Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9603172 · Journal of Physics A Mathematical and General

The asymmetric exclusion model with sequential update

Nikolaus Rajewsky, Andreas Schadschneider, Michael Schreckenberg

Published Jun 21, 1996• 60 citations
arXivHTMLPDF

Abstract

We present a solution for the stationary state of an asymmetric exclusion model with sequential update and open boundary conditions. We solve the model exactly for random hopping in both directions by applying a matrix-product formalism which was recently used to solve the model with sublattice-parallel update (Hinrichsen H 1995 Weizmann Inst. Preprint cond-mat/9512172). It is shown that the matrix-algebra describing the sequential update and sublattice-parallel update are identical and can be mapped onto the random sequential case treated by Derrida et al (Derrida B, Evans M R, Hakim V and Pasquier V 1993 J. Phys. A: Math. Gen. 26 1493).

DOI
10.1088/0305-4470/29/12/004

Related papers

Multi-species asymmetric exclusion process in ordered sequential update

M. Ebrahim Fouladvand, Farhad H Jafarpour

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.

arXiv:cond-mat/9901007•Journal of Physics A Mathematical and General•Jul 29, 1999•9 citations

Temporal evolution of product shock measures in the totally asymmetric simple exclusion process with sublattice-parallel update

Farhad H Jafarpour, Seyedeh Raziyeh Masharian

It is known that when the steady state of a one-dimensional multispecies system, which evolves via a random-sequential updating mechanism, is written in terms of a linear combination of Bernoulli shock measures with random-walk dynamics, it can be equivalently expressed as a matrix-product state. In this case the quadratic algebra of the system always has a two-dimensional matrix representation. Our investigations show that this equivalence exists at least for the systems with deterministic sublattice-parallel update. In this paper we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries and sublattice-parallel update as an example.

arXiv:0902.4047•Physical Review E•May 22, 2009•19 citations

Exact shock profile for the ASEP with sublattice-parallel update

Farhad H Jafarpour, F E Ghafari, S R Masharian

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.

arXiv:cond-mat/0504397•Journal of Physics A Mathematical and General•May 10, 2005•16 citations

Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring

Boyka Aneva, J.G. Brankov

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, $p$ 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.

Physical review. E•Aug 25, 2016•11 citations

The totally asymmetric exclusion process with generalized update

A E Derbyshev, S S Poghosyan, Alexander M. Povolotsky +1 more

We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for N particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.

arXiv:1203.0902•Journal of Statistical Mechanics Theory and Experiment•May 16, 2012•21 citations

Exact results for a fully asymmetric exclusion process with sequential dynamics and open boundaries

J.G. Brankov, Nina Pesheva, Nikola Valkov

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.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 1, 2000•19 citations
We present a solution for the stationary state of an asymmetric exclusion model with sequential update and open boundary conditions. We solve the model exactly for random hopping in both directions by applying a matrix-product formalism which was recently used to solve the model with sublattice-parallel update (Hinrichsen H 1995 Weizmann Inst. Preprint cond-mat/9512172). It is shown that the matrix-algebra describing the sequential update and sublattice-parallel update are identical and can be mapped onto the random sequential case treated by Derrida et al (Derrida B, Evans M R, Hakim V and Pasquier V 1993 J. Phys. A: Math. Gen. 26 1493).
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.
It is known that when the steady state of a one-dimensional multispecies system, which evolves via a random-sequential updating mechanism, is written in terms of a linear combination of Bernoulli shock measures with random-walk dynamics, it can be equivalently expressed as a matrix-product state. In this case the quadratic algebra of the system always has a two-dimensional matrix representation. Our investigations show that this equivalence exists at least for the systems with deterministic sublattice-parallel update. In this paper we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries and sublattice-parallel update as an example.
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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{p}$. 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 consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for N particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.
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.