Bird Flock Simulations Reveal Surprising Reversal of Particle Clustering Behaviour

Researchers investigating motility-induced phase separation have observed a surprising phenomenon: that increasing activity can sometimes lead to a return to a homogeneous state, known as reentrance. Letian Chen and Luke K. Davis, both from the School of Mathematics and Maxwell Institute for Mathematical Sciences at the University of Edinburgh, demonstrate this reentrance within a Hamiltonian flocking model, a system designed to mimic the collective behaviour of bird flocks. Their work, establishing a collaborative link between equilibrium and non-equilibrium materials, identifies a competition between drive amplitude and kinetic frustration as the key mechanism, revealing that strong spin-velocity coupling effectively suppresses diffusion and prevents the formation of clusters. This finding offers a novel Hamiltonian framework for understanding reentrant behaviour not only in active matter but also across a broader range of physical systems.

Researchers have uncovered a surprising phenomenon in simulations of self-propelled particles, demonstrating that clustering, a hallmark of active matter, can unexpectedly diminish with increasing drive. This counterintuitive finding challenges conventional understanding of how energy input dictates collective behaviour in these systems. The work centres on motility-induced phase separation (MIPS), where self-motile and repulsive particles spontaneously organise into dense clusters. Typically, enhancing the particles’ self-propulsion intensifies this clustering, but this study reveals a reentrance to a homogeneous state at sufficiently high drive strengths. This reentrance emerges from a simplified, conservative model inspired by the collective behaviour of bird flocks, replicating the clustering seen in active systems and offering a new perspective on the underlying mechanisms. Numerical simulations demonstrate that the reentrance is driven by a delicate balance between the strength of spin-velocity coupling and limitations imposed by particle mobility. Strong spin-velocity coupling effectively restricts particle movement, suppressing the transverse diffusion necessary for cluster formation and ultimately preventing phase separation. This suppression of diffusion acts as a form of ‘arrest’, halting the system’s ability to form clusters and providing a crucial link between reentrant behaviour observed in both equilibrium and non-equilibrium materials, suggesting a common underlying principle governing their phase transitions. By establishing a conservative framework for understanding MIPS, this work opens avenues for exploring analogous phenomena in a wider range of physical systems and potentially informing the design of materials with tailored collective behaviours. The researchers developed a parameter-free scaling ansatz, a predictive relationship, that accurately captures the observed non-monotonic phase boundaries in their simulations, further solidifying their mechanistic understanding. A numerical investigation of particle dynamics forms the basis of this work, employing the Euler-Maruyama algorithm to integrate equations of motion with a time step of 10−4. This algorithm is well-suited for simulating stochastic differential equations, accurately capturing the interplay of deterministic and random forces governing particle behaviour. Simulations were conducted within a two-dimensional box of size L2, utilising periodic boundary conditions to minimise edge effects and effectively simulate an infinite system. The packing fraction, maintained at a constant value of η = Nπ(σ/2)2/(L2), dictated the density of particles within the simulation domain. To systematically examine liquid-gas phase coexistence, slab simulations were implemented, extending the simulation box to a rectangular geometry (Lx = 3Ly). Particles were initially arranged in a dense strip, creating a controlled asymmetry that promotes the formation of stable liquid-gas interfaces aligned perpendicular to the x-axis, circumventing high nucleation barriers and accelerating the observation of phase separation. Each parameter set underwent simulations for a total duration of 4 × 107 time steps, allowing ample time for the system to reach a steady state. A relaxation period, constituting the initial 30% of the trajectory, was discarded to ensure that all subsequent analyses were based on equilibrated data. Observables, such as density histograms, were monitored to confirm saturation and validate the equilibration process, with reported data representing averages calculated from 5 independent simulation replicas, enhancing the statistical robustness of the findings. Simulations reveal a striking reentrant phase behaviour in this Hamiltonian flocking model, with the spin-velocity coupling strength, K, acting as the primary control parameter. As K increases, the system transitions from a homogeneous gas to a phase-separated coexistence state, and then unexpectedly back to a homogeneous phase. This reentrance is quantified using the difference in high- and low-density peaks of the local density distribution, denoted as ∆ρ, which exhibits a non-monotonic dependence on K. Slab simulations corroborate these findings, demonstrating that ∆ρ(K) is indeed non-monotonic, confirming a genuine transition through a clustered phase. Further analysis demonstrates the robustness of this reentrance to variations in several dynamical parameters. The order parameter, ∆ρ, remains non-monotonic even with changes in the product of the translational and rotational friction coefficients, γt × γr, though shifts in the peak location are observed. The peak of the order parameter is consistently found around Kpeak ≈ √3γtγr, suggesting a direct relationship between the coupling strength and frictional forces. Investigations into finite size effects, using particle numbers ranging from N = 250 to N = 5000 at a fixed packing fraction, show qualitatively consistent binodal lines, indicating a true phase transition rather than a finite-size artifact. The maximum density contrast, ∆ρσ2, increases with system size, aligning with recent observations of reentrant behaviour in active Brownian particles. Varying the ferromagnetic strength, J, at a fixed K = 3/2 reveals that ∆ρ(J/T) transitions from zero below a critical value of J/T ≈ 1.2, where alignment is disrupted by thermal noise, to a rapidly rising and saturating value above this point. This confirms that ferromagnetic alignment is a prerequisite for coupling-induced clustering and, consequently, for reentrance. Altering the global packing fraction between 0.24 and 0.34 produces only minor shifts in overall densities, without impeding the reentrant behaviour. Researchers have long sought to understand how order emerges from seemingly chaotic systems, and this work offers a compelling new perspective on the delicate balance between movement and organisation. The re-emergence of a uniform state in a system of self-propelled particles, previously thought to inevitably cluster, is a surprising and potentially significant finding. For years, the challenge has been to reconcile the tendency of active elements, from bird flocks to cellular components, to self-organise with the need to maintain stability and prevent runaway aggregation. This research suggests a mechanism, rooted in the interplay of movement and constraint, that can actively dissolve such clusters. The key lies in a strong coupling between the particles’ direction of travel and their responsiveness to neighbours. This isn’t simply about stronger interactions, but about a fundamental shift in how particles diffuse and respond to crowding. By suppressing sideways movement, the system effectively ‘freezes’ degrees of freedom, preventing the formation of the dense, localised clusters characteristic of motility-induced phase separation. The fact that this behaviour arises from a relatively simple, Hamiltonian model, one that conserves energy, is particularly noteworthy, offering a potentially unifying framework for understanding similar phenomena in diverse physical and biological contexts. However, the simulations reveal a sensitivity to geometry, with the shape of the simulation space influencing the alignment of particles even as their overall distribution remains uniform. This suggests that real-world systems will likely exhibit more complex behaviours, shaped by the constraints of their environment. Future work could explore how these findings translate to three-dimensional systems, or how external forces and asymmetries might disrupt the re-entrant behaviour. Ultimately, this research doesn’t just explain a specific phenomenon; it highlights the power of subtle interactions to sculpt order from disorder, and opens new avenues for designing self-organising systems with predictable and controllable properties.