Asymmetric exclusion model with two species: Spontaneous symmetry breaking

Evans, Martin R.; Foster, D. P.; Godrèche, Claude; Mukamel, David

doi:10.1007/bf02178354

Cited by 177 publications

(189 citation statements)

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“…In the mathematical literature the ASEP is usually defined in continuous time or, equivalently for the purposes of simulation, by a random sequential updating scheme where for each update a particle is selected at random. In contrast, when simulating traffic flow parallel updating is usually employed for reasons both idealistic-parallel dynamics provides a perhaps more faithful representation of real traffic-and pragmatic-parallel dynamics yields economy of random numbers.For random sequential dynamics a relative wealth of exact results on the ASEP are now available [2,3,13,14,15,16,17,18,19], in particular the steady states of various models have been constructed using a matrix product ansatz [3,14,20,21,22,23,24]. This technique has been extended to a sublattice parallel updating scheme [25,26,27] and, in the case of open boundary conditions, to an ordered sequential scheme [28,29].…”

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confidence: 99%

Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

Evans

1997

J. Phys. A: Math. Gen.

118

138

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A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backward ordered sequential dynamics it occurs at a higher density. In both cases the critical density is higher than that for random sequential dynamics. In all the models studied the steady state velocity is related to the fugacity of a Bose system suggesting a principle of minimisation of velocity. A generalisation of the dynamics where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.Date: 29/4/1997 Submitted to Journal of Physics A. Key words:asymmetric exclusion process, random rates, steady state, parallel dynamics, phase transition, Bose condensation, traffic flow IntroductionThe asymmetric simple exclusion process (ASEP) is an archetypal example of a driven diffusive system [1, 2] for which analytical results are possible, particularly in one dimension [3]. The model comprises particles which hop stochastically in a preferred direction with hard core exclusion imposed. The model has a natural interpretation as a simplistic description of traffic flow on a one lane road and indeed forms the basis for more sophisticated traffic flow models [4,5]. In particular one may cite variations of the model proposed originally by Nagel and Schreckenberg [6,7,8,9,10,11]. However, a basic difference between the original ASEP and traffic flow models lies in the updating scheme. In the mathematical literature the ASEP is usually defined in continuous time or, equivalently for the purposes of simulation, by a random sequential updating scheme where for each update a particle is selected at random. In contrast, when simulating traffic flow parallel updating is usually employed for reasons both idealistic-parallel dynamics provides a perhaps more faithful representation of real traffic-and pragmatic-parallel dynamics yields economy of random numbers.For random sequential dynamics a relative wealth of exact results on the ASEP are now available [2,3,13,14,15,16,17,18,19], in particular the steady states of various models have been constructed using a matrix product ansatz [3,14,20,21,22,23,24]. This technique has been extended to a sublattice parallel updating scheme [25,26,27] and, in the case of open boundary conditions, to an ordered sequential scheme [28,29]. However, for fully par...

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confidence: 99%

Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

Evans

1997

J. Phys. A: Math. Gen.

118

138

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“…there would have found a change in the phase diagram of our model in (26). It can be shown that in this case an extra phase exists for α > v 1 which is produced by the singularity of (10) at v 1 4 :…”

Section: Exact Results For the Physical Quantities And The Phase Diagrammentioning

confidence: 79%

“…As we mentioned above, the analytical results for the case d du S u=v1 > 0 suggest that the phase diagram in the case 0 < α < +∞ is made up of two different phases given by (26) and (30). The Monte Carlo simulations for this case have also been carried out.…”

Section: Remarkmentioning

confidence: 73%

“…For n > ρv1 1−ρ−v1 the sign of (35) becomes positive and the phase diagram should be obtained from (26). Taking n = 1 and ρ = 0.6 we will obtain the total density profile of the ordinary particles in each phase using the Monte Carlo simulation for v 1 = 0.6 and v 1 = 0.2.…”

Section: Special Cases and Simulation Resultsmentioning

confidence: 99%

“…Following [26] we write A as |V W | which simplifies the above algebra to the one in [24] for the study of the multi-species ASEP with open boundaries.…”

Section: Algebraic Preliminariesmentioning

confidence: 99%

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A multi-species asymmetric exclusion model with an impurity

Jafarpour

2002

Physica A: Statistical Mechanics and its Applications

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A multi-species generalization of the Asymmetric Simple Exclusion Process (ASEP) has been considered in the presence of a single impurity on a ring. The model describes particles hopping in one direction with stochastic dynamics and hard core exclusion condition. The ordinary particles hop forward with their characteristic hopping rates and fast particles can overtake slow ones with a relative rate. The impurity, which is the slowest particle in the ensemble of particles on the ring, hops in the same direction of the ordinary particles with its intrinsic hopping rate and can be overtaken by ordinary particles with a rate which is not necessarily a relative rate. We will show that the phase diagram of the model can be obtained exactly. It turns out that the phase structure of the model depends on the density distribution function of the ordinary particles on the ring so that it can have either four phases or only one. The mean speed of impurity and also the total current of the ordinary particles are explicitly calculated in each phase. Using Monte Carlo simulation, the density profile of the ordinary particles is also obtained. The simulation data confirm all of the analytical calculations.

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