Matrix Product Steady States as Superposition of Product Shock Measures in 1D Driven Systems
چکیده
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 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 traveling wave solutions have been obtained for three families of two-states driven-diffusive 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 traveling wave solution is not limited to the two-states systems with nearest-neighbor interactions. Shocks in three-states lattice models with nearest-neighbor interactions [10, 11, 12, 13, 14] and also two-states lattice models with next nearest-neighbors interactions [15] have also been studied recently. As long as the time evolution of Corresponding author’s e-mail:farhad@ipm.ir 1 the product shock measure is equivalent to that of a random walker on a discrete lattice with homogeneous hopping rates in the bulk and special reflection rates at the boundary, the steady state of the model can be written as a linear superposition of these product shock measures. The steady states of these systems, on the other hand, can be obtained using the Matrix Product Formalism (MPF) first introduced in [9] and then developed in [16] (for a recent review see [17]). According to the MPF the steady state weight of a given configuration of a two-states stochastic non-equilibrium onedimensional lattice models with open boundaries can be written as expectation value of product of non-commuting operators associated with different types of particles and vacancies in that configuration. These operators should satisfy an associative algebra. By using a matrix representation of this algebra or using the commutation relations of the operators one can in principle calculate the unnormalized steady state weights. Whether or not this algebra has a finiteor infinite-dimensional representation in not known in general; however, it is known that some quadratic algebras have finite-dimensional representations on special manifolds in their parameters space. For instance it has been shown that the quadratic algebra of PASEP has finite-dimensional representations given that one is restricted to some special values of the reaction rates through some constraints [18, 19]. Interestingly, it has been shown that the constraints for the existence of a two-dimensional representation are exactly those necessary for a single product shock measure as an initial configuration to have a simple random walk dynamics on the lattice [20]. It has also been shown that if one takes a product measure of N − 1 consecutive product shocks, the shock positions have simple random walk dynamics provided that the constraints for the existence of an N -dimensional representation for the quadratic algebra of the PASEP are fulfilled. To see this one should adopt a slightly different definition of the shock measure. According to this new definition the shock is a product measure with density 1 at the shock positions and intermediate densities between these sites [20]. Hence the N -dimensional representations of the stationary algebra describe the stationary linear combination of such shock measures. The same is true for both the BCRW and the AKGP [6]. In fact it is known that the quadratic algebras associated with the BCRW and that of the AKGP have two-dimensional representations under some conditions on the reaction rates. Surprisingly, it has been shown that a single product measure in these systems will have a random walk dynamics under exactly the same constraints [21]. In [11] a three-states model with open boundary has been studied and it has been shown that the traveling shock solutions exist under some constraints on the reaction rates. Later a generalization of this model was introduced and studied in [13]. The authors in [10] have shown that the quadratic algebra of this generalized model can be mapped into the quadratic algebra of the PASEP provided that the constraints under which the traveling shock solutions exist, are fulfilled. This means that the quadratic algebra of this generalized model has two dimensional representation under the same constraints. Our major motivation in this paper is to investigate how the steady states of one-dimensional driven-diffusive systems with open boundaries when written in terms of linear superposition of Bernoulli shocks are related to the matrix product steady states. In this direction, we will consider those (n + 1)-states stochastic non-equilibrium one-dimensional lattice models with open boundaries and nearest-neighbors interactions in which the time evolution equation of 2 a single product shock measure is similar to the evolution equation for a biased single-particle random walk moving on a finite lattice with reflecting boundaries. Since the dynamics of the shock position, as a single-particle excitation, is simply a random walk it is easy to find the steady state probability distribution function of these systems in terms of linear superposition of single shocks. We will then show that the matrix product steady state of these systems can always be obtained using two-dimensional representations of their quadratic algebras. These matrix representations have always a generic form regardless of the details of the microscopic reactions. In order to confirm our assertion we will give a couple of example for both n = 1 and n = 2 cases. In what follows we will first review very quickly the dynamics of a biased singleparticle random walk on a discrete lattice of finite length and show how the steady states probability distribution of this system can be obtained. The time evolution of the probability distribution function of any configuration of a Markovian interacting particle system |P (t)〉 is governed by a master equation which can be written as a Schrödinger like equation in imaginary time d dt |P (t)〉 = H |P (t)〉 (1) in which H is a stochastic Hamiltonian [8]. The matrix elements of the Hamiltonian are the transition rates between different configurations. For the onedimensional systems defined on a lattice of length L with nearest neighbors interactions the Hamiltonian H has the following general form H = L−1 ∑ k=1 hk,k+1 + h1 + hL. (2) in which hk,k+1 = I ⊗(k−1) ⊗ h⊗ I (3) h1 = h (l) ⊗ I (4) hL = I ⊗(L−1) ⊗ h (5) For an (n + 1)-states one-dimensional driven diffusive system, in which these states are associated with n different types of particles and empty sites, I is an (n+ 1)× (n+ 1) identity matrix and h is an (n+ 1) × (n+ 1) matrix for the bulk interactions. The two matrices h and h are both (n + 1) × (n + 1) square matrices and determine the interactions at the boundaries. Let us assume that at t = 0 the probability distribution function of the an (n + 1)-states one-dimensional driven diffusive system is given by a product shock measure defined on a lattice of length L as follows |k〉 =