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

Farhad H Jafarpour, Seyedeh Raziyeh Masharian

Published Nov 27, 2012• 28 citations
OpenAlex

Abstract

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

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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
It is known that exact traveling 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 always has a two-dimensional representation with a generic structure. A couple of examples for the n = 1 and 2 cases are presented.
We consider a generalized coagulation-decoagulation system on a one-dimensional discrete lattice with reflecting boundaries. It is known that a Bernoulli shock measure with two shock fronts might have a simple random-walk dynamics, provided that some constraints on the microscopic reaction rates of this system are fulfilled. Under these constraints the steady state of the system can be written as a linear superposition of such shock measures. We show that the coefficients of this expansion can be calculated using the finite-dimensional representation of the quadratic algebra of the system obtained from a matrix-product approach.
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.
Abstract A one-dimensional, driven lattice gas with a freely moving, driven defect particle is studied. Although the dynamics of the defect are simply biased diffusion, it disrupts the local density of the gas, creating nontrivial nonequilibrium steady states. The phase diagram is derived using mean field theory and comprises three phases. In two phases, the defect causes small localized perturbations in the density profile. In the third, it creates a shock, with two regions at different bulk densities. When the hopping rates satisfy a particular condition (that the products of the rates of the gas and defect are equal), it is found that the steady state can be solved exactly using a two-dimensional matrix product ansatz. This is used to derive the phase diagram for that case exactly and obtain exact asymptotic and finite size expressions for the density profiles and currents in all phases. In particular, the front width in the shock phase on a system of size L is found to scale as L 1/2, which is not predicted by mean field theory. The results are found to agree well with Monte Carlo simulations.
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 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.