Non-equilibrium phase transitions in active rank diffusions

TL;DR

This paper studies the complex phase behavior of active particles with Coulomb interactions in one dimension, revealing multiple phase transitions and clustering phenomena through analytical and numerical methods.

Contribution

It introduces a large N limit analytical framework for active rank diffusion with Coulomb interactions, uncovering diverse phase transitions and clustering behaviors.

Findings

Identification of multiple phase transitions including smooth, shock, and clustered phases.

Derivation of coupled Burger's equations describing density evolution.

Validation of theoretical predictions through extensive numerical simulations.

Abstract

We consider N run and tumble particles in one dimension interacting via a linear 1D Coulomb potential, an active version of the rank diffusion problem. It was solved previously for N = 2 leading to a stationary bound state in the attractive case. Here the evolution of the density fields is obtained in the large N limit in terms of two coupled Burger's type equations. In the attractive case the exact stationary solution describes a non-trivial N-particle bound state, which exhibits transitions between a phase where the density is smooth with infinite support, a phase where the density has finite support and exhibits "shocks", i.e. clusters of particles, at the edges, and a fully clustered phase. In presence of an additional linear potential, the phase diagram, obtained for either sign of the interaction, is even richer, with additional partially expanding phases, with or without shocks. Finally, a general self-consistent method is introduced to treat more general interactions. The predictions are tested through extensive numerical simulations.