Moderator: Sergio Simonella (Roma)
Sergio Simonella (Roma),
Lorenzo Bertini (Roma),
Pierre Degond (Toulouse),
Rossana Marra (Roma),
Sylvia Serfaty (New York),
Herbert Spohn (München),
Raphael Winter (Cardiff)
Sergio Simonella (Roma): Foundations of kinetic theory: recent progress and open directions. pdf
We consider deterministic, time-reversible dynamics with random initial data, in a low-density scaling. Under suitable assumptions on the initial measure, a strong chaos property is propagated in time, which also encodes the transition to irreversibility. This result is complemented by large deviation estimates and by a theory of small fluctuations, allowing to establish the connection between microscopic and hydrodynamic scales, for perturbations of a global equilibrium. Many of the open problems left require a deeper understanding of the coupling mechanisms between deterministic and stochastic dynamics.
Lorenzo Bertini (Roma): TBA.
TBA
Pierre Degond (Toulouse): Swarming rigid bodies: geometry and topology. pdf
Collective dynamics in systems of self-propelled particles has stimulated intense mathematical research in the recent years. Many different models have been proposed but most of them rely on point particles. In practice, particles often have more complex geometrical structures. Here, we will consider particles as rigid bodies whose body attitude is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. A hydrodynamic model will be derived when the number of particles is large. It will be used to exhibit solutions having non-trivial topology. We will investigate whether topology provides enhanced stability against perturbations, as observed in other systems such as topological insulators. This talk is based on recent results issued from collaborations with Antoine Diez, Amic Frouvelle, Sara Merino-Aceituno, Mingye Na and Ariane Trescases.
Rossana Marra (Roma): On the derivation of new non-classical hydrodynamic equations for a Boltzmann gas and for Hamiltonian particle systems. pdf
The derivation of hydrodynamic equations for a Hamiltonian system of particles is a challenging and open problem. Partial results are available for stochastic system of particles on the lattice. In kinetic theory, there are many results on the derivation of Euler or incompressible Navier-Stokes-Fourier equations from the Boltzmann equation, in different cases. When the density and temperature at initial time and/or the temperature on the boundary have gradients of order 1 the limiting equations (called “ghost effect equations”) are different and cannot be predicted by the classical fluid theory. In this talk I will discuss the rigorous derivation of this non standard hydrodynamic behaviour for the stationary Boltzmann equation in a bounded domain with diffuse reflection boundary condition. Moreover, I will also discuss the formal derivation, under the same scaling, of analogous (and completely new) equations for a Hamiltonian particle system in the time-dependent setting and discuss some stochastic model that could be useful to see this new hydrodynamic limit.
Sylvia Serfaty (New York): Mean-field limits for log/Coulomb/Riesz interacting diffusions. pdf
We consider a system of N points in singular interaction of logarithmic, Coulomb or Riesz type, evolving by gradient flow or conservative flow (such as the point vortex system in 2D) with or without noise. We discuss local and global-in-time convergence to the mean-field limit by a modulated energy method.
Herbert Spohn (München): Kinetic theory of weakly nonlinear wave equations. pdf
Reviewed is the kinetic limit for weakly nonlinear wave equations with spatially homogeneous Gaussian random initial data. Our guiding example will be the nonlinear Schroedinger equation. Touched upon are also the formation of a condensate and very recent advances.
Raphael Winter (Cardiff): Kinetic scaling limits in plasma physics. pdf
The kinetic theory of plasma features a surprising variety of equations such as the Landau equation, the Vlasov-Poisson equation and the Balescu-Lenard equation. This wealth of different dynamics is due to the onset of collective effects and the scale-invariance of the Coulomb potential. In this talk, we discuss the theory in the framework of scaling limits of interacting particle systems and present recent advances.