Bulk topological states in a new collective dynamics model

Abstract

In this paper, we demonstrate the existence of topological states in a new collective dynamics model. This individual-based model (IBM) describes self-propelled rigid bodies moving with constant speed and adjusting their rigid-body attitude to that of their neighbors. In previous works, a macroscopic model has been derived from this IBM in a suitable scaling limit. In the present work, we exhibit explicit solutions of the macroscopic model characterized by a non-trivial topology. We show that these solutions are well approximated by the IBM during a certain time but then the IBM transitions towards topologically trivial states. Using a set of appropriately defined topological indicators, we reveal that the breakage of the non-trivial topology requires the system to go through a phase of maximal disorder. We also show that similar but topologically trivial initial conditions result in markedly different dynamics, suggesting that topology plays a key role in the dynamics of this system.

Body-attitude coordination in arbitrary dimension

“…A question is whether new behavior could be observed if more complex control variables are introduced. Recently, a new model in which the agents control their motion by their body attitude has been proposed in dimension n = 3 [24,26,29,30,31,40]. In this model, a reference frame describing the body attitude is attached to each agent.…”

“…There are similarities with the Vicsek model except in the alignment process, which operates on the full body frame rather than on the single velocity vector. Consequently, heterogeneities in the body frame distribution affect motion and alter the dynamics compared with the Vicsek model [26]. So far, previous works were restricted to dimension n = 3.…”

“…In these hydrodynamic models, the agents control variables have been replaced by their local means which only depend on position. This results in considerable information savings and make the models amenable to both intensive numerical simulations and theoretical qualitative analysis [9,10,26,63].…”

“…For the body orientation model, the passage from kinetic to hydrodynamic equations has been realized in [24,26,29,30,31,40] in dimension n = 3 only. Specifically, in [29], the particle dynamics was described by a stochastic differential equation leading to a kinetic equation of Fokker-Planck type.…”

“…In [24,40] this kinetic equation was studied from the viewpoint of phase transitions. Numerical comparisons of the PDMP with the hydrodynamic models were performed in [26]. In the present paper, we aim to derive the macroscopic equations corresponding to the same PDMP and Boltzmann-BGK kinetic equation as in [26] but in arbitrary dimensions n ≥ 3.…”