Inferring the size of a collective of self-propelled Vicsek particles from the random motion of a single unit

Inferring the number of interacting components in a complex, noisy, possibly non-equilibrium, system is of interest to various fields of science. Here, the authors present a method to extract the size of an ensemble of self-propelled Vicsek particles starting from the motion of a single unit by exploiting the analogy with the classical kinetic theory of gases. Inferring the size of a collective from the motion of a few accessible units is a fundamental problem in network science and interdisciplinary physics. Here, we recognize stochasticity as the commodity traded in the units' interactions. Drawing inspiration from the work of Einstein-Perrin-Smoluchowski on the discontinuous structure of matter, we use the random motion of one unit to identify the footprint of every other unit. Just as the Avogadro's number can be determined from the Brownian motion of a suspended particle in a liquid, the size of the collective can be inferred from the random motion of any unit. For self-propelled Vicsek particles, we demonstrate an inverse proportionality between the diffusion coefficient of the heading of any particle and the size of the collective. We provide a rigorous method to infer the size of a collective from measurements of a few units, strengthening the link between physics and collective behavior.