A particle-field approach bridges phase separation and collective motion in active matter

Active matter, systems of interacting self-propelled particles, exhibit a wide range of collective behaviors far from thermodynamic equilibrium. Two key phenomena are motility-induced phase separation (MIPS), observed in self-propelled hard discs due to self-propulsion and isotropic repulsion, and orientational ordering and collective motion in self-propelled rods due to velocity alignment. While theoretical understanding exists for these phenomena individually, a unified framework is lacking. Existing models for anisotropic particles are complex, hindering analytical studies and systematic coarse-graining. Experimental evidence for MIPS in self-propelled discs remains scarce, with recent findings suggesting that torques disrupt MIPS. Furthermore, the sources of alignment interactions are diverse and include inelastic collisions, hydrodynamic interactions, or particle shape. The collective dynamics of cells, often modeled using soft deformable particles or phase fields, constitute a different class due to the coupling of shape and activity. The goal of this research is to develop a novel theoretical framework that encompasses both MIPS and orientational ordering, bridging the gap between scalar and vectorial active matter and providing a more comprehensive understanding of active systems.

The literature extensively explores both MIPS and collective motion in active matter. MIPS, initially described theoretically for run-and-tumble bacteria and later for self-propelled hard discs, is characterized by the formation of dense aggregates in a dilute environment. This phenomenon has been linked to density-dependent speed reduction upon collisions and active pressure. However, experimental validation of MIPS in self-propelled discs is lacking. Studies on aligning active matter focus on the emergence of collective motion and orientational order due to velocity alignment interactions. These interactions arise from diverse mechanisms, including inelastic collisions, hydrodynamic interactions, and anisotropic particle shape. Self-propelled rod-like particles exhibit a rich variety of collective phenomena, including mesoscale turbulence, band formation, and complex phase diagrams. Previous models have focused on either alignment-based models which are analytically tractable, or realistic models explicitly incorporating particle shape but often lacking analytical accessibility. This paper attempts to combine the benefits of both approaches.

The authors introduce a novel particle-field approach where individual active particles are represented by anisotropic Gaussian fields. The interaction between particles is derived from minimizing their mutual overlap, leading to an analytical expression for the interaction force and torque. The anisotropy of the Gaussian field, controlled by a parameter ε, determines the particle shape, ranging from circular (ε=0) to needle-like (ε→∞). The model incorporates self-propulsion, anisotropic repulsion, translational and rotational mobilities, and noise terms to account for fluctuations in position and orientation. The overdamped dynamics of the particles are described by coupled equations for the particle position and orientation. The interaction force and torque are derived from minimizing the overlap energy, which explicitly depends on the particle shape. The key control parameters are particle anisotropy (ε) and self-propulsion speed (v₀). Numerical simulations using a stochastic Euler scheme are performed to investigate the collective behavior. For the analytical studies, a Fokker-Planck equation is derived for the one-particle density, and a mode expansion is performed to obtain equations for coarse-grained order parameters (density ρ, polar order parameter **p**, and nematic order parameter Q). Linear stability analysis of the homogeneous state is performed to identify the conditions for MIPS.

The study reveals a continuous transition from MIPS to collective motion by varying particle anisotropy. For self-propelled discs (ε=0), MIPS is observed, consistent with previous findings. However, MIPS breaks down for even small anisotropies (ε > 0) due to an effective torque that dissolves the polar boundary layer crucial for maintaining aggregates. This destabilization of MIPS is attributed to the combined action of anisotropic repulsion and self-propulsion. Increasing anisotropy leads to the emergence of orientational order. Initially, the system exhibits global disorder with local nematic order. Further increase in anisotropy results in the formation of large-scale polar domains, even though the interaction potential is strictly nematic. The system's symmetry is a dynamic property, not fixed by the interaction potential. The emergence of polar order is not captured by mean-field theory but is linked to collision kinetics and the formation of polar clusters, indicated by an enhanced probability of parallel motion. A bistable coexistence region of polar and nematic order is observed for intermediate self-propulsion speeds, where the system switches stochastically between polar domains and nematic bands. This bistability, despite purely nematic interactions, highlights the emergent nature of the ordered states and is influenced by the balance between self-propulsion and repulsive forces. Analytical calculations based on a mode expansion of the Fokker-Planck equation reveal how the particle anisotropy destabilizes MIPS and promotes the emergence of orientational order. A necessary condition for MIPS is derived, showing that the critical speed for MIPS increases with anisotropy. Numerical measurements of the pair distribution function support the theoretical analysis and show the enhancement of parallel motion for anisotropic particles.

The findings of this study offer a comprehensive understanding of the transition from MIPS to collective motion in active matter, driven by changes in particle anisotropy. The particle-field model successfully unifies seemingly disparate phenomena within a single framework, bridging the realms of scalar (MIPS) and vectorial (orientational ordering) active matter. The destabilization of MIPS for anisotropic particles is explained by the emergence of a torque that disrupts the polar boundary layer around aggregates. The coexistence of polar and nematic order, despite purely nematic interactions, underscores the emergent and dynamic nature of symmetry in active systems. The dependence of emergent order on both particle shape and the balance between self-propulsion and repulsion is clearly demonstrated. The study advances our understanding of active matter by providing detailed mechanistic explanations for observed phenomena and highlighting the importance of both scalar and vector order parameters.

This research presents a novel particle-field model that successfully bridges phase separation and collective motion in active matter. The model accounts for the transition from motility-induced phase separation (MIPS) to orientational ordering by varying particle anisotropy and self-propulsion speed. The results demonstrate that the symmetry of the emergent patterns is not solely determined by the microscopic interaction but is an emergent property arising from the interplay of particle shape, self-propulsion, and collision kinetics. Future work could focus on extending the model to include more complex interactions, such as hydrodynamic interactions or flexible filaments, and investigating the influence of external fields or confinement.

While the model provides a comprehensive framework, there are limitations to consider. The analytical calculations rely on approximations, such as linearizing the Fokker-Planck equation and performing a mode expansion. The numerical simulations are computationally intensive, limiting the system size and simulation time. The model focuses on overdamped dynamics, neglecting inertial effects. Finally, the study primarily considers homogeneous systems, neglecting the possible influence of boundaries or external fields.