Cooling mechanism controls motility-induced phase separation in inertial active liquids

TL;DR

The paper investigates how inertia modifies motility-induced phase separation (MIPS) in active matter, showing that a cooling mechanism arising from density-dependent collisions couples density, polarization, and temperature and can drive MIPS even when volume exclusion alone is not the main driver. It develops an active Direct Simulation Monte Carlo (ADSMC) method and a kinetic theory based on Boltzmann–Fokker–Planck dynamics with Enskog corrections, establishing a density-temperature-polarization instability controlled by a density-dependent collision rate. The key result is that MIPS in inertial active systems requires Enskog-type corrections (not just molecular chaos), and the phase behavior smoothly connects to granular-like cooling in the high-density limit, vanishing in the high-inertia and overdamped limits unless excluded volume is present. The work provides a fast simulation framework and a concrete kinetic mechanism linking inertial active matter to granular physics, with implications for bridging micro- and macroscopic active systems.

Abstract

Motility-induced phase separation (MIPS) is a central collective phenomenon in active matter, theoretically established in the overdamped regime. We discover that the dynamical origin of MIPS is fundamentally altered by inertia, which induces a cooling mechanism absent in overdamped active matter. This conclusion is supported by an active variant of the direct simulation Monte Carlo method and by a kinetic theory for inertial self-propelled hard spheres derived from the microscopic dynamics. In contrast to the overdamped case, both analyses demonstrate that inertial MIPS does not rely on volume exclusion but on a cooling mechanism involving density, polarization, and temperature fields. This mechanism emerges from the competition between activity and a density dependent collision rate, arising from spatial correlations between colliding particles. These findings open a pathway to fundamentally connect inertial active matter with granular physics.

Figures (4)

• Figure 1: Inertia-induced cooling in active systems. (a) Particles are endowed with both a velocity ${\bf v}$ (black arrow) and an active force $\gamma v_0 \mathbf{e}$ directed along the direction normal to the hemisphere, going from gray to color. (b) After a collision, velocities are instantaneously changed while active forces remains unchanged. (c) In ADSMC, time and space are discretized. At each time step the first phase is free motion (red arrow), the second phase is interaction: pair collisions in the same cell (orange star), and (when implemented) "excluded volume events" of single particles with overcrowded cells (purple star). (d) Collisional cooling effect. Cap colors identify the same particles before and after the collision. The modulus of the colored arrows represents the "active power" $\gamma v_{0} (\mathbf{v} \cdot \mathbf{e})$ (in $d=2$), which decreases after impact (from red to blue). Since $\gamma v_{0} (\mathbf{v} \cdot \mathbf{e})$ determines the value of the local temperature field, see Eq. \ref{['eq:kinen']}, inertial active particles undergo an effective cooling mechanism during collisions.
• Figure 2: Numerical results obtained using ADSMC with the Enskog collision rate. (a), (c) Snapshot phase diagrams in the plane of Péclet number $\ell$ and packing fraction $\phi$ at Stokes number $St = 1$. (b), (d) Snapshot phase diagrams in the $(\phi, St)$ plane for $\ell = 20$. Panels (a)–(b) correspond to simulations without excluded-volume interactions, while (c)–(d) include excluded volume. Red lines serve as guides to the eye. The remaining simulation parameter is $T_0 = 1$.
• Figure 3: Steady-state temperature $\tilde{T}$, normalized by its dilute-limit value $\tilde{T}_{\phi \to 0}$, as a function of the packing fraction $\phi$ for homogeneous configurations, demonstrating the cooling mechanism. Simulations are performed using both the Molecular Chaos (red) and Enskog (green) collision rates. Colored dashed lines indicate the corresponding theoretical predictions, while the light-blue dashed line serves as a guide to the eye, for the scaling $\tilde{T} \approx z \sim \phi^{-2/3}$. Inset: theoretical prediction for the $k^2$-order contribution to the density eigenvalue $\lambda_0^{(2)}$ as a function of $\phi$, for Enskog (solid lines) and Molecular Chaos (dashed lines) collision rates. The colored numbers in the plot denote the Stokes number values. In all data, $\ell = 100$ and $\tilde{T}_0 = 2 \times 10^{-4}$. Note that $\mathrm{St}$ is varied by keeping $m = 1$ and $\tau = 1$ fixed while changing $\gamma$.
• Figure 4: Theoretical phase diagrams from Eq. \ref{['lambda']}. (a) phase diagram in the plane of Péclet number $\ell$ and packing fraction $\phi$ at Stokes number $St=1$. (b) Phase diagram in the plane of $\phi$ and $St$ at $\ell=20$. The violet region (MIPS) highlights the region where the homogeneous density is unstable ($\lambda_0^{(2)}>0$).