Diagnostic of spatial organization in the Vicsek Model

Collective motion is the name given to the complex behavior exhibited by a set of moving elements interacting with each other. The moving elements are usually called ``self-propelled particles'' and normally represent living organisms with some kind of simple interaction rules with their neighbors. Schooling, swarming, herding and flocking are examples of a wide variety of collective behaviors exhibited by groups of animals, insects or bacteria. The interest on models of self-propelled particle is to understand how and why individuals become unified groups. We propose a method to measure the cohesion of collective motion exhibited by self-propelled particles and compare to the already existent methods. We perform this study using the well-known Vicsek model. In this model, all the elements move with a constant absolute velocity and at each time step they assume the average direction of their neighbors into a chosen radial distance. A random angle is added to the average direction of each element, which are considered to be a natural perturbation caused by the many stochastic and deterministic factors affecting the motion of the living organisms. The most common diagnostic for measure the self-organization of the system is the average normalized vectorial velocity. This order parameter is capable to quantify the directional organization took by the particles but omits the spatial distribution of the set. We introduce an order parameter based on Informational Entropy defined by Claude Shannon applied separately on the positions and directions distributions. This method gives a normalized quantification of spatial and directional organization of the system. The results obtained by this order parameter reveal an unexpected behavior not detected by the well-established order parameters. In some cases, the maximum cohesion does not occurs for trivial values of noise and interaction radius.