Pierre Degond, Collective dynamics and topology
Collective dynamics occurs when a system of self-propelled particles organizes itself into a coherent motion, such as a flock, a vortex, etc. Recently, the question of the existence of collective states with non-trivial topology has been raised. Topologically non-trivial states are extremely robust i.e. « topologically protected » against localized perturbations. In this work, we consider a system of self-propelled solid bodies interacting through local full body alignment up to some noise. In the large-scale limit, this system can be described by hydrodynamic equations with topologically non-trivial explicit solutions. We will investigate to what extent topologically non-trivial collective dynamics states are ‘protected’ against perturbations. In passing, we will raise interesting mathematical questions underpinning the analysis of collective dynamics systems.
Amic Frouvelle, Rigid body alignment : phase transition and links with quaternions and rodlike polymer suspensions
We are interested in an alignment model of rigid bodies (for instance birds aligning their movement direction and their wings), submitted to angular noise. To this aim we consider a kinetic model of alignment-diffusion for which the “velocity variable” is a rotation matrix. We present an interesting link between this model and a generalization, in dimension four, of the Maier—Saupe model for alignment in suspensions of rodlike polymers. This is due to the fact that a rotation may be represented by a unit quaternion (or its opposite, linked to the fact that “rods” has no preferred direction). We obtain the phase diagram of this model:
when the alignment force is low, the unique equilibrium is the uniform distribution,
when the force is sufficiently high, there exists a unique family of stable equilibria (concentrated around a given rotation matrix),
in between, we have stability of these two types of stationary states (“disordered” and “aligned”).
This talk comes from works in collaboration with Pierre Degond, Antoine Diez, Sara Merino-Aceituno and Ariane Trescases.
Angelika Manhart, Aggregation without attraction: Modelling how an elastic environment influences collective cell dynamics
Aggregation phenomena in biology and beyond are often attributed to attraction between individuals. In this work we study how elastically tethered obstacles interacting with the swimmers impact the macroscopically created patterns. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model, for which we assume large tether stiffness. The result is a coupled system of non-linear, non-local partial differential equations. We use linear stability analysis to predict pattern size from model parameters. Further analysis of the macroscopic equations reveal that, surprisingly, the obstacle interactions induce short-ranged swimmer aggregation, irrespective of whether obstacles and swimmers are attractive or repulsive.