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/0211472 · Journal of Physics A Mathematical and General

Matrix product approach for the asymmetric random average process

Frank Zielen, Andreas Schadschneider

Published Mar 19, 2003• 22 citations
arXivHTMLPDF

Abstract

We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.

DOI
10.1088/0305-4470/36/13/306

Related papers

Matrix product representation of the stationary state of the open zero range process

Éric Bertin, Matthieu Vanicat

Abstract Many one-dimensional lattice particle models with open boundaries, like the paradigmatic asymmetric simple exclusion process (ASEP), have their stationary states represented in the form of a matrix product, with matrices that do not explicitly depend on the lattice site. In contrast, the stationary state of the open 1D zero-range process (ZRP) takes an inhomogeneous factorized form, with site-dependent probability weights. We show that in spite of the absence of correlations, the stationary state of the open ZRP can also be represented in a matrix product form, where the matrices are site-independent, non-commuting and determined from algebraic relations resulting from the master equation. We recover the known distribution of the open ZRP in two different ways: first, using an explicit representation of the matrices and boundary vectors; second, from the sole knowledge of the algebraic relations satisfied by these matrices and vectors. Finally, an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden Markov chains.

arXiv:1803.08275•Journal of Physics A Mathematical and Theoretical•May 1, 2018•4 citations

Towards a matrix product Ansatz in two dimensions

Chandraniva Guha Ray, Aikya Banerjee, P. K. Mohanty

Abstract Matrix product ansatz (MPA) is a powerful framework for constructing exact steady state weights of one dimensional non-equilibrium stochastic processes; but its generalization to higher dimensions is limited. Here, we introduce the MPA formalism for two dimensions (2D). As a concrete application, we introduce and exactly solve a non-conserved assisted exclusion model (NAEM) in one and two dimensions with constrained hopping and local birth–death dynamics: a particle can hop to a neighbouring site only when exactly one of its neighbouring sites is vacant, while creation and annihilation occur exclusively at sites whose neighbours are all occupied. The MPA yields exact steady-state weights and provides a systematic method to compute observables such as density moments and particle currents. In the particle-conserving limit, the system undergoes an absorbing phase transition at the critical density ρ c =1/2 with order-parameter exponent β=3. We further show that the steady state of the NAEM maps exactly onto the well-studied hard-square lattice gas with nearest-neighbour exclusion, thereby providing a nonequilibrium dynamical route to realizing equilibrium states of constrained lattice gases. Our work generalizes matrix-product methods beyond one dimension, establishing a systematic approach to exact solutions of interacting stochastic systems in 2D.

arXiv:2605.20160•Journal of Physics A Mathematical and Theoretical•Oct 23, 2026•0 citations

Exact results for an asymmetric annihilation process with open boundaries

Arvind Ayyer, Kirone Mallick

We consider a nonequilibrium reaction–diffusion model on a finite one-dimensional lattice with bulk and boundary dynamics inspired by the Glauber dynamics of the Ising model. We show that the model has a rich algebraic structure that we use to calculate its properties. In particular, we show that the Markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes. This remark leads us to devise a technique which we call the transfer matrix Ansatz that allows us to determine the steady-state distribution and correlation functions. Furthermore, we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of Markov matrices. Finally, we compare the transfer matrix Ansatz used here with the matrix product representation of the steady state of one-dimensional stochastic models.

arXiv:0910.0693•Journal of Physics A Mathematical and Theoretical•Jan 4, 2010•22 citations

Matrix product steady states as superposition of product shock measures in 1D driven systems

Farhad H Jafarpour, Seyedeh Raziyeh Masharian

It is known that exact travelling wave solutions exist for families of (n + 1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a twodimensional representation with a generic structure. A couple of examples for n = 1 and n = 2 cases are presented. On the macroscopic level nonlinear hydrodynamic equations, such as the Burgers equation or the Fisher equation, might exhibit shocks in some cases [1, 2]. On the microscopic level some stochastic non-equilibrium one-dimensional lattice models with open boundaries might also develop shocks provided that some constrained on the reaction rates are fulfilled [3, 4, 5]. In [6] exact travelling wave solutions have been obtained for three families of two-states drivendiffusive models in one-dimension with nearest-neighbors interactions. These three models are the Partially Asymmetric Simple Exclusion Process (PASEP) [7, 8, 9], the Branching-Coalescing Random Walk (BCRW) and the Asymmetric Kawasaki-Glauber Process (AKGP). By taking a Bernoulli shock distribution as initial distribution, the shock distribution in these models evolves in time into a linear combination of similar distributions with different shock positions. The time evolution of the shock position in these models is similar to the dynamics of a random walker moving on a one-dimensional lattice with reflecting boundaries [6]. This property of the travelling wave solution is not limited to the two-states systems with nearest-neighbor interactions. Shocks in threestates

•Nov 27, 2012•28 citations

On the ubiquity of matrix-product states in one-dimensional stochastic processes with boundary interactions

Kai Klauck, Andreas Schadschneider

Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.

•Jan 1, 1999•31 citations

Matrix product states for interacting particles without hardcore constraints

Amit Kumar Chatterjee, P. K. Mohanty

Abstract We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.

arXiv:1707.06123•Journal of Physics A Mathematical and Theoretical•Sep 27, 2017•3 citations
We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.
Abstract Many one-dimensional lattice particle models with open boundaries, like the paradigmatic asymmetric simple exclusion process (ASEP), have their stationary states represented in the form of a matrix product, with matrices that do not explicitly depend on the lattice site. In contrast, the stationary state of the open 1D zero-range process (ZRP) takes an inhomogeneous factorized form, with site-dependent probability weights. We show that in spite of the absence of correlations, the stationary state of the open ZRP can also be represented in a matrix product form, where the matrices are site-independent, non-commuting and determined from algebraic relations resulting from the master equation. We recover the known distribution of the open ZRP in two different ways: first, using an explicit representation of the matrices and boundary vectors; second, from the sole knowledge of the algebraic relations satisfied by these matrices and vectors. Finally, an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden Markov chains.
Abstract Matrix product ansatz (MPA) is a powerful framework for constructing exact steady state weights of one dimensional non-equilibrium stochastic processes; but its generalization to higher dimensions is limited. Here, we introduce the MPA formalism for two dimensions (2D). As a concrete application, we introduce and exactly solve a non-conserved assisted exclusion model (NAEM) in one and two dimensions with constrained hopping and local birth–death dynamics: a particle can hop to a neighbouring site only when exactly one of its neighbouring sites is vacant, while creation and annihilation occur exclusively at sites whose neighbours are all occupied. The MPA yields exact steady-state weights and provides a systematic method to compute observables such as density moments and particle currents. In the particle-conserving limit, the system undergoes an absorbing phase transition at the critical density ρ c =1/2 with order-parameter exponent β=3. We further show that the steady state of the NAEM maps exactly onto the well-studied hard-square lattice gas with nearest-neighbour exclusion, thereby providing a nonequilibrium dynamical route to realizing equilibrium states of constrained lattice gases. Our work generalizes matrix-product methods beyond one dimension, establishing a systematic approach to exact solutions of interacting stochastic systems in 2D.
We consider a nonequilibrium reaction–diffusion model on a finite one-dimensional lattice with bulk and boundary dynamics inspired by the Glauber dynamics of the Ising model. We show that the model has a rich algebraic structure that we use to calculate its properties. In particular, we show that the Markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes. This remark leads us to devise a technique which we call the transfer matrix Ansatz that allows us to determine the steady-state distribution and correlation functions. Furthermore, we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of Markov matrices. Finally, we compare the transfer matrix Ansatz used here with the matrix product representation of the steady state of one-dimensional stochastic models.
It is known that exact travelling wave solutions exist for families of (n + 1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a twodimensional representation with a generic structure. A couple of examples for n = 1 and n = 2 cases are presented. On the macroscopic level nonlinear hydrodynamic equations, such as the Burgers equation or the Fisher equation, might exhibit shocks in some cases [1, 2]. On the microscopic level some stochastic non-equilibrium one-dimensional lattice models with open boundaries might also develop shocks provided that some constrained on the reaction rates are fulfilled [3, 4, 5]. In [6] exact travelling wave solutions have been obtained for three families of two-states drivendiffusive models in one-dimension with nearest-neighbors interactions. These three models are the Partially Asymmetric Simple Exclusion Process (PASEP) [7, 8, 9], the Branching-Coalescing Random Walk (BCRW) and the Asymmetric Kawasaki-Glauber Process (AKGP). By taking a Bernoulli shock distribution as initial distribution, the shock distribution in these models evolves in time into a linear combination of similar distributions with different shock positions. The time evolution of the shock position in these models is similar to the dynamics of a random walker moving on a one-dimensional lattice with reflecting boundaries [6]. This property of the travelling wave solution is not limited to the two-states systems with nearest-neighbor interactions. Shocks in threestates
Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.
Abstract We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.