The totally asymmetric exclusion process with generalized update

TL;DR

This paper extends the totally asymmetric exclusion process by introducing a control parameter that interpolates between known update rules, revealing new flow behaviors including jamming tendencies, and provides exact solutions via Bethe ansatz.

Contribution

It introduces a generalized update rule with a control parameter, unifies known updates, and derives exact non-stationary solutions for the process.

Findings

Flow exhibits jamming with attractive interactions

Exact solutions obtained for arbitrary initial conditions

Unified framework for different update schemes

Abstract

We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for $N$ particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.