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arXiv:0903.5447 · Journal of Statistical Mechanics Theory and Experiment

Exact matrix-product states for parallel dynamics: open boundaries and excess mass on the ring

Marko Woelki, Michael Schreckenberg

Published May 21, 2009• 19 citations
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Abstract

In this paper it is shown that the steady state weights of the asymmetric simple exclusion process (ASEP) with open boundaries and parallel update can be written as a product of a scalar pair-factorized and a matrix-product state. This type of state is also obtained for an ASEP on a ring in which particles can move one or two sites. The dynamics leads to the formation of an excess hole that plays the role of a defect. In the continuous-time limit the process recovers the ASEP with a single defect particle. The phase diagram shows a first-order phase transition between two phases with different defect velocities. These are calculated exactly from the process-generating function.

DOI
10.1088/1742-5468/2009/05/p05014

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In this paper it is shown that the steady state weights of the asymmetric simple exclusion process (ASEP) with open boundaries and parallel update can be written as a product of a scalar pair-factorized and a matrix-product state. This type of state is also obtained for an ASEP on a ring in which particles can move one or two sites. The dynamics leads to the formation of an excess hole that plays the role of a defect. In the continuous-time limit the process recovers the ASEP with a single defect particle. The phase diagram shows a first-order phase transition between two phases with different defect velocities. These are calculated exactly from the process-generating function.
A totally asymmetric exclusion process on a ring is investigated in which particles can move one or two sites. Special attention is spent to the high-speed case where particles are not allowed to move a single site if they could move two sites. The stationary state is calculated exactly in the framework of the matrix-product ansatz. Independently of the update this process evolves into subspace of the configuration space and can lead to the formation of an excess hole. One observes two phases where its velocity takes different values which are calculated exactly from the normalization-generating function. Numerically computed density profiles show an interesting algebraic form as a limit of a shock profile. For continuous time the process turns out to be related to the ASEP with a single defect particle and for synchronous update it leads to a natural defect dynamics. For the general definition of the process with maximum velocity two, from an exact analysis of the two-particle sector we find that the distribution of inter-particle distance can oscillate. This is underlined in the thermodynamic limit by an improved mean-field theory which shows a remarkably good agreement with simulations. The main focus of this work is the exact solution of lattice models by matrix-product ansatz. We finally obtain stationary states also for related processes. Especially considerable is a new formulation of steady states for parallel dynamics like the ASEP with open boundaries as a product of a pair-factorized and a matrix-product state.
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 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.
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 analytically study the one-dimensional asymmetric simple exclusion process with open boundaries under sublattice-parallel updating scheme. We investigate the stationary state properties of this model conditioned on finding a given particle number in the system. Recent numerical investigations have shown that the model possesses three different phases in this case. Using a matrix product method we calculate both the exact canonical partition function and also density profiles of the particles in each phase. Application of the Yang–Lee theory reveals that the model undergoes two second-order phase transitions at critical points. These results confirm the correctness of our previous numerical studies.
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.