The Impact of Initial Velocity and Repulsion Radius on Phase Transitions in Self-Propelled Particle Systems

Phase transitions in the collective motions of self-propelled particles are a fascinating area of study that bridges the boundaries of biology, physics, and computational science. These phenomena occur in systems where particles, endowed with the ability to move independently, transition from disordered to ordered states. The transitions resemble phase changes observed in physical systems, such as the shift from liquid to solid or the onset of magnetism in materials. Despite significant advancements in understanding these transitions, many of the mechanisms influencing them remain elusive, particularly the roles played by the initial velocity of particles and the radius within which they repel one another to avoid collisions.

In recent research, Salma Moushi and her colleagues at the University of Hassam II, Morocco, explored these two critical factors. Their findings, published in The European Physical Journal E, reveal how the initial velocity and repulsive radius surrounding each particle significantly influence the conditions under which phase transitions occur in self-propelled particle systems. The team’s work sheds light on collective motion in a range of natural systems, from bacterial colonies and animal flocks to human crowds.

Self-propelled particle systems are ubiquitous in nature. They include flocks of birds coordinating in flight, schools of fish maneuvering as one, swarms of insects, and even microscopic entities such as bacteria forming collective movements. Despite the diversity of these systems, they share common traits: the ability to transition from random, uncoordinated motion to synchronized, collective behavior. These transitions often depend on a balance between alignment forces that pull particles into a coherent group and disruptive factors, such as noise, that introduce randomness.

The researchers focused on understanding how these transitions are influenced by two specific parameters: the initial speed at which particles move and the repulsion radius. The repulsion radius defines the distance within which particles steer away from their neighbors to avoid overlapping or colliding. Moushi’s team demonstrated that both parameters play a significant role in determining the critical conditions for collective motion.

In their study, the researchers used a simulation technique based on the classical Vicsek model. This model is a cornerstone in the study of self-propelled particle systems and offers a framework for analyzing collective dynamics. It describes particles that move at constant speed in random directions but adjust their headings to align with their neighbors within a defined radius. By introducing parameters for initial velocity and repulsion radius into the Vicsek model, the researchers created a virtual system in which they could closely observe phase transitions.

Through their simulations, the team discovered that two primary factors influence the emergence of collective motion: particle density and critical noise levels. Density refers to the number of particles per unit area, while noise describes the level of random fluctuations affecting particle movement. For the system to form an ordered, coherent structure, it must achieve a sufficiently high density while keeping noise below a critical threshold. However, Moushi and her colleagues found that these thresholds are not fixed values; instead, they vary depending on the initial velocity and repulsion radius of the particles.

The researchers observed that higher initial velocities tend to promote more robust transitions to ordered motion. When particles move faster, their interactions with neighbors become more frequent and impactful, enhancing the likelihood of synchronization. Conversely, slower-moving particles have weaker interactions, making it more challenging to achieve collective motion. Similarly, the repulsion radius also plays a key role. A larger repulsion radius means particles interact over greater distances, creating a wider field of influence and encouraging alignment. A smaller radius, on the other hand, limits interactions, potentially reducing the cohesiveness of the group.

The interplay between these two factors has profound implications for the behavior of natural systems. For instance, in a school of fish, individuals that swim faster or maintain larger personal spaces may play a critical role in shaping the overall dynamics of the group. Similarly, in a crowd of humans, walking speed and personal space preferences might influence how efficiently the group moves in unison, particularly in crowded environments like concerts or evacuation scenarios.

This study also opens the door to better understanding the behavior of microorganisms such as bacteria. Bacteria often exhibit collective behaviors, such as biofilm formation or swarming, which are crucial to their survival and pathogenicity. By manipulating factors like movement speed and repulsion range, scientists might influence bacterial behavior, potentially offering novel strategies for controlling harmful microbial populations.

In addition to biological systems, the findings have applications in designing autonomous systems and robotics. In swarms of drones, for example, ensuring collective motion depends on parameters like the speed of individual drones and the communication range between them. Insights from Moushi’s research could help engineers fine-tune these parameters to create more efficient and resilient systems for tasks such as search-and-rescue missions, environmental monitoring, or large-scale agriculture.

Furthermore, the research contributes to the theoretical understanding of non-equilibrium systems in physics. Unlike equilibrium systems, which settle into a stable state over time, non-equilibrium systems like self-propelled particle systems are inherently dynamic. The emergence of order in such systems is a fundamental problem in physics, and the insights gained from studying phase transitions add depth to this complex field.

Moushi’s findings also underscore the importance of initial conditions in collective behaviors. The fact that the starting velocity and repulsion radius of individual particles can drastically alter the nature of phase transitions suggests that seemingly small changes at the individual level can have significant ripple effects at the collective level. This concept resonates beyond the realm of physics and biology, offering analogies to social systems where individual behaviors or preferences can influence group dynamics.

Reference: I. Tarras et al, Effect of repulsive interaction and initial velocity on collective motion process, The European Physical Journal E (2024). DOI: 10.1140/epje/s10189-024-00455-2