Run-and-tumble particles with 1D Coulomb interaction: the active jellium model and the non-reciprocal self-gravitating gas

TL;DR

This paper extends the analysis of 1D run-and-tumble particles with Coulomb interactions by exploring their behavior in a harmonic trap and under non-reciprocal forces, revealing phase transitions and density shock phenomena.

Contribution

It introduces the active jellium model with harmonic confinement and analyzes a non-reciprocal interaction case, providing explicit stationary density expressions and identifying phase transitions.

Findings

Bounded support density in harmonic confinement

Transitions between different edge behaviors including shocks

Explicit stationary density with broken symmetry in non-reciprocal case

Abstract

Recently we studied $N$ run-and-tumble particles in one dimension - which switch with rate $\gamma$ between driving velocities $\pm v_0$ - interacting via the long range 1D Coulomb potential (also called rank interaction), both in the attractive and in the repulsive case, with and without a confining potential. We extend this study in two directions. First we consider the same system, but inside a harmonic confining potential, which we call "active jellium". We obtain a parametric representation of the particle density in the stationary state at large $N$, which we analyze in detail. Contrary to the linear potential, there is always a steady-state where the density has a bounded support. However, we find that the model still exhibits transitions between phases with different behaviors of the density at the edges, ranging from a continuous decay to a jump, or even a shock (i.e. a cluster of particles, which manifests as a delta peak in the density). Notably, the interactions forbid a divergent density at the edges, which may occur in the non-interacting case. In the second part, we consider a non-reciprocal version of the rank interaction: the $+$ particles (of velocity $+v_0$) are attracted towards the $-$ particles (of velocity $-v_0$) with a constant force $b/N$, while the $-$ particles are repelled by the $+$ particles with a force of same amplitude. In order for a stationary state to exist we add a linear confining potential. We derive an explicit expression for the stationary density at large $N$, which exhibits an explicit breaking of the mirror symmetry with respect to $x=0$. This again shows the existence of several phases, which differ by the presence or absence of a shock at $x=0$, with one phase even exhibiting a vanishing density on the whole region $x>0$. Our analytical results are complemented by numerical simulations for finite $N$.