Non-equilibrium phase transitions in active rank diffusions

Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & PSL University, Sorbonne Université, Université Paris Cité - 75005 Paris, France

Received: 11 October 2023
Accepted: 24 January 2024

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 the 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.