On the blockage problem and the non-analyticity of the current for the parallel TASEP on a ring

TL;DR

This paper rigorously derives the stationary measure for parallel TASEP, analyzes the blockage problem, and explores the non-analyticity of current with respect to blockage intensity, especially highlighting the special case of rule-184 cellular automaton.

Contribution

It provides a rigorous derivation of the stationary measure for parallel TASEP and exact current expressions for the rule-184 automaton, while investigating non-analyticity in blockage effects.

Findings

Exact current expression for rule-184 cellular automaton.

Rigorous derivation of stationary measure for parallel TASEP.

Numerical evidence of non-analyticity in current for other parallel update rules.

Abstract

The Totally Asymmetric Simple Exclusion Process (TASEP) is an important example of a particle system driven by an irreversible Markov chain. In this paper we give a simple yet rigorous derivation of the chain stationary measure in the case of parallel updating rule. In this parallel framework we then consider the blockage problem (aka slow bond problem). We find the exact expression of the current for an arbitrary blockage intensity $\varepsilon$ in the case of the so-called rule-184 cellular automaton, i.e. a parallel TASEP where at each step all particles free-to-move are actually moved. Finally, we investigate through numerical experiments the conjecture that for parallel updates other than rule-184 the current may be non-analytic in the blockage intensity around the value $\varepsilon = 0$.