Crystal to liquid cross-over for active particles with inverse-square power-law interaction

TL;DR

This paper investigates the transition from crystal-like to liquid-like density profiles in a one-dimensional system of active particles with inverse-square interactions, combining numerical simulations and analytical methods.

Contribution

It introduces an analytical approach to describe the crystal-liquid crossover in active particles with inverse-square interactions, supported by numerical simulations.

Findings

Density profile transitions from crystal to liquid with increasing activity

Identification of an intermediate Wigner semi-circle density profile

Analytical covariance calculations corroborated by simulations

Abstract

We consider a one-dimensional system comprising of $N$ run-and-tumble particles confined in a harmonic trap interacting via a repulsive inverse-square power-law interaction. We numerically compute the global density profile in the steady state which shows interesting crossovers between three different regimes: as the activity increases, we observe a change from a density with sharp peaks characteristic of a crystal region to a smooth bell-shaped density profile, passing through the intermediate stage of a smooth Wigner semi-circle characteristic of a liquid phase. We also investigate analytically the crossover between the crystal and the liquid regions by computing the covariance of the positions of these particles in the steady state in the weak noise limit. It is achieved by using the method introduced in Touzo {\it et al.} [Phys. Rev. E {\bf 109}, 014136 (2024)] to study the active Dyson Brownian motion. Our analytical results are corroborated by thorough numerical simulations.