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

TL;DR

This work analyzes two extended 1D active-particle models with rank (Coulomb) interactions. First, the active jellium uses RTPs in a harmonic trap and yields a parametric stationary-density representation, revealing bounded support and multiple edge-phase transitions including shocks and delta peaks; a complete phase diagram is established and validated numerically. Second, a non-reciprocal rank-diffusion model coupled to a linear potential is solved exactly in the large-N limit, showing explicit breaking of spatial parity and a richer phase structure with phases lacking or containing shocks at the origin. The analysis relies on Dean–Kawasaki-type hydrodynamics for rank fields and a carefully constructed parametric solution, with special tractable cases (e.g., a=1/2) providing explicit densities. Together, these results offer exact insights into how confinement and non-reciprocity shape stationary states, edge behavior, and clustering in active matter with long-range interactions, and they open paths to more general potentials and multi-species extensions.

Abstract

Recently we studied $N$ run-and-tumble particles in one dimension - which switch with rate $γ$ 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 $N$.

Figures (6)

• Figure 1: Left panel: phase diagram of the active jellium model, (i.e. the active rank diffusion in a harmonic potential $V(x)=\frac{\mu}{2} x^2$) in the plane $(\frac{\kappa}{v_0}, \frac{\mu}{\gamma})$, with repulsive interactions for $\kappa>0$ and attractive for $\kappa<0$. Right panel : schematic representation of the total density $\rho_s(x)$ for each phase. The up arrows represent delta functions in the density, i.e. shocks/clusters of particles. In phases I, IIa and IIb there are no shocks. In phase I (which extends on either side of $\kappa=0$) the density vanishes at the edge (with an exponent $\frac{\gamma}{\mu}-1$ identical to the non-interacting case $\kappa=0$). In phases IIa and IIb the density has a finite jump at the edge (except for $\kappa=0$ where it diverges). The dotted line between phases IIa and IIb represents only a crossover where the density changes from concave to convex (not a true phase transition). In phase III there are shocks at the two edges, with a cluster of $+$ only particles at the right edge, and $-$ only at the left edge. In phase IV all the particles belong to a single cluster. In the text more explicit expressions are obtained on the special line $\frac{\mu}{\gamma}=2$, represented as a dotted red line on the phase diagram.
• Figure 2: Top: Comparison of the rank field $r(x)$ in the stationary state computed using numerical simulations with different values of $N$, with the analytical prediction for $a=1/2$ i.e. $\mu=2 \gamma$. In all cases $v_0=1$, $\mu=1$ and $\gamma=0.5$, and $\kappa$ varies to explore the 3 non-trivial regimes: $\kappa>v_0$, i.e. phase IIb (left), $0<\kappa<v_0$, i.e. phase IIa (center) and $-2v_0<\kappa<0$, i.e. phase III (right). In $r(x)$, the shocks appear as jumps for $N\to+\infty$ in the right panel. The dashed black lines correspond to the predictions of Eqs. \ref{['r_repulsive2_intro']}, \ref{['r_repulsive1_intro']} and \ref{['r_repulsive1_intro']}-\ref{['eq_edge_attractive_intro']} respectively. Bottom: Plot of the density $\rho_s(x)=r'(x)$ for the same parameters, obtained through numerical simulations. The dashed black lines correspond to the prediction \ref{['rhos_param']} (the delta peaks are not shown on the right panel).
• Figure 3: Left: Phase diagram of the non-reciprocal active rank diffusions. The phase diagram is symmetric upon $b \to - b$ and a parity transformation $x \to - x$. Right: the behavior of the total particle density in the four phases. The arrows denote delta function peaks in the density.
• Figure 4: Rank field $r(x)$ (top) and total density $\rho_s(x)=r'(x)$ (bottom) obtained from simulations with $\gamma=1$ and $v_0=1$, for increasing values of $N$. The delta peaks are not shown on the densities for the phases II and III, but they are visible as discontinuities in $r(x)$. Left: $a=0.4$ and $b=1$ (phase I). The dashed black line corresponds to the prediction for infinite $N$\ref{['rhos_nr_intro']}. Note the discontinuity of the density at $x=0$, with however no shock (no delta peak). Center: $a=1$ and $b=1$ (phase II). The dashed black line corresponds to \ref{['rhos_nr_neg2_intro']}. The density is zero for $x>0$ for any $N$ (with a delta peak at $x=0$). Right: $a=0.4$ and $b=3$ (phase III) the dashed black line corresponds to \ref{['rhos_shock_intro']}. Larger values of $N$ are used here since the convergence in $N$ is slower than in the other phases.
• Figure 5: Sign of the integration constant $C$. It vanishes on the line $\frac{\mu}{\gamma}=1 + \frac{\kappa}{v_0}$.