Séminaire MACS (Modélisation, Analyse et Calcul Scientifique). Commun ICJ & UMPA.

Topological states in collective dynamics

by Pierre Degond (Toulouse)

States of matter (such as solid, liquid, etc) are characterized
by different types of order associated with local invariances under
different transformation groups. Recently, a new notion of topological
order, popularized by the 2016 physics nobel prize awarded to Haldane,
Kosterlitz and Thouless, has emerged. It refers to the global rigidity of
the system arising in some circumstances from topological constraints.
Topologically ordered states are extremely robust i.e. « topologically
protected » against localized perturbations. 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 realizing
topologically protected collective states has been raised. 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. At the particle level, these solutions persist for a
certain time but eventually, for some of them, decay towards a topologically
trivial state, due to the noise induced by the stochastic nature of the
particle system. We numerically analyse these topological phase transitions
and investigate to what extent topologically non-trivial states are
‘protected’ against perturbations. To our knowledge, it is the first time
that a hydrodynamic model guides the design of topologically non-trivial
states of a particle system and allows for their quantitative analysis and
understanding. In passing, we will raise interesting mathematical questions
underpinning the analysis of collective dynamics systems.