The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process

Abstract

We describe the translation invariant stationary states of the one dimensional discrete-time facilitated totally asymmetric simple exclusion process (F-TASEP). In this system a particle at site $j$ in $Z$ jumps, at integer times, to site $j+1$, provided site $j-1$ is occupied and site $j+1$ is empty. This defines a deterministic noninvertible dynamical evolution from any specified initial configuration on $\{0,1{\}}^Z$. When started with a Bernoulli product measure at density $\rho$ the system approaches a stationary state, with phase transitions at $\rho =1/2$ and $\rho =2/3$. We discuss various properties of these states in the different density regimes $0<\rho <1/2$, $1/2<\rho <2/3$, and $2/3<\rho <1$; for example, we show that the pair correlation $g(j)=⟨\eta (i)\eta (i+j)⟩$ satisfies, for all $n\in Z$, ${\sum }_{j=kn+1}^{k(n+1)}g(j)=k{\rho }^2$, with $k=2$ when $0\le \rho \le 1/2$ and $k=3$ when $2/3\le \rho \le 1$, and conjecture (on the basis of simulations) that the same identity holds with $k=6$ when $1/2\le \rho \le 2/3$. The $\rho <1/2$ stationary state referred to above is also the stationary state for the deterministic discrete-time TASEP at density $\rho$ (with Bernoulli initial state) or, after exchange of particles and holes, at density $1-\rho$.

arXiv comment36 pages, 4 figures. Treatment of Lemma 2.2 and Corollary 2.3 revised

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Cited by in corpus (2)

• Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process
• Stationary States of the One-dimensional Facilitated Asymmetric Exclusion Process