Finite-dimensional representation of the quadratic algebra of a generalized coagulation-decoagulation model

The steady-state of a generalized coagulation-decoagulation model on a one-dimensional lattice with reflecting boundaries is studied using a matrix-product approach. It is shown that the quadratic algebra of the model has a four-dimensional representation provided that some constraints on the microscopic reaction rates are fulfilled. The dynamics of a product shock measure with two shock fronts, generated by the Hamiltonian of this model, is also studied. It turns out that the shock fronts move on the lattice as two simple random walkers which repel each other provided that the same constraints on the microscopic reaction rates are satisfied.Comment: Minor revision

Repelling Random Walkers in a Diffusion-Coalescence System

We have shown that the steady state probability distribution function of a diffusion-coalescence system on a one-dimensional lattice of length L with reflecting boundaries can be written in terms of a superposition of double shock structures which perform biased random walks on the lattice while repelling each other. The shocks can enter into the system and leave it from the boundaries. Depending on the microscopic reaction rates, the system is known to have two different phases. We have found that the mean distance between the shock positions is of order L in one phase while it is of order 1 in the other phase.Comment: 5 pages, 1 EPS figure, Accepted for publication in PRE (2008

Discontinuous Phase Transition in an Exactly Solvable One-Dimensional Creation-Annihilation System

An exactly solvable reaction-diffusion model consisting of first-class particles in the presence of a single second-class particle is introduced on a one-dimensional lattice with periodic boundary condition. The number of first-class particles can be changed due to creation and annihilation reactions. It is shown that the system undergoes a discontinuous phase transition in contrast to the case where the density of the second-class particles is finite and the phase transition is continuous.Comment: Revised, 8 pages, 1 EPS figure. Accepted for publication in Journal of Statistical Mechanics: theory and experimen

Exact Solution of a Reaction-Diffusion Model with Particle Number Conservation

We analytically investigate a 1d branching-coalescing model with reflecting boundaries in a canonical ensemble where the total number of particles on the chain is conserved. Exact analytical calculations show that the model has two different phases which are separated by a second-order phase transition. The thermodynamic behavior of the canonical partition function of the model has been calculated exactly in each phase. Density profiles of particles have also been obtained explicitly. It is shown that the exponential part of the density profiles decay on three different length scales which depend on total density of particles.Comment: 7 pages, REVTEX4, Contents updated and new references added, to appear in Physical Review

Connection between matrix-product states and superposition of Bernoulli shock measures

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.Comment: 5 page

Exact Solution of an Exclusion Model in the Presence of a moving Impurity

We study a recently introduced model which consists of positive and negative particles on a ring. The positive (negative) particles hop clockwise (counter-clockwise) with rate 1 and oppositely charged particles may swap their positions with asymmetric rates q and 1. In this paper we assume that a finite density of positively charged particles $\rho$ and only one negative particle (which plays the role of an impurity) exist on the ring. It turns out that the canonical partition function of this model can be calculated exactly using Matrix Product Ansatz (MPA) formalism. In the limit of infinite system size and infinite number of positive particles, we can also derive exact expressions for the speed of the positive and negative particles which show a second order phase transition at $q_c=2\rho$. The density profile of the positive particles on the ring has a shock structure for $q \leq q_c$ and an exponential behaviour with correlation length $\xi$ for $q \geq q_c$. It will be shown that the mean-field results become exact at q=3 and no phase transition occurs for q>2.Comment: 9 pages,4 EPS figures. To be appear in JP

The Study of Shocks in Three-States Driven-Diffusive Systems: A Matrix Product Approach

We study the shock structures in three-states one-dimensional driven-diffusive systems with nearest neighbors interactions using a matrix product formalism. We consider the cases in which the stationary probability distribution function of the system can be written in terms of superposition of product shock measures. We show that only three families of three-states systems have this property. In each case the shock performs a random walk provided that some constraints are fulfilled. We calculate the diffusion coefficient and drift velocity of shock for each family.Comment: 15 pages, Accepted for publication in Journal of Statistical Mechanics: Theory and Experiment (JSTAT

First Order Phase Transition in a Reaction-Diffusion Model With Open Boundary: The Yang-Lee Theory Approach

A coagulation-decoagulation model is introduced on a chain of length L with open boundary. The model consists of one species of particles which diffuse, coagulate and decoagulate preferentially in the leftward direction. They are also injected and extracted from the left boundary with different rates. We will show that on a specific plane in the space of parameters, the steady state weights can be calculated exactly using a matrix product method. The model exhibits a first-order phase transition between a low-density and a high-density phase. The density profile of the particles in each phase is obtained both analytically and using the Monte Carlo Simulation. The two-point density-density correlation function in each phase has also been calculated. By applying the Yang-Lee theory we can predict the same phase diagram for the model. This model is further evidence for the applicability of the Yang-Lee theory in the non-equilibrium statistical mechanics context.Comment: 10 Pages, 3 Figures, To appear in Journal of Physics A: Mathematical and Genera

Relaxation time in a non-conserving driven-diffusive system with parallel dynamics

We introduce a two-state non-conserving driven-diffusive system in one-dimension under a discrete-time updating scheme. We show that the steady-state of the system can be obtained using a matrix product approach. On the other hand, the steady-state of the system can be expressed in terms of a linear superposition Bernoulli shock measures with random walk dynamics. The dynamics of a shock position is studied in detail. The spectrum of the transfer matrix and the relaxation times to the steady-state have also been studied in the large-system-size limit.Comment: 10 page