Geometric and Nonequilibrium Criticality in Run-and-Tumble Particles with Competing Motility and Attraction

TL;DR

The paper investigates run-and-tumble particles with explicit nearest-neighbor attraction (IRTP) on a square lattice to understand how motility and attraction shape motility-induced phase separation (MIPS). By analyzing geometric percolation and using finite-size scaling and Binder cumulants, the authors locate a critical line in the ω–J plane where percolation and MIPS occur simultaneously, with continuously varying exponents along this line. They show that percolation in IRTPs belongs to the Z2P universality class while the accompanying MIPS transition falls into the Ising superuniversality class, with certain scaling functions remaining identical to equilibrium lattice-gas behavior. The findings connect nonequilibrium active matter criticality to classical Ising-like critical behavior, revealing a robust Ising-like superuniversality despite nonequilibrium dynamics and activity-driven modifications of microscopic interactions.

Abstract

Self-propulsion in run-and-tumble particles (RTPs) generates effective attractive interactions that can drive motility-induced phase separation (MIPS), a phenomenon absent in passive systems. Here, we investigate RTPs in the presence of explicit attractive interactions and show that, at high motility, such interactions can suppress MIPS, yielding a homogeneous phase. Upon further increasing the attraction strength, phase separation reappears, giving rise to a re-entrant transition. We characterize this transition by analyzing the percolation properties of dense clusters, which provide geometric signatures of phase separation. Along the resulting critical line, we find continuously varying critical exponents, while certain scaling functions remain unchanged and coincide with those of equilibrium lattice gas models undergoing interacting percolation, which is in the Ising-percolation universality class. These results reveal that the MIPS transition in interacting RTP systems exhibit Ising superuniversality, thereby establishing a connection between nonequilibrium active matter and classical critical behavior.

Figures (13)

• Figure 1: (Color online) Dynamics of IRTP model: An RTP (the central particle, marked red), chosen out of $N$, with site index $\mathbf{i}$ and internal orientation $\theta_k= \frac{\pi}{2}$ (pointing up) can (a) run or (b) tumble with indicated rates. Here $r={\rm min} (1, e^{-\Delta E})$ is the Metropolis rate with respect to Eq. \ref{['eq: E']}, the parameter $0\le p<1$ generates translational diffusion, and $\omega>0$ is a constant tumble rate. Note that the hardcore nature of RTPs does not allow the particle at ${\bf i}$ to move to its left, as this site is already occupied by another RTP.
• Figure 2: A snapshot of the steady-state of the IRTP system for $\omega=0.015$ and $J=0.0$, on a $6\ell\times2\ell$ simulation box (with periodic boundary conditions on both directions) with $\ell=36$. The four square sub-boxes of size $\ell\times \ell$ are placed in dense and dilute regions to determine coexisting densities $\rho_+$ (liquid) and $\rho_-$ (gas), respectively.
• Figure 3: (Color online) Density plot of the percolation order parameter $\phi= \langle s_{\text{max}} \rangle/L^2$ in the $\omega$-$J$ plane. The estimated critical points (circles) are joined by a line; this critical line separates the percolating phase from a nonpercolating one, and it coincides with the critical line of MIPS transition shown later in Fig. \ref{['fig:phase']}. The system size is $64\times64$ and $\rho = 0.5$.
• Figure 4: (Color online) Finite-size scaling of percolation transition in IRTP model at $J_{c}=0.6,~\rho_{c}=\rho=\frac{1}{2}$ on a $L\times L$ square lattice. (a) Estimation of critical point $\omega_{c} = 0.0225(2)$ from the crossing point of $B_L$ vs $\omega$ curve for different system sizes: $L=48,~64,~96,~128,~256$. Panels (b), (c), and (d) provide scaling collapse of $B_L,~\phi,~\chi$ according to Eq. \ref{['eq:FSS']} with $\nu = 0.82(1)$, $\beta = 0.042(2)$ and $\gamma = 1.556(20).$
• Figure 5: (Color online) Density plot of order parameter $\tilde{\phi}$ [Eq. \ref{['eq: MIPS_Order_Par']}], for a $6\ell\times2\ell$ system to study phase separation transition with $\ell=36$. Estimated critical points (circles) are shown along with the critical line of percolation transition obtained in Fig. \ref{['fig:Phaseplot']}. Within error limits they match well, suggesting that the percolation transition and phase separation transition occurs simultaneously and they share the same critical points along the critical line. Note that the conserved particle density taken for this case is $\rho=0.5$.