Understanding the edge effect in TASEP with mean-field theoretic approaches

TL;DR

This paper investigates the edge effect in TASEP with a defect site near the boundary using mean-field approaches, providing analytical and numerical insights into the steady state current and density profiles.

Contribution

It develops a finite segment mean-field framework to analyze TASEP with a defect, deriving an analytical expression for the current when the defect is at the entry.

Findings

Analytical expression for current at the entry defect matches simulations.

Refined approximation scheme for bulk defect density profiles.

Comparison of methods highlights strengths and limitations.

Abstract

We study a totally asymmetric simple exclusion process (TASEP) with one defect site, hopping rate $q<1$, near the system boundary. Regarding our system as a pair of uniform TASEP's coupled through the defect, we study various methods to match a \emph{finite} TASEP and an \emph{infinite} one across a common boundary. Several approximation schemes are investigated. Utilizing the finite segment mean-field (FSMF) method, we set up a framework for computing the steady state current $J$ as a function of the entry rate $% \alpha $ and $q$. For the case where the defect is located at the entry site, we obtain an analytical expression for $J(\alpha, q) $ which is in good agreement with Monte Carlo simulation results. When the defect is located deeper in the bulk, we refined the scheme of MacDonald, et.al. [Biopolymers, \textbf{6}, 1 (1968)] and find reasonably good fits to the density profiles before the defect site. We discuss the strengths and limitations of each method, as well as possible avenues for further studies.