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.
Bibliographical noteFunding Information:
We thank Irene Giardina and Stefania Melillo for fruitful discussions. This publication has been produced as a part of the Italian-Israeli scientific cooperation project “IDA-CReBS”: A.C., A.P., and M.V. were supported by Ministero degli Affari Esteri e della Cooperazione Internazionale, and D.L. was supported by the Israel Ministry of Science and Technology (Grant No. 3-15150). A.C. and M.V. were supported by European Research Council (Advanced Grant “RG.BIO” No. 785932). A.P. acknowledges the financial support of Regione Lazio through the “Progetti Gruppi di Ricerca” (Grant No. 85-2017-15257) and from the MIUR PRIN 2017 (Project No. 201798CZLJ). D.L. was supported by the Israel Science Foundation (Grant No. 1866/16) and the U.S.-Israel Binational Science Foundation (Grant No. 2014713). P.M.C., D.L., and S.M. were supported by the National Science Foundation Physics of Living Systems program (Grant No. 1504867). P.M.C. was partially supported by the Materials Research Science and Engineering Center program of the National Science Foundation (Award No. DMR-1420073).
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