Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.Comment: 9 pages, 2 figure

Exactly solvable models through the generalized empty interval method: multi-species and more-than-two-site interactions

Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E^{\mathbf a}_n(t)$'s, the expectation value of the product of certain linear combination of the number operators on $n$ consecutive sites at time $t$.Comment: 10 pages, LaTe

Cluster approximation solution of a two species annihilation model

A two species reaction-diffusion model, in which particles diffuse on a one-dimensional lattice and annihilate when meeting each other, has been investigated. Mean field equations for general choice of reaction rates have been solved exactly. Cluster mean field approximation of the model is also studied. It is shown that, the general form of large time behavior of one- and two-point functions of the number operators, are determined by the diffusion rates of the two type of species, and is independent of annihilation rates.Comment: 9 pages, 7 figure

Static- and dynamical-phase transition in multidimensional voting models on continua

A voting model (or a generalization of the Glauber model at zero temperature) on a multidimensional lattice is defined as a system composed of a lattice each site of which is either empty or occupied by a single particle. The reactions of the system are such that two adjacent sites, one empty the other occupied, may evolve to a state where both of these sites are either empty or occupied. The continuum version of this model in a Ddimensional region with boundary is studied, and two general behaviors of such systems are investigated. The stationary behavior of the system, and the dominant way of the relaxation of the system toward its stationary state. Based on the first behavior, the static phase transition (discontinuous changes in the stationary profiles of the system) is studied. Based on the second behavior, the dynamical phase transition (discontinuous changes in the relaxation-times of the system) is studied. It is shown that the static phase transition is induced by the bulk reactions only, while the dynamical phase transition is a result of both bulk reactions and boundary conditions.Comment: 10 pages, LaTeX2

Phase transition in an asymmetric generalization of the zero-temperature Glauber model

An asymmetric generalization of the zero-temperature Glauber model on a lattice is introduced. The dynamics of the particle-density and specially the large-time behavior of the system is studied. It is shown that the system exhibits two kinds of phase transition, a static one and a dynamic one.Comment: LaTeX, 9 pages, to appear in Phys. Rev. E (2001

Exactly solvable models through the empty interval method

The most general one dimensional reaction-diffusion model with nearest-neighbor interactions, which is exactly-solvable through the empty interval method, has been introduced. Assuming translationally-invariant initial conditions, the probability that $n$ consecutive sites are empty ($E_n$), has been exactly obtained. In the thermodynamic limit, the large-time behavior of the system has also been investigated. Releasing the translational invariance of the initial conditions, the evolution equation for the probability that $n$ consecutive sites, starting from the site $k$, are empty ($E_{k,n}$) is obtained. In the thermodynamic limit, the large time behavior of the system is also considered. Finally, the continuum limit of the model is considered, and the empty-interval probability function is obtained.Comment: 12 pages, LaTeX2

Phase transition in an asymmetric generalization of the zero-temperature q-state Potts model

An asymmetric generalization of the zero-temperature q-state Potts model on a one dimensional lattice, with and without boundaries, has been studied. The dynamics of the particle number, and specially the large time behavior of the system has been analyzed. In the thermodynamic limit, the system exhibits two kinds of phase transitions, a static and a dynamic phase transition.Comment: 11 pages, LaTeX2

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

Autonomous multispecies reaction-diffusion systems with more-than-two-site interactions

Autonomous multispecies systems with more-than-two-neighbor interactions are studied. Conditions necessary and sufficient for closedness of the evolution equations of the $n$-point functions are obtained. The average number of the particles at each site for one species and three-site interactions, and its generalization to the more-than-three-site interactions is explicitly obtained. Generalizations of the Glauber model in different directions, using generalized rates, generalized number of states at each site, and generalized number of interacting sites, are also investigated.Comment: 9 pages, LaTeX2

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