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arXiv:hep-lat/9303011 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Time-dependent correlation functions in a one-dimensional asymmetric exclusion process

Gunter Markus Schutz

Published Jun 1, 1993• 89 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.47.4265

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The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.

Physical Review E•Mar 22, 2005•12 citations
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.
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.
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.
We consider an exclusion process on a ring in which a particle hops to an empty neighboring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well-known mapping of this model to the zero-range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n)=1+(b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b>2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b-2. We also calculate the two-point correlation function above the critical density and find that the correlation length diverges with a critical exponent ν=1/(b-2) for b<3 and 1 for b>3. These results are compared with those obtained using an exact series expansion for finite systems.
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
The applicability of the concepts of finite-size scaling and universality to nonequilibrium phase transitions is considered in the framework of the one-dimensional totally asymmetric simple-exclusion process with open boundaries. In the thermodynamic limit there are boundary-induced transitions both of the first and second order between steady-state phases of the model. We derive finite-size scaling expressions for the current near the continuous phase transition and for the local density near the first-order transition under different stochastic dynamics and compare them to establish the existence of universal functions. Next we study numerically the finite-size behavior of the Lee-Yang zeros of the normalization factor for the different steady-state probabilities.