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arXiv:cond-mat/9910270 · Journal of the Physical Society of Japan

Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries

Tomohiro Sasamoto

Published Apr 15, 2000• 43 citations
arXivHTMLPDF

Abstract

The one-dimensional partially asymmetric simple exclusion process with open boundaries is considered. The stationary state, which is known to be constructed in a matrix product form, is studied by applying the theory of q-orthogonal polynomials. Using a formula of the q-Hermite polynomials, the average density profile is computed in the thermodynamic limit. The phase diagram for the correlation length, which was conjectured in the previous work[J. Phys. A {\bf 32} (1999) 7109], is confirmed.

DOI
10.1143/jpsj.69.1055

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The one-dimensional partially asymmetric simple exclusion process with open boundaries is considered. The stationary state, which is known to be constructed in a matrix product form, is studied by applying the theory of q-orthogonal polynomials. Using a formula of the q-Hermite polynomials, the average density profile is computed in the thermodynamic limit. The phase diagram for the correlation length, which was conjectured in the previous work[J. Phys. A {\bf 32} (1999) 7109], is confirmed.
The stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered. The so-called matrix product ansatz is employed. This enables us to construct the stationary state in the form of the product of the matrices D and E . Noticing the fact that the matrix C ( = D + E ) for the model is related to certain q -orthogonal polynomials, the model is analysed for a wide range of parameters. The current and the correlation length are evaluated in the thermodynamic limit. It turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case.
An exact and rigorous calculation of the current and density profile in the steady state of the one-dimensional fully asymmetric simple-exclusion process with open boundaries and forward-ordered sequential dynamics is presented. The method is based on a matrix product representation of the steady-state probability distribution. The main idea is to choose a suitable representation in which the scalar products describing the current and local density profile for a chain of arbitrary finite size depend only on the elements in a finite number of rows and columns. This makes possible the use of a truncated finite-dimensional representation of the matrices and vectors involved. After performing the calculations, we lift the truncation by letting its dimensionality go to infinity. In this limit the results become exact for any size of the chain. By rescaling one of the infinite-dimensional matrix representations found in the work of Derrida et al. [J. Phys. A 26, 1493 (1993)] for their algebra, we obtain a symmetric ``propagator'' matrix. Its truncated version is diagonalized by orthogonal transformation for easy calculation of the relevant scalar products. An interpretation of the phase transitions between the different phases is given in terms of eigenvalue splitting from a bounded quasicontinuous spectrum. A precise description of the local density profiles is given for all values of the parameters. It is shown that the leading-order asymptotic form of the position-dependent terms in the local density changes within the low- and high-density phases, signaling the presence of a higher-order transition.
We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida et al the stationary probability measure can be expressed as a matrix-product state involving two matrices forming a Fock-like representation of a general quadratic algebra. We obtain the representations of this algebra, which were unknown in the mathematical literature and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin- Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction - diffusion models are discussed.
We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the "slower" species but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.
In recent years, one-dimensional driven diffusion models consisting of particles on a lattice have been studied. The most famous topics of these models are solvability and phase transitions. For example, the exact stationary-state solution of the asymmetric exclusion process (ASEP) with an open boundary condition can be obtained as a matrix product form and its physical quantities exhibit boundary-induced phase transitions in the thermodynamic limit. There are three regimes, namely, maximal-current (MC) phase, low-density (LD) phase and high-density (HD) phase in the phase diagram. The two-species models or three-state models are also interesting. The stationary-state solutions of the two-species totally asymmetric exclusion process (TASEP) with a periodic boundary condition and an open boundary condition can be obtained as matrix product forms. The model in ref. 6 exhibits two phases in which symmetry breaks spontaneously. Furthermore, the model in ref. 7 exhibits phase separation. In this note, the two-species TASEP with a new open boundary condition different from the condition in ref. 6 is introduced and we obtain the stationary-state solution exactly as a matrix product form and calculate the physical quantities in the thermodynamic limit. Let us consider a stochastic process on an L-site chain with two-species particles. We note j 1⁄4 0 if site j is empty, j 1⁄4 1 if site j is occupied by the first-class particle and j 1⁄4 2 if site j is occupied by the second-class particle. The exchange rule in the bulk of the chain is expressed as 10 ! 01 with a rate 1; ð1Þ 20 ! 02 with a rate 1; ð2Þ 12 ! 21 with a rate 1: ð3Þ
We continue our studies [J. Stat. Phys. 71, 485 (1993)] of a one-dimensional anisotropic exclusion process with parallel dynamics describing particles moving to the right on a chain of L sites. Instead of considering periodic boundary conditions with a defect as in our studies, we study open boundary conditions with injection of particles with rate \ensuremath{\alpha} at the origin and absorption of particles with rate \ensuremath{\beta} at the boundary. We construct the steady state and compute the density profile as a function of \ensuremath{\alpha} and \ensuremath{\beta}. In the large-L limit we find a high-density phase (\ensuremath{\alpha}>\ensuremath{\beta}) and a low-density phase (\ensuremath{\alpha}). In both phases, the density distribution along the chain approaches its respective constant bulk value exponentially on a length scale \ensuremath{\xi}. They are separated by a phase-transition line where \ensuremath{\xi} diverges and where the density increases linearly with the distance from the origin. Furthermore, we present exact expressions for all equal-time n-point density correlation functions and for the time-dependent two-point function in the steady state. We compare our results with predictions from local dynamical scaling and discuss some conjectures for other exclusion models.