Spontaneous Breaking of Translational Invariance in One-Dimensional Stationary States on a Ring

We consider a model in which positive and negative particles diffuse in an asymmetric, CP-invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise and oppositely-charged adjacent particles may swap positions. Monte-Carlo simulations and analytic calculations suggest that the model has three phases; a "pure" phase in which one has three pinned blocks of only positive, negative particles and vacancies, and in which translational invariance is spontaneously broken, a "mixed" phase with a non-vanishing current in which the three blocks are positive, negative and neutral, and a disordered phase without blocks.Comment: 7 pages, LaTeX, needs epsf.st

Symmetry breaking and phase coexistence in a driven diffusive two-channel system

We consider classical hard-core particles moving on two parallel chains in the same direction. An interaction between the channels is included via the hopping rates. For a ring, the stationary state has a product form. For the case of coupling to two reservoirs, it is investigated analytically and numerically. In addition to the known one-channel phases, two new regions are found, in particular the one, where the total density is fixed, but the filling of the individual chains changes back and forth, with a preference for strongly different densities. The corresponding probability distribution is determined and shown to have an universal form. The phase diagram and general aspects of the problem are discussed.Comment: 12 pages, 10 figures, to appear in Phys.Rev.

On Matrix Product States for Periodic Boundary Conditions

The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m-state quantum spin Hamiltonian with nearest neighbour interaction and periodic boundary condition is considered. The quadratic algebra used for this representation is generated by 2m operators which fulfil m^2 quadratic relations and is endowed with a trace. It is shown that {\em not} every eigenstate with energy and momentum zero can be written as matrix product state. An explicit counter-example is given. This is in contrast to the case of open boundary conditions where every zero energy eigenstate can be written as a matrix product state using a Fock-like representation of the same quadratic algebra.Comment: 7 pages, late

Exact solution and asymptotic behaviour of the asymmetric simple exclusion process on a ring

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.Comment: 10 page

One-Dimensional Partially Asymmetric Simple Exclusion Process on a Ring with a Defect Particle

The effect of a moving defect particle for the one-dimensional partially asymmetric simple exclusion process on a ring is considered. The current of the ordinary particles, the speed of the defect particle and the density profile of the ordinary particles are calculated exactly. The phase diagram for the correlation length is identified. As a byproduct, the average and the variance of the particle density of the one-dimensional partially asymmetric simple exclusion process with open boundaries are also computed.Comment: 23 pages, 1 figur

Symmetry breaking through a sequence of transitions in a driven diffusive system

In this work we study a two species driven diffusive system with open boundaries that exhibits spontaneous symmetry breaking in one dimension. In a symmetry broken state the currents of the two species are not equal, although the dynamics is symmetric. A mean field theory predicts a sequence of two transitions from a strongly symmetry broken state through an intermediate symmetry broken state to a symmetric state. However, a recent numerical study has questioned the existence of the intermediate state and instead suggested a single discontinuous transition. In this work we present an extensive numerical study that supports the existence of the intermediate phase but shows that this phase and the transition to the symmetric phase are qualitatively different from the mean-field predictions.Comment: 19 pages, 12 figure

Criterion for phase separation in one-dimensional driven systems

A general criterion for the existence of phase separation in driven one-dimensional systems is proposed. It is suggested that phase separation is related to the size dependence of the steady-state currents of domains in the system. A quantitative criterion for the existence of phase separation is conjectured using a correspondence made between driven diffusive models and zero-range processes. Several driven diffusive models are discussed in light of the conjecture

Competition of coarsening and shredding of clusters in a driven diffusive lattice gas

We investigate a driven diffusive lattice gas model with two oppositely moving species of particles. The model is motivated by bi-directional traffic of ants on a pre-existing trail. A third species, corresponding to pheromones used by the ants for communication, is not conserved and mediates interactions between the particles. Here we study the spatio-temporal organization of the particles. In the uni-directional variant of this model it is known to be determined by the formation and coarsening of ``loose clusters''. For our bi-directional model, we show that the interaction of oppositely moving clusters is essential. In the late stages of evolution the cluster size oscillates because of a competition between their `shredding' during encounters with oppositely moving counterparts and subsequent "coarsening" during collision-free evolution. We also establish a nontrivial dependence of the spatio-temporal organization on the system size

Ergodicity breaking in one-dimensional reaction-diffusion systems

We investigate one-dimensional driven diffusive systems where particles may also be created and annihilated in the bulk with sufficiently small rate. In an open geometry, i.e., coupled to particle reservoirs at the two ends, these systems can exhibit ergodicity breaking in the thermodynamic limit. The triggering mechanism is the random motion of a shock in an effective potential. Based on this physical picture we provide a simple condition for the existence of a non-ergodic phase in the phase diagram of such systems. In the thermodynamic limit this phase exhibits two or more stationary states. However, for finite systems transitions between these states are possible. It is shown that the mean lifetime of such a metastable state is exponentially large in system-size. As an example the ASEP with the A0A--AAA reaction kinetics is analyzed in detail. We present a detailed discussion of the phase diagram of this particular model which indeed exhibits a phase with broken ergodicity. We measure the lifetime of the metastable states with a Monte Carlo simulation in order to confirm our analytical findings.Comment: 25 pages, 14 figures; minor alterations, typos correcte

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