Poissonian approximation for the tagged particle in asymmetric simple exclusion

1996 ◽  
Vol 33 (2) ◽  
pp. 411-419 ◽  
Author(s):  
P. A. Ferrari ◽  
L. R. G. Fontes
Keyword(s):  
Nearest Neighbor ◽  
Exclusion Process ◽  
Initial Distribution ◽  
Zero Range Process ◽  
Tagged Particle ◽  
In Series ◽  
Asymmetric Simple Exclusion ◽  
Net Flux ◽  
Simple Exclusion ◽  
The One

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt = Nt − Bt + B0, where {Nt} is a Poisson process of parameter (p – q)(1– λ) and {Bt} is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt} converges to Brownian motion. A previous result says that in the scale t1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.

Poissonian approximation for the tagged particle in asymmetric simple exclusion

1996 ◽  
Vol 33 (02) ◽  
pp. 411-419
Author(s):  
P. A. Ferrari ◽  
L. R. G. Fontes
Keyword(s):  
Nearest Neighbor ◽  
Exclusion Process ◽  
Initial Distribution ◽  
Zero Range Process ◽  
Tagged Particle ◽  
In Series ◽  
Asymmetric Simple Exclusion ◽  
Net Flux ◽  
Simple Exclusion ◽  
The One

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p &gt; q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt = Nt − Bt + B 0 , where {N t} is a Poisson process of parameter (p – q)(1– λ) and {Bt } is a stationary process satisfying E exp (θ | B, |) &lt; ∞ for some θ &gt; 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt } converges to Brownian motion. A previous result says that in the scale t 1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.

First displacement time of a tagged particle in a stochastic cluster in a simple exclusion process with random slow bonds

2019 ◽  
Vol 51 (03) ◽  
pp. 717-744
Author(s):  
Adriana Uquillas ◽  
Adilson Simonis
Keyword(s):  
Poisson Process ◽  
Exclusion Process ◽  
Upper And Lower Bounds ◽  
Density Profiles ◽  
Tagged Particle ◽  
Simple Exclusion Process ◽  
Discrete Torus ◽  
Simple Exclusion ◽  
The One ◽  
Initial Measure

AbstractWe consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus $\mathbb{T}_N=\mathbb{Z}/N\mathbb{Z}$ , with random rates $c_N=\{c_{x,N}\colon x \in \mathbb{T}_N\}$ defined in terms of a homogeneous Poisson process on $\mathbb{R}$ with intensity $\lambda$ . Given a realization of the Poisson process, the jump rate along the edge $\{x,x+1\}$ is 1 if there is not any Poisson mark in $ (x,x+1) $ ; otherwise, it is $\lambda/N,\, \lambda \in( 0,1]$ . The density profile of this process with initial measure associated to an initial profile $\rho_0\colon \mathbb{R} \rightarrow [0,1]$ , evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If $\lambda=1$ then $\rho$ is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of $T_j$ , where $T_j$ is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then $\mathbb{P}\{T_j \geq t\}$ decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected.

scholarly journals Dynamical Transitions in a One-Dimensional Katz–Lebowitz–Spohn Model

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1028 ◽  
Author(s):  
Alessandro Pelizzola ◽  
Marco Pretti ◽  
Francesco Puccioni
Keyword(s):  
Finite Size ◽  
Exclusion Process ◽  
Asymmetric Simple Exclusion Process ◽  
Pair Approximation ◽  
Fundamental Diagram ◽  
One Dimensional ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
The One

Dynamical transitions, already found in the high- and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition in the NESS itself. We investigate dynamical transitions in the one-dimensional Katz–Lebowitz–Spohn model, a further generalization of the Totally Asymmetric Simple Exclusion Process where the hopping rate depends on the occupation state of the 2 nodes adjacent to the nodes affected by the hop. Following previous work, we choose Glauber rates and bulk-adapted boundary conditions. In particular, we consider a value of the repulsion which parameterizes the Glauber rates such that the fundamental diagram of the model exhibits 2 maxima and a minimum, and the NESS phase diagram is especially rich. We provide evidence, based on pair approximation, domain wall theory and exact finite size results, that dynamical transitions also occur in the one-dimensional Katz–Lebowitz–Spohn model, and discuss 2 new phenomena which are peculiar to this model.

scholarly journals Mapping TASEP back in time

2021 ◽  
Author(s):  
Leonid Petrov ◽  
Axel Saenz
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Exclusion Process ◽  
Discrete Space ◽  
Baxter Equation ◽  
Simple Exclusion Process ◽  
Open Questions ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Hammersley Process

AbstractWe obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

A stochastic airplane boarding model in a framework of ASEP with distinguishable particles

2018 ◽  
Vol 29 (10) ◽  
pp. 1850093
Author(s):  
ShengJie Qiang ◽  
Bin Jia ◽  
QingXia Huang
Keyword(s):  
Removal Rate ◽  
Exclusion Process ◽  
System Size ◽  
Nonequilibrium Systems ◽  
One Dimensional ◽  
Simple Exclusion Process ◽  
Maximum Current ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
The Difference

The asymmetric simple exclusion process (ASEP) is a paradigmatic model for nonequilibrium systems and has been used in many applications. Airplane boarding provides another interesting example where this framework can be applied. We propose a simple model for boarding process, in which a particle moves along a one-dimensional aisle after being injected, and finally is removed at a reserved site. Different from the typical ASEP model, particles are removed in a disorderly or a parallel way. Detailed calculations and discussions of some related characteristics, such as mean boarding time and parallelism indicator, are provided based on Monte-Carlo simulations. Results show that three phases exist in the boarding process: free-flow, jamming and maximum current. Transitions between these phases are governed by the difference between the injection and removal rate. Further analysis shows how the scaling behavior depends on the system size and the boarding conditions. Those results emphasize the importance of utilizing the whole length of the aisle to reduce the boarding time when designing an efficient boarding strategy.

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