Thierry Goudon - Université Côte d'Azur, Inria, CNRS, LJAD - Keywords: Statistical Physics, Many Particles Systems, Hydrodynamic Regimes

Thierry Goudon - Université Côte d'Azur, Inria, CNRS, LJAD - Keywords: Statistical Physics, Many Particles Systems, Hydrodynamic Regimes

Contribution title: Kinetic models for interacting "particles"

Kinetic equations are  standard models arising in statistical physics. They appear naturally for describing the dynamics of many interacting particles, as an intermediate modeling between a description of the individual behavior of each particle, and a description based of the principles of fluid mechanics. For instance, the Boltzmann equation of rarefied gas dynamics or the Vlasov equation of plasma physics are classical kinetic models, like the radiative energy equation in radiative transfer theory.

More recently kinetic equations have been applied in life sciences, to describe chemotactic mechanisms or macroscopic organisation in populations of several interacting individuals.
Beyond the usual questions of existence-uniqueness of solutions, which are always a motivation for mathematicians, the asymptotic issues play a fundamental role. For instance the rigorous derivation of the Boltzmann equation from a N particles system, or the derivation of Euler and Navier-Stokes equations form the Boltzmann equation remain largely open questions.
The analysis of the stability of specific solutions has also motivated many interesting researches, with recent breakthrough on the understanding of the Landau damping. Finally, a very important activity is concerned with the design of efficient numerical methods, that preserve the main features of the model (conservation and dissipation properties), and that are able to handle the stiffness induced by the values of the relevant physical parameters. Progress on these problems have been at the origin of the Fields medals awarded by P.-L. Lions and C. Villani.

Let us describe a few examples recently investigated with mathematicians of LJAD:
* Directly motivated by experimental results obtained by the experts of Cold Atoms, we have considered simple mathematical models describing Magneto-Optical Traps (MOT), which are experimental devices used to trap cold atoms. Our findings justify that it can be relevant to replace the Vlasov-Poisson system by a model of macroscopic nature, that can be numerically investigated for a reduced numerical cost. But the surprising result is that the shape of the domain on which the limit equation is posed depends on a highly non trivial way on the applied strong external field.

* The interaction of particles with their environment can be thought of through a dynamic of momentum and energy exchanges between the particles and oscillating scatterers. This point of view leads to original models, that have interesting features. It is then physically relevant to investigate several asymptotic issues (fast speed of energy evacuation, large time behavior, etc).

* We set up hierarchies of models, ranging form individual-agents ODE to hydrodynamic like systems, describing populations that interact through attraction-repulsion mechanisms. For instance it can be applied as a model to describe ants foraging and the analysis might provide interesting interpretations, in particular related to the angle of vision of the individuals.