Vicsek model by time-interlaced compression: A dynamical computable information density

Collective behavior, both in real biological systems and in theoretical models, often displays a rich combination of different kinds of order. A clear-cut and unique definition of "phase" based on the standard concept of the order parameter may therefore be complicated, and made even trickier by the lack of thermodynamic equilibrium. Compression-based entropies have been proved useful in recent years in describing the different phases of out-of-equilibrium systems. Here, we investigate the performance of a compression-based entropy, namely, the computable information density, within the Vicsek model of collective motion. Our measure is defined through a coarse graining of the particle positions, in which the key role of velocities in the model only enters indirectly through the velocity-density coupling. We discover that such entropy is a valid tool in distinguishing the various noise regimes, including the crossover between an aligned and misaligned phase of the velocities, despite the fact that velocities are not explicitly used. Furthermore, we unveil the role of the time coordinate, through an encoding recipe, where space and time localities are both preserved on the same ground, and find that it enhances the signal, which may be particularly significant when working with partial and/or corrupted data, as is often the case in real biological experiments.