Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

Eßler, Fabian H. L.; Rittenberg, V.

doi:10.1088/0305-4470/29/13/013

Cited by 163 publications

(253 citation statements)

References 34 publications

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“…Several representations have been used to solve (3.3a-3.3c) [14][15][16][17]29]. We choose in this section a particular representation which will be convenient for the remaining of our derivation.…”

Section: A Representation For D and Ementioning

confidence: 99%

Large Deviation Functional of the Weakly Asymmetric Exclusion Process

Derrida¹,

Enaud²

2004

Journal of Statistical Physics

123

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We obtain the large deviation functional of a density profile for the asymmetric exclusion process of L sites with open boundary conditions when the asymmetry scales like 1 L . We recover as limiting cases the expressions derived recently for the symmetric (SSEP) and the asymmetric (ASEP) cases. In the ASEP limit, the non linear differential equation one needs to solve can be analysed by a method which resembles the WKB method.

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Section: A Representation For D and Ementioning

confidence: 99%

Large Deviation Functional of the Weakly Asymmetric Exclusion Process

Derrida¹,

Enaud²

2004

Journal of Statistical Physics

123

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“…In particular, in an open system with two boundaries at which particles are injected and extracted with given rates, the bulk behaviour in the stationary state is strongly dependent on the injection and extraction rates. Over the last decade many stationary state properties of the PASEP with open boundaries have been determined exactly [1,2,[8][9][10][11]. On the other hand, much less is known about its dynamics.…”

mentioning

confidence: 99%

“…Bethe's Ansatz: It is well known that the transition matrix M is related to the Hamiltonian H of the open spin-1/2 XXZ quantum spin chain through a similarity transformation M = − √ pq U λ HU −1 λ where H is given by [10] …”

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confidence: 99%

Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries

2005

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We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. For totally asymmetric diffusion we calculate the spectral gap, which characterizes the approach to stationarity at large times. We observe boundary induced crossovers in and between massive, diffusive and KPZ scaling regimes. The partially asymmetric simple exclusion process (PASEP) describes the asymmetric diffusion of particles along a one-dimensional chain with L sites. It is one of the most studied models of non-equilibrium statistical mechanics, see [1,2] for recent reviews. This is in part due to the fact that is one of the simplest lattice gas models, but also because of its applicability to molecular diffusion in zeolites [3], bioploymers [4], traffic flow [5] and other one-dimensional complex systems [6].At large times the PASEP exhibits a relaxation towards a nonequilibrium stationary state. An interesting feature of the PASEP is the presence of boundary induced phase transitions [7]. In particular, in an open system with two boundaries at which particles are injected and extracted with given rates, the bulk behaviour in the stationary state is strongly dependent on the injection and extraction rates. Over the last decade many stationary state properties of the PASEP with open boundaries have been determined exactly [1,2,[8][9][10][11]. On the other hand, much less is known about its dynamics. This is in contrast to the PASEP on a ring for which exact results using Bethe's ansatz have been available for a long time [12,13]. For open boundaries there have been several studies of dynamical properties based mainly on numerical and phenomenological methods [14,15]. In this Letter we employ Bethe's ansatz to obtain exact results for the approach to stationarity at large times in the PASEP with open boundaries. Upon varying the boundary rates, we find crossovers in massive regions, with dynamic exponents z = 0, and between massive and scaling regions with diffusive (z = 2) and KPZ (z = 3/2) behaviour.The dynamical rules of the PASEP are as follows. At any given time t each site is either occupied by a particle or empty and the system evolves subject to the following rules. In the bulk of the system (i = 2, . . . , L − 1) a particle attempts to hop one site to the right with rate p and one site to the left with rate q. The hop is executed unless the neighbouring site is occupied, in which case nothing happens. On the first and last sites these rules are modified. If site i = 1 is empty, a particle may enter the system with rate α. If on the other hand site 1 is occupied by a particle, the latter will leave the system with rate γ. Similarly, at i = L particles are injected and extracted with rates δ and β respectively. With every site i we associate a Boolean variable τ i , indicating whether a particle is present (τ i = 1) or not (τ i = 0). The state of the system at time t is then characterized by the probability distribu...

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“…It is known that the density profile of the particles has a shock structure 2 on the coexistance line between LD and HD phase where a first order phase transtion takes place. A second order phase transition also exists from LD and HD phases to the MC phase [14,15,16]. This phase structure seems to be generic for all those models in which the fundamental digram (the current of particles as a function of the mean density of them) has a single local maximum.…”

Section: Introductionmentioning

confidence: 68%

“…On the other hand, the convergence radius of (8) is the absolute value of its nearest singularity to the origin; therefore, from (10), (15) and (18) we arrive at the following expression which gives u m only as a function of…”

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

confidence: 99%

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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Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

Cited by 163 publications

References 34 publications

Large Deviation Functional of the Weakly Asymmetric Exclusion Process

Large Deviation Functional of the Weakly Asymmetric Exclusion Process

Bethe Ansatz Solution of the Asymmetric Exclusion Process with Open Boundaries

A multi-species asymmetric exclusion model with an impurity

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