Chikashi Arita

Matrix product solution of an inhomogeneous multi-species TASEP

We study a multi-species exclusion process with inhomogeneous hopping rates and find a matrix product representation for the stationary state of this model. The matrices belong to the tensor algebra of the fundamental quadratic algebra associated with the exclusion process. We show that our matrix product representation is equivalent to a graphical construction proposed by Ayyer and Linusson (2012 arXiv:1206.0316), which generalizes an earlier probabilistic construction due to Ferrari and Martin (2007 Ann. Prob. 35 807).

Generalized matrix Ansatz in the multispecies exclusion process—the partially asymmetric case

We investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case, operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by Arita et al (2011 J. Phys. A: Math. Theor. 44 335004). The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In this work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case.

Density profiles of the exclusive queuing process

The exclusive queuing process (EQP) incorporates the exclusion principle into classic queuing models. It is characterized by, in addition to the entrance probability ? and exit probability ?, a third parameter: the hopping probability p. The EQP can be interpreted as an exclusion process of variable system length. Its phase diagram in the parameter space (?,?) is divided into a convergent phase and a divergent phase by a critical line which consists of a curved part and a straight part. Here we extend previous studies of this phase diagram. We identify subphases in the divergent phase, which can be distinguished by means of the shape of the density profile, and determine the velocity of the system length growth. This is done for EQPs with different update rules (parallel, backward sequential and continuous time). We also investigate the dynamics of the system length and the number of customers on the critical line. They are diffusive or subdiffusive with non-universal exponents that also depend on the update rules.

Dynamical analysis of the exclusive queueing process.

Recently, the stationary state of a parallel-update totally asymmetric simple exclusion process with varying system length, which can be regarded as a queueing process with excluded-volume effect (exclusive queueing process), was obtained [C Arita and D Yanagisawa, J. Stat. Phys. 141, 829 (2010)]. In this paper, we analyze the dynamical properties of the number of particles [N(t)] and the position of the last particle (the system length) [L(t)], using an analytical method (generating function technique) as well as a phenomenological description based on domain-wall dynamics and Monte Carlo simulations. The system exhibits two phases corresponding to linear convergence or divergence of [N(t)] and [L(t)]. These phases can both further be subdivided into high-density and maximal-current subphases. The predictions of the domain-wall theory are found to be in very good agreement quantitively with results from Monte Carlo simulations in the convergent phase. On the other hand, in the divergent phase, only the prediction for [N(t)] agrees with simulations.

Remarks on the multi-species exclusion process with reflective boundaries

We investigate one of the simplest multi-species generalizations of the one-dimensional exclusion process with reflective boundaries. The Markov matrix governing the dynamics of the system splits into blocks (sectors) specified by the number of particles of each kind. We find matrices connecting the blocks in a matrix product form. The procedure (generalized matrix ansatz) to verify that a matrix intertwines blocks of the Markov matrix was introduced in the periodic boundary condition (Arita et al 2011 J. Phys. A 44 335004), which starts with a local relation. The solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition.

Exact dynamical state of the exclusive queueing process with deterministic hopping.

The exclusive queueing process (EQP) has recently been introduced as a model for the dynamics of queues that takes into account the spatial structure of the queue. It can be interpreted as a totally asymmetric exclusion process of varying length. Here we investigate the case of deterministic bulk hopping p=1 that turns out to be one of the rare cases where exact nontrivial results for the dynamical properties can be obtained. Using a time-dependent matrix product form we calculate several dynamical properties, e.g., the density profile of the system.

Exclusive queueing processes and their application to traffic systems

The dynamics of pedestrian crowds has been studied intensively in recent years, both theoretically and empirically. However, in many situations pedestrian crowds are rather static, e.g. due to jamming near bottlenecks or queueing at ticket counters or supermarket checkouts. Classically such queues are often described by the M/M/1 queue that neglects the internal structure (density profile) of the queue by focussing on the system length as the only dynamical variable. This is different in the Exclusive Queueing Process (EQP) in which the queue is considered on a microscopic level. It is equivalent to a Totally Asymmetric Exclusion Process (TASEP) of varying length. The EQP has a surprisingly rich phase diagram with respect to the arrival probability alpha and the service probability beta. The behavior on the phase transition line is much more complex than for the TASEP with a fixed system length. It is nonuniversal and depends strongly on the update procedure used. In this article, we review the main properties of the EQP. We also mention extensions and applications of the EQP and some related models.

Critical behavior of the exclusive queueing process

The exclusive queueing process (EQP) is a generalization of the classical M/M/1 queue. It is equivalent to a totally asymmetric exclusion process (TASEP) of varying length. Here we consider two discrete-time versions of the EQP with parallel and backward-sequential update rules. The phase diagram (with respect to the arrival probability ? and the service probability ?) is divided into two phases corresponding to divergence and convergence of the system length. We investigate the behavior on the critical line separating these phases. For both update rules, we find diffusive behavior for small service probability . However, for \beta_c

Two dimensional outflows for cellular automata with shuffle updates

SRC=http://ej.iop.org/images/0295-5075/104/3/30004/epl15871ieqn2.gif/> it becomes sub-diffusive and nonuniversal: the critical exponents characterizing the divergence of the system length and the number of customers are found to depend on the update rule. For the backward-update case, they also depend on the hopping parameter p, and remain finite when p is large, indicating a first-order transition.

Variational calculation of transport coefficients in diffusive lattice gases

In this paper, we explore the two-dimensional behavior of cellular automata with shuffle updates. As a test case, we consider the evacuation of a square room by pedestrians modeled by a cellular automaton model with a static floor field. Shuffle updates are characterized by a variable associated to each particle and called phase, that can be interpreted as the phase in the step cycle in the frame of pedestrian flows. Here we also introduce a dynamics for these phases, in order to modify the properties of the model. We investigate in particular the crossover between low- and high-density regimes that occurs when the density of pedestrians increases, the dependency of the outflow in the strength of the floor field, and the shape of the queue in front of the exit. Eventually we discuss the relevance of these results for pedestrians.