Matrix product states of three families of one-dimensional interacting particle systems

Abstract

The steady states of three families of one-dimensional nonequilibrium models with open boundaries, first proposed in [22], are studied using a matrix product formalism. It is shown that their associated quadratic algebras have two-dimensional representations, provided that the transition rates lie on specific manifolds of parameters . Exact expressions for the correlation functions of each model have also been obtained. We have also studied the steady state properties of one of these models, first introduced in [23], with more details. By introducing a canonical ensemble we calculate the canonical partition function of this model exactly. Using the Yang-Lee theory of phase transitions we spot a second-order phase transition from a power-law to a jammed phase. The density profile of particles in each phase has also been studied. A simple generalization of this model in which both the left and the right boundaries are open has also been introduced. It is shown that double shock structures may evolve in the system under certain conditions.

Particle-current fluctuations in a variant of the asymmetric Glauber model

“…It turns out that the dynamics of the position of a shock front is similar to that of a biased random walker moving on a finite lattice with reflecting boundaries. This steady-state probability distribution vector has also been obtained using a matrix product method in [14]. By associating the operators E and D to the presence of a vacancy and a particle in a given lattice site, the steady-state weight of any configuration…”

“…In (4), |V and W | are two auxiliary vectors. It has been shown that these two operators and vectors have a two-dimensional matrix representation given by [14] …”

“…This means that the large deviation function for the total particle current fluctuations, which can be obtained using (14), is a linear function of J for J a ≤ J ≤ J b [6] …”

“…The minimum eigenvalue of the modified Hamiltonian for λ ≤ 0 generates, using (14), the large deviation function for the total particle current fluctuations for J ≥ J .…”

“…In the bulk of the lattice, on the other hand, the particles are subjected to totally asymmetric branching and death processes. The steady-state of this system has already been calculated using the matrix product method [14]. It is known that this steady-state can be written as a linear combination of product shock measures with one shock front and that these shock fronts have simple random walk dynamics [15].…”