Phase Transition in a Three-States Reaction-Diffusion System

A one-dimensional reaction-diffusion model consisting of two species of particles and vacancies on a ring is introduced. The number of particles in one species is conserved while in the other species it can fluctuate because of creation and annihilation of particles. It has been shown that the model undergoes a continuous phase transition from a phase where the currents of different species of particles are equal to another phase in which they are different. The total density of particles and also their currents in each phase are calculated exactly. Recently one-dimensional reaction-diffusion systems have received much attention because they show a variety of interesting critical phenomena such as outof-equilibrium phase transitions [1, 2]. A simple system of this type, which has been studied widely in related literatures, is the Asymmetric Simple Exclusion Process (ASEP) [3]. In this two-states model the particles are injected from the left site of an open discreet lattice of length L. They diffuse in the system and at the end of the lattice are extracted from the system. It is known that depending on the injection and the extraction rates the ASEP shows different boundary induced phase transitions. Non-equilibrium phase transition may also happen in the systems with non-conserving dynamics [4, 5]. For instance in [5] the authors investigate a three-states model consists of two species of particles besides vacancies on a lattice with ring geometry. The dynamics of this model consists of diffusion, creation and annihilation of both species of the particles. They have found that the phase diagram of the model highly depends on the annihilation rate of the particles. By changing the annihilation rate of the particles, the system transfers from a maximal current phase to a fluid phase. The density of the vacancies changes discontinuously from one phase to the other phase. In present paper we introduce and study a reaction-diffusion model on a discrete lattice of length L with periodic boundary condition. Besides the vacancies there are two different types of particles in the system. Throughout this paper the vacancies and the particles are denoted by E, A and B. The dynamics of the system is not conserving. The particles of type A and type B hop to the left and to the right respectively. The total number of particles of type B is a ∗Corresponding author’s e-mail:[email protected] 1 conserved quantity and assumed to be equal to M . The density of these particles is defined as ρB = M L . In contrast, the total number of particles of type A is not a conserved quantity due to the creation and annihilation of them. Only the nearest neighbors interactions are allowed and the model evolves through the following processes A∅ −→ ∅A with rate 1 ∅B −→ B∅ with rate 1 AB −→ BA with rate 1 A∅ −→ ∅∅ with rate ω ∅∅ −→ A∅ with rate 1 (1) As can be seen the parameter ω determines the annihilation rate for the particles of type A which besides the number of the particles of type B i.e. ρB are the free parameters of the model. One should note that the annihilation in our model only takes place for one species of particles. Our main aim in the present work is to study the phase diagram of the model in terms of ω and the density of the B particles. In our model if one starts with a lattice without any vacancies the dynamics of the model prevents it from evolving into other configurations consisting of vacancies. In this case the system remains in its initial configuration and the steady state of the system is trivial. In order to study the non-trivial case we consider those configurations which have at least one vacancy. In order to find the stationary probability distribution function of the system we apply the Matrix Product Formalism (MPF) first introduced in [3] and then generalized in [6]. According to this formalism the stationary probability for any configuration of a system with periodic boundary condition is proportional to the trace of product of non-commuting operators which satisfy a quadratic algebra. In our model we have three different states at each site of the lattice associated with the presence of vacancies, the A particles and the B particles. We assign three different operators E, A and B to each state. Now the unnormalized steady state probability of a configuration C is given by P (C) = 1 ZL Tr[ L ∏ i=1 Xi] (2) in which Xi = E if the site i is empty, Xi = A if the site i is occupied by a particle of type A and Xi = B if it is occupied by a particle of type B. The normalization factor ZL in the denominator of (2) is called the partition function of the system and is given by the sum of unnormalized weights of all accessible configurations. By applying the MPF one finds the following quadratic algebra for our model AB = A+ B AE = E EB = E E 2 = ωE. (3) Now by defining E = ω|V 〉〈W | in which 〈W |V 〉 = 1 one can simply find AB = A+ B A|V 〉 = |V 〉 〈W |B = 〈W | E = ω|V 〉〈W |. (4)