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arXiv:1308.5893 · Journal of Statistical Mechanics Theory and Experiment

A new family of exactly solvable disordered reaction–diffusion systems

Mohammad Ghadermazi, Farhad H Jafarpour

Published Sep 30, 2013• 4 citations
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Abstract

Using a matrix product method the steady-state of a family of disordered reaction-diffusion systems consisting of different species of interacting classical particles moving on a lattice with periodic boundary conditions is studied. A new generalized quadratic algebra and its matrix representations is introduced. The steady-states of two members of this exactly solvable family of systems are studied in detail.

DOI
10.1088/1742-5468/2013/09/p09023

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Using a matrix product method the steady-state of a family of disordered reaction-diffusion systems consisting of different species of interacting classical particles moving on a lattice with periodic boundary conditions is studied. A new generalized quadratic algebra and its matrix representations is introduced. The steady-states of two members of this exactly solvable family of systems are studied in detail.
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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.
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
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.
We consider a process in which there are two types of particles, A and B, on an infinite one-dimensional lattice. The particles hop to their adjacent sites, like the totally asymmetric exclusion process (ASEP), and have also the following interactions: A+B-->B+B and B+A-->B+B, which all occur with equal rate. We study this process by imposing four boundary conditions on the ASEP master equation. It is shown that this model is integrable, in the sense that its N-particle S matrix is factorized into a product of two-particle S matrices and, more importantly, the two-particle S matrix satisfies the quantum Yang-Baxter equation. Using the coordinate Bethe-ansatz, the N-particle wave functions and the two-particle conditional probabilities are found exactly. Further, by imposing four reasonable physical conditions on two-species diffusion-reaction processes (where the most important ones are the equality of the reaction rates and the conservation of the number of particles in each reaction), we show that among the 4096 types of interactions which have these properties and can be modeled by a master equation and an appropriate set of boundary conditions, there are only 28 independent interactions which are integrable. We find all these interactions and also their corresponding wave functions. Some of these may be new solutions of the quantum Yang-Baxter equation.