The Blockage Problem
by O. Costin   J. L. Lebowitz   E. R. Speer   A. Troiani  

Vol. 8 No. 1 (2013) P.49~P.72

ABSTRACT

We investigate the totally asymmetric exclusion process on $\mathbb{Z}$, with the jump rate at site $i$ given by $r_i=1$ for $i\ne0$, $r_0=r$. It is easy to see that the maximal stationary current $j(r)$ is nondecreasing in $r$ and that $j(r)=1/4$ for $r\ge1$; it is a long outstanding problem to determine whether or not the critical value $r_c$ of $r$ such that $j(r)=1/4$ for $r>r_c$ is strictly less than 1. Here we present a heuristic argument, based on the analysis of the first sixteen terms in a formal power series expansion of $j(r)$ obtained from finite volume systems, that $r_c=1$ and that for $r\lessapprox1$, $j(r)\simeq{1/4}-\gamma\exp[-{a/(1-r)}]$ with $a\approx2$. We also give some new exact results about this system; in particular we prove that $j(r)=J_{\rm max}(r)$, with $J_{\rm max}(r)$ the hydrodynamic maximal current defined by Sepp\"al\"ainen, and thus establish continuity of $j(r)$. Finally we describe a related exactly solvable model, a semi-infinite system in which the site $i=0$ is always occupied. For that system, $r^{\text{s-i}}_c=1/2$ and the analogue $j^{\text{s-i}}(r)$ of $j(r)$ satisfies $j^{\text{s-i}}(r)=r(1-r)$ for $r \le r^{\text{s-i}}_c$; $j^{\text{s-i}}(r)$ is the limit of finite volume currents inside the curve $|r(1-r)|=1/4$ in the complex $r$ plane and we suggest that analogous behavior may hold for the original system.

KEYWORDS
Blockage problem, asymmetric exclusion process, maximal current

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 82C20, 82C22, 82C26

MILESTONES

Received: 2012-07-17
Revised :
Accepted: 2013-02-08

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