion in an open system with two species $A,B$ of particles. At the boundaries we assume different reservoir densities which drive the system into a non-equilibrium steady state. As a model we use an one-dimensional two-component simple symmetric exclusion process with two different hopping rates $D_A,D_B$ and open boundaries. For investigating the dynamics in the hydrodynamic limit we derive a system of coupled non-linear diffusion equations for the coarse-grained particle densities. The relaxation of the initial density profile is analyzed by numerical integration…
A class of generalized exclusion processes parametrized by the maximal occupancy, $k\geq 1$, is investigated. For these processes with symmetric nearest-neighbor hopping, we compute the diffusion coefficient and show that it is independent on the spatial dimension. In the extreme cases of $k=1$ (simple symmetric exclusion process) and $k=\infty$ (non-interacting symmetric random walks) the diffusion coefficient is constant; for $2\leq k<\infty$, the diffusion coefficient depends on the density and the maximal occupancy $k$. We also study the evolution of a tagged particle. It exhibits a diffus…
We investigate the growth of the total number of particles in a symmetric exclusion process driven by a localized source. The average total number of particles entering an initially empty system grows with time as t^{1/2} in one dimension, t/log(t) in two dimensions, and linearly in higher dimensions. In one and two dimensions, the leading asymptotic behaviors for the average total number of particles are independent on the intensity of the source. We also discuss fluctuations of the total number of particles and determine the asymptotic growth of the variance in one dimension.
We study the one dimensional partially asymmetric simple exclusion process (ASEP) with open boundaries, that describes a system of hard-core particles hopping stochastically on a chain coupled to reservoirs at both ends. Derrida, Evans, Hakim and Pasquier [J. Phys. A 26, 1493 (1993)] have shown that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra, closely related to the deformed oscillator-algebra. We construct all finite dimensional irreducible representations of this algebra. This enables us to compute the stationary bulk density as …
We give numerical evidence that the location of the first order phase transition between the low and the high density phases of the one dimensional asymmetric simple exclusion process with open boundaries becomes sample dependent when quenched disorder is introduced for the hopping rates.
The Symmetric Exclusion Process (SEP), in which particles hop symmetrically on a discrete line with hard-core constraints, is a paradigmatic model of subdiffusion in confined systems. This anomalous behavior is a direct consequence of strong spatial correlations induced by the requirement that the particles cannot overtake each other. Even if this fact has been recognised qualitatively for a long time, up to now there is no full quantitative determination of these correlations. Here we study the joint probability distribution of an arbitrary number of tagged particles in the SEP. We determine …
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…
The dynamics of an asymmetric tracer in the symmetric simple exclusion process (SEP) is mapped, in the continuous scaling limit, to the local current through the origin in the zero-range process (ZRP) with a biased bond. This allows us to study the hydrodynamics of the SEP with an asymmetric tracer with a step initial condition, leading to the average displacement as a function of the bias and the densities on both sides. We then derive the cumulant generating function of the process in the high-density limit, by using the Macroscopic Fluctuation Theory and obtain agreement with the microscopi…
An exact mapping is established between sequence alignment, one of the most commonly used tools of computational biology, and the asymmetric exclusion process, one of the few exactly solvable models of nonequilibrium physics. The statistical significance of sequence alignments is characterized through studying the total hopping current of the discrete time and space version of the asymmetric exclusion process.
The relation between noise and Fick's diffusion coefficient in barrier limited transport associated with hopping or tunneling mechanisms of particles obeying the asymmetric simple exclusion processes (ASEP) is physically assessed by Monte Carlo simulations. For a closed ring consisting of a large number of barriers the diffusion coefficient is related explicitly to the current noise thus revealing the existence of a generalized Nyquist-Einstein relation. Both diffusion and noise are confirmed to decrease as the square root of the number of barriers as a consequence of the correlation induced b…
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 …
The Symmetric Exclusion Process (SEP), where particles hop on a 1D lattice with the restriction that there can only be one particle per site, is a paradigmatic model of interacting particle systems. Recently, it has been shown that the nature of the initial conditions - annealed or quenched - has a quantitative impact on the long-time properties of tracer diffusion. However, so far, all the studies in the quenched case focused on the low-density limit of the SEP. Here, we derive the cumulant generating function of the tracer position in the dense limit with quenched initial conditions. Importa…
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 …
A lattice gas with infinite repulsion between particles separated by $\leq 1$ lattice spacing, and nearest-neighbor hopping dynamics, is subject to a drive favoring movement along one axis of the square lattice. The equilibrium (zero drive) transition to a phase with sublattice ordering, known to be continuous, shifts to lower density, and becomes discontinuous for large bias. In the ordered nonequilibrium steady state, both the particle and order-parameter densities are nonuniform, with a large fraction of the particles occupying a jammed strip oriented along the drive. The relaxation exhibit…
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…
The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non--equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non--equilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so called quasi potential,…
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…
In the boundary driven symmetric simple exclusion process consider an open set $\ms O$ of density profiles which does not contain the stationary density profile. We prove that the first time the empirical measure visits the set $\ms O$ converges to an exponential distribution.
We examine the entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes. In contrast with the Gibbs--Shannon entropy \cite{B, DLS2}, the entropy of nonequilibrium stationary states differs from the entropy of local equilibrium states.
In this article, we consider a one-dimensional symmetric exclusion process in weak contact with reservoirs at the boundary. In the diffusive time-scaling the empirical measure evolves according to the heat equation with Robin boundary conditions. We prove the associated dynamical large deviations principle.
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…
Using the Bethe ansatz we obtain in a determinant form the exact solution of the master equation for the conditional probabilities of the totally asymmetric exclusion process with particle-dependent hopping rates on Z. From this we derive a determinant expression for the time-integrated current for a step-function initial state.
Tracer dynamics in the Symmetric Exclusion Process, where hardcore particles diffuse on an infinite one-dimensional lattice, is a paradigmatic model of anomalous diffusion. While the equilibrium situation has received a lot of attention, the case where the tracer is driven by an external force, which provides a minimal model of nonequilibrium transport in confined crowded environments, remains largely unexplored. Indeed, the only available analytical results concern the means of both the position of the tracer and the lattice occupation numbers in its frame of reference, and higher-order momen…
We consider the totally asymmetric exclusion process on a ring in discrete time with the backward-ordered sequential update and particle-dependent hopping probabilities. Using a combinatorial treatment of the Bethe ansatz, we derive the determinant expression for the non-stationary probability of transitions between particle configurations. In the continuous-time limit, we find a generalization of the recent result, obtained by A. Rákos and G.M. Schütz for infinite lattice, to the case of ring geometry.
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.
In the universality class of the one dimensional Kardar-Parisi-Zhang surface growth, Derrida and Lebowitz conjectured the universality of not only the scaling exponents, but of an entire scaling function. Since Derrida and Lebowitz's original publication [PRL 80 209 (1998)] this universality has been verified for a variety of continuous time, periodic boundary systems in the KPZ universality class. Here, we present a numerical method for directly examining the entire particle flux of the asymmetric exclusion process (ASEP), thus providing an alternative to more difficult cumulant ratios studie…
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.
