An Infinite Set of Integral Formulae for Polar, Nematic, and Higher Order Structures at the Interface of Motility-Induced Phase Separation

TL;DR

This paper derives an infinite set of integral formulae to describe the emergence of polar, nematic, and higher order structures at the interface of motility-induced phase separation, validated through simulations.

Contribution

It introduces a novel analytical framework of integral formulae for interfacial structures in MIPS, with some being exact for active Brownian particles.

Findings

Half of the integral formulae are exact for a broad class of active systems.

The integral formulae accurately predict interfacial ordering in simulations.

The approach enhances understanding of non-equilibrium phase separation phenomena.

Abstract

Motility-induced phase separation (MIPS) is a purely non-equilibrium phenomenon in which self-propelled particles phase separate without any attractive interactions. One surprising feature of MIPS is the emergence of polar, nematic, and higher order structures at the interfacial region, whose underlying physics remains poorly understood. Starting with a model of MIPS in which all many-body interactions are captured by an effective speed function and an effective pressure function that depend solely on the local particle density, I derive analytically an infinite set of integral formulae (IF) for the ordering structures at the interface. I then demonstrate that half of these IF are in fact exact for a wide class of active Brownian particle systems. Finally, I test the IF by applying them to numerical data from direct particle dynamics simulation and find that all the IF remain valid to a great extent.