Nonexistence of motility induced phase separation transition in one dimension

TL;DR

This paper models one-dimensional run-and-tumble particles and demonstrates analytically that motility-induced phase separation cannot occur in such systems due to the nature of their coarse-grained dynamics.

Contribution

It introduces a new one-dimensional run-and-tumble particle model and analytically proves the nonexistence of motility-induced phase separation in one dimension.

Findings

Coarse-grained dynamics map to a beads-in-urn model.

Hop rates depend on urn occupancy and do not support phase separation.

Tumbling destabilizes jamming, preventing phase separation.

Abstract

We introduce and study a model of hardcore particles obeying run-and-tumble dynamics on a one-dimensional lattice, where particles run in either +ve or -ve $x$-direction with an effective speed $v$ and tumble (change their direction of motion) with a constant rate $\omega.$ We show that the coarse-grained dynamics of the system can be mapped to a beads-in-urn model called misanthrope process where particles are identified as urns and vacancies as beads that hop to a neighbouring urn situated in the direction opposite to the current. The hop rate, same as the magnitude of the current, depends on the total number of beads present in the departure and the arrival urn; we calculate it analytically and show that it does not satisfy the criteria required for a phase separation transition. Tumbling is generally detrimental to the stability of jamming; thus, our results for this restricted tumbling model strongly suggest that motility induced phase separation transition can not occur in one dimension.