The totally asymmetric exclusion process on a ring: Exact relaxation dynamics and associated model of clustering transition

TL;DR

This paper provides an exact analysis of the relaxation dynamics and clustering transition in the totally asymmetric exclusion process on a ring, including explicit formulas and phase transition evidence.

Contribution

It introduces a translation-invariant update rule, derives exact expressions for key quantities, and reveals a clustering phase transition through zeroes analysis.

Findings

Exact stationary velocity for arbitrary ring size and particle filling

Explicit propagator and generating function derivations

Evidence of a clustering transition at low fugacities

Abstract

The totally asymmetric simple exclusion process in discrete time is considered on finite rings with fixed number of particles. A translation-invariant version of the backward-ordered sequential update is defined for periodic boundary conditions. We prove that the so defined update leads to a stationary state in which all possible particle configurations have equal probabilities. Using the exact analytical expression for the propagator, we find the generating function for the conditional probabilities, average velocity and diffusion constant at all stages of evolution. An exact and explicit expression for the stationary velocity of TASEP on rings of arbitrary size and particle filling is derived. The evolution of small systems towards a steady state is clearly demonstrated. Considering the generating function as a partition function of a thermodynamic system, we study its zeros in planes of complex fugacities. At long enough times, the patterns of zeroes for rings with increasing size provide evidence for a transition of the associated two-dimensional lattice paths model into a clustered phase at low fugacities.