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Journal of Physics A Mathematical and General

A matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process

Cédric Boutillier, Paul Fran ois, Kirone Mallick, Shamlal Mallick

Published Nov 7, 2002• 27 citations
Publisher

Abstract

We study the fluctuations of the position of an impurity in the asymmetric exclusion process on a ring with an arbitrary number of particles and holes. The steady state of this model is exactly known and four different phases appear in the limit of a large system. We calculate the diffusion constant of the impurity by using a matrix product method and also obtain a representation for unequal time correlation functions. We show that our results found by the matrix ansatz agree with those obtained previously by the Bethe ansatz.

DOI
10.1088/0305-4470/35/46/301

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We study the fluctuations of the position of an impurity in the asymmetric exclusion process on a ring with an arbitrary number of particles and holes. The steady state of this model is exactly known and four different phases appear in the limit of a large system. We calculate the diffusion constant of the impurity by using a matrix product method and also obtain a representation for unequal time correlation functions. We show that our results found by the matrix ansatz agree with those obtained previously by the Bethe ansatz.
The asymmetric exclusion process on a ring in one dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second-class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover, the variance (diffusion constant) grows as L 1/2 where L is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.
Abstract: The asymmetric exclusion process on a ring in one-dimension is considered with a single defect particle. The steady state has previously been solved by a matrix product method. Here we use the Bethe ansatz to solve exactly for the long time limit behaviour of the generating function of the distance travelled by the defect particle. This allows us to recover steady state properties known from the matrix approach such as the velocity, and obtain new results such as the diffusion constant of the defect particle. In the case where the defect particle is a second class particle we determine the large deviation function and show that in a certain range the distribution of the distance travelled about the mean is Gaussian. Moreover the variance (diffusion constant) grows as L 1/2 where L is the system size. This behaviour can be related to the superdiffusive spreading of excess mass fluctuations on an infinite system. In the case where the defect particle produces a shock, our expressions for the velocity and the diffusion constant coincide with those calculated previously for an infinite system by Ferrari and Fontes.
We study a recently introduced model by Arndt et al which consists of positive and negative particles on a ring. The positive (negative) particles hop clockwise (anticlockwise) with unit rate and oppositely charged particles may swap their positions with asymmetric rates q and unity. In this paper we assume that a finite density of positively charged particles ρ and only one negative particle (which plays the role of an impurity) exist on the ring. It turns out that the canonical partition function of this model can be calculated exactly using matrix product ansatz formalism. In the limit of infinite system size and infinite number of positive particles, we can also derive exact expressions for the speed of the positive and negative particles, which show a second-order phase transition at q c = 2ρ. The density profile of the positive particles on the ring has a shock structure for q ⩽ q c and an exponential behaviour with correlation length ξ for q ⩾ q c . It will be shown that the mean-field results become exact at q = 3 and no phase transition occurs for q >2.
A totally asymmetric exclusion process on a ring with ν nonconserved internal degrees of freedom, where particles hop forward with a rate that depends on their internal state, has been studied. We show, using a mapping of the model to a zero range process with ν different kinds of boxes, that steady-state weights can be written in a matrix product form and calculate the spatial correlations exactly. A comparison of the model with an equivalent conserved system reveals that unequal hopping rates of particles belonging to different internal states are responsible for the nontrivial correlations.
We study a generalized two-species model on a ring. The original model (Mallick K 1996 J. Phys. A: Math. Gen. 29 5375) describes ordinary particles hopping exclusively in one direction in the presence of an impurity. The impurity hops with a rate different from that of ordinary particles and can be overtaken by them. Here we let the ordinary particles also hop backward with rate q . Using the matrix product ansatz we obtain the relevant quadratic algebra. A finite-dimensional representation of this algebra enables us to compute the stationary bulk density of the ordinary particles, as well as the speed of impurity on a set of special surfaces of the parameter space. We will obtain the phase structure of this model in the accessible region and show how the phase structure of the original model is modified. In the infinite-volume limit this model presents a shock in one of its phases.
The one-dimensional totally asymmetric simple exclusion process, a Markov process describing classical hard-core particles hopping in the same direction, is considered on a periodic lattice of L sites. The relaxation to the nonequilibrium steady state, which occurs on the time scale t∼L^{3/2} for large L, is studied for the half-filled system with N=L/2 particles. Using large L asymptotics of Bethe ansatz formulas for the eigenstates, exact expressions depending explicitly on the rescaled time t/L^{3/2} are obtained for the average and two-point function of the local density, and for the current fluctuations for simple (stationary, flat and step) initial conditions, relating previous results for the infinite system to stationary large deviations. The final formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.