Exclusion processes in higher dimensions: Stationary measures and convergence

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on $Z$. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on $Z^d$ and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on $Z^d$. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as $t\to\infty$ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Distribution of a Second Class Particle in the Asymmetric Simple Exclusion Process

We give an exact expression for the distribution of the position X(t) of a single second class particle in the asymmetric simple exclusion process (ASEP) where initially the second class particle is located at the origin and the first class particles occupy the sites {1,2,...}

Ultra-discrete Optimal Velocity Model: a Cellular-Automaton Model for Traffic Flow and Linear Instability of High-Flux Traffic

In this paper, we propose the ultra-discrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultra-discrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations, and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.Comment: 9 pages, 6 figure

Queueing process with excluded-volume effect

We introduce an extension of the M/M/1 queueing process with a spatial structure and excluded- volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary-state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We also consider a generalization of the model with backward hopping of particles allowed and an alternate joined system of the M/M/1 queueing process and the open TASEP.Comment: 9 figure

A new model of binary opinion dynamics: coarsening and effect of disorder

We propose a model of binary opinion in which the opinion of the individuals change according to the state of their neighbouring domains. If the neighbouring domains have opposite opinions, then the opinion of the domain with the larger size is followed. Starting from a random configuration, the system evolves to a homogeneous state. The dynamical evolution show novel scaling behaviour with the persistence exponent $\theta \simeq 0.235$ and dynamic exponent $z \simeq1.02 \pm 0.02$. Introducing disorder through a parameter called rigidity coefficient $\rho$ (probability that people are completely rigid and never change their opinion), the transition to a heterogeneous society at $\rho = 0^{+}$ is obtained. Close to $\rho =0$, the equilibrium values of the dynamic variables show power law scaling behaviour with $\rho$. We also discuss the effect of having both quenched and annealed disorder in the system.Comment: 4 pages, 6 figures, Final version of PR

Phase transitions in a two parameter model of opinion dynamics with random kinetic exchanges

Recently, a model of opinion formation with kinetic exchanges has been proposed in which a spontaneous symmetry breaking transition was reported [M. Lallouache et al, Phys. Rev. E, {\bf 82} 056112 (2010)]. We generalise the model to incorporate two parameters, $\lambda$, to represent conviction and $\mu$, to represent the influencing ability of individuals. A phase boundary given by $\lambda=1-\mu/2$ is obtained separating the symmetric and symmetry broken phases: the effect of the influencing term enhances the possibility of reaching a consensus in the society. The time scale diverges near the phase boundary in a power law manner. The order parameter and the condensate also show power law growth close to the phase boundary albeit with different exponents. Theexponents in general change along the phase boundary indicating a non-universality. The relaxation times, however, become constant with increasing system size near the phase boundary indicating the absence of any diverging length scale. Consistently, the fluctuations remain finite but show strong dependence on the trajectory along which it is estimated.Comment: Version accepted for PRE; text modified, new figures and references adde

Evolutionary dynamics on degree-heterogeneous graphs

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model (VM) dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process (IP) dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for VM dynamics and to 1/k for IP dynamics.Comment: 4 pages, 4 figures, 2 column revtex4 format. Revisions in response to referee comments for publication in PRL. The version on arxiv.org has one more figure than the published PR

Site-bond representation and self-duality for totalistic probabilistic cellular automata

We study the one-dimensional two-state totalistic probabilistic cellular automata (TPCA) having an absorbing state with long-range interactions, which can be considered as a natural extension of the Domany-Kinzel model. We establish the conditions for existence of a site-bond representation and self-dual property. Moreover we present an expression of a set-to-set connectedness between two sets, a matrix expression for a condition of the self-duality, and a convergence theorem for the TPCA.Comment: 11 pages, minor corrections, journal reference adde

Bulk and surface transitions in asymmetric simple exclusion process: Impact on boundary layers

In this paper, we study boundary-induced phase transitions in a particle non-conserving asymmetric simple exclusion process with open boundaries. Using boundary layer analysis, we show that the key signatures of various bulk phase transitions are present in the boundary layers of the density profiles. In addition, we also find possibilities of surface transitions in the low- and high- density phases. The surface transition in the low-density phase provides a more complete description of the non-equilibrium critical point found in this system.Comment: 9 pages including figure

Noise driven dynamic phase transition in a a one dimensional Ising-like model

The dynamical evolution of a recently introduced one dimensional model in \cite{biswas-sen} (henceforth referred to as model I), has been made stochastic by introducing a parameter $\beta$ such that $\beta =0$ corresponds to the Ising model and $\beta \to \infty$ to the original model I. The equilibrium behaviour for any value of $\beta$ is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent $z$,that for any $\beta \neq 0$, the system belongs to the dynamical class of model I indicating a dynamic phase transition at $\beta = 0$. On the other hand, the persistence probabilities in a system of $L$ spins saturate at a value $P_{sat}(\beta, L) = (\beta/L)^{\alpha}f(\beta)$, where $\alpha$ remains constant for all $\beta \neq 0$ supporting the existence of the dynamic phase transition at $\beta =0$. The scaling function $f(\beta)$ shows a crossover behaviour with $f(\beta) = \rm{constant}$ for $\beta <<1$ and $f(\beta) \propto \beta^{-\alpha}$ for $\beta >>1$.Comment: 4 pages, 5 figures, accepted version in Physical Review