A quantitative analysis of the original Vicsek model establishes an explicit relationship between the coupled structural dynamics of the particles and the degree of collective order that emerges in the system. The work addresses an open question in the physics of active systems: how correlations in the position and orientation of the agents translate into the onset of ordered collective motion.

The Vicsek model, proposed in 1995, describes particles moving at constant speed that adjust their direction towards the average of their neighbours within a given radius, with added angular noise. Despite its simplicity, it reproduces the transition between a disordered state and a state of collectively aligned motion, and has become the reference paradigm for studying active matter. Quantifying precisely how structural correlations feed the global order has, however, proved difficult.

The study examines the Vicsek model in its original formulation and quantifies the coupling between the structural dynamics of the system and the order parameter that measures the alignment of the velocities. The resulting relationship allows the collective behaviour to be predicted from the coupled structural fluctuations of the particles.

The significance of the result lies in providing a quantitative framework to connect microscopic and macroscopic descriptions in active systems, a bridge that could be extended to bird flocks, fish schools, bacterial suspensions and autonomous robots. Understanding how collective order emerges from local interactions remains one of the central questions of active matter.