The dynamics in dispersing billiards are mostly characterized by two competitive effects: strong expansion and the presence of singularities. In my talk I would like to give an overview on the geometry of singularity manifolds, focusing in particular on some surprising phenomena in the multidimensional case, which have been - and remain - the main obstacle for proving optimal results on the rate of mixing
In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium.
This control of the collision process can be obtained in fluctuation regimes. In a joint work with Thierry Bodineau and Isabelle Gallagher, we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.
What is the equation of motion describing the motion of swarms of animals and can one think of them as ”social fluids”? The interactions between individuals are usually assumed to be local, in which case an active version of the Navier-Stokes equation can be proposed. Recent observations of orientation correlation functions in starling flocks are only local in a weak, topological sense. We propose a natural “most simple” form for interactions that are consistent with the use of vision and therefore non-local. Using an agent-based model we compare with experimental data on bird flocks and identify several emergent phenotypes. In an attempt to move towards a continuum model I draw an analogy with photo-thermophoretic colloids that respond to light in a way that shares some similarities with vision. In our work there is a focussed external light source and the system undergoes a first order transition to a compact state that shares one of the primary organisational features seen in bird flocks.