Bulk topological states in a new collective dynamics model

Pierre Degond (1) , Antoine Diez (2) , Mingye Na (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Department of Mathematics [Imperial College London]
3 SUSTech - Southern University of Science and Technology

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.

Mots clés: individual-based model; macroscopic model; self-organization; topological phase transition; winding number; order parameter

Domaines: Mathématiques [math]

Pierre Degond, Antoine Diez, Mingye Na. Bulk topological states in a new collective dynamics model. SIAM Journal on Applied Dynamical Systems, 2022, 21 (2), pp.1455-1494. ⟨10.1137/21M1393935⟩. ⟨hal-03119172v4⟩