Abstracts Mathematics

How Self-Propelled Particles (SPP) Models can explain the Origin of Collective Behaviour

by Hajar Abutuqayqah

In this work we explore the collective behavior of self–driven particles in active matter. We analyze the mechanisms of the emerging circular motion through Vicsek–like models with a leader. We consider both a circular trajectory of the leader (”shepherd” – a hierarchical leader that orbits a fixed circle with a constant speed) and linearly–moving leader (i.e. a hierarchical leader that moves with a constant speed in a fixed direction) or a group of leaders. We show that the circular motion may not be supported by Vicsek-like models. We also analyze models based on the inter–particle interaction forces. We considered rather large systems which allows to analyze thermodynamic–like behavior with different phases. We report the existence of different phases of the system, gaseous, liquid, swirlonic, collapsed and solid phases, depending on the system parameters. Here the new swirlonic phase is composed of new quasi-particles, comprised of active particles orbiting a common center of mass. Surprisingly, we do not observe the coexistence of different phases, although it is common for conventional matter. Finally, to explore a possible coexistence of phases we analyze systems, comprised of active particles of two different types, which differ either by their motility of by the interaction potential. We observed a rich behavior of these system, with numerous phases and phase coexistence. For instance, we found that the novel swirlonic phase might coexist in such systems with gaseous and liquid phase, or even be dissolved in these phases. Several theoretical approaches are developed and extensive numerical simulations are pereformed in this work. Our theoretical predictions are in a good quantitative and qualitative agreement with the numerical results.