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arXiv:cond-mat/9908327 · Journal of Physics A Mathematical and General

Partially asymmetric simple exclusion model in the presence of an impurity on a ring

Farhad H Jafarpour

Published Feb 28, 2000• 11 citations
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Abstract

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.

DOI
10.1088/0305-4470/33/9/306

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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.
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.
We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, ppp and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}$. The model contains as special cases the TASEP with parallel update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=0$, and with sequential backward-ordered update, when $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}=p$. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}>p$ or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}<p$. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}\ensuremath{\rightarrow}1$, when the stationary configurations contain just one cluster.
We apply the Matrix Product Ansatz to study the Totally Asymmetric Simple Exclusion Process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, ppp and p~\tilde{p}p~​. The model contains as special cases the TASEP with parallel update, when p~=0\tilde{p} =0p~​=0, and with sequential backward-ordered update, when p~=p\tilde{p} =pp~​=p. We construct a two-dimensional matrix-product representation and use it to obtain exact finite-size expressions for the partition function, the current of particles and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether p~>p\tilde{p} >pp~​>p or p~<p\tilde{p} < pp~​<p. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation p~→1\tilde{p}\rightarrow 1p~​→1, when the stationary configurations contain just one cluster.
We investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case, operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by Arita et al (2011 J. Phys. A: Math. Theor. 44 335004). The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In this work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case.
A multi-species generalization of the asymmetric simple exclusion process is studied with ordered sequential and sub-lattice parallel updating schemes. In this model, particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability. Using the matrix product ansatz technique, we obtain the relevant algebra and study the uncorrelated stationary state of the model both for an open system and on a ring. A complete comparison between the physical results in these updates and those of random sequential introduced in Karimipour V 1999 A multi-species ASEP and its relation to traffic flow Phys. Rev. E 59 205 and Karimipour V 1998 A multi-species ASEP, steady state and correlation functions on a periodic lattice Preprint cond-mat/9809193, is made.
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.