2024-25

This seminar presents a derivation of the Vlasov equation from the fermionic many-body Schrödinger system, employing the Husimi measure as a liking tool. Our exploration begins with a heuristic understanding of the Vlasov equation's derivation. This is followed by a short review of the many-body Schrödinger equation. Then we will talk about the methodology of linking the solutions of the many-body Schrödinger equation with the Vlasov equation, namely the Wigner measure and the Husimi measure. Attendees will gain insights into the formalism of this approach and explore strategies for controlling residual terms appearing in the derivations. 

Bardeen-Cooper-Schrieffer (BCS) theory is a successful model of superconductivity. 

In this talk, I will begin by providing an overview of how BCS theory connects with other models of superconductivity, highlighting major open questions and recent developments in the field. 

Second, I will explain recent results on the BCS critical temperature in the presence of a boundary in more detail and discuss the mathematical methods behind.

The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford’s theorem). In the case of large systems of particles that interact inelastically (sand, snow, interstellar dust), the derivation of the inelastic Boltzmann equation is still open. One major difficulty, already at the microscopic level, comes from the phenomenon of inelastic collapse, when infinitely many collisions take place in finite time.

Assuming that the restitution coefficient r is constant, we obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles: either the particles arrange in a nearly-linear chain (studied in [3]), or they form a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct explicit initial configurations leading to a nearly-linear collapse in a stable way, such that the angle between the particles at the time of collapse can be chosen a priori, with arbitrary precision.

The results are taken from [1] and [2], obtained in collaboration with Juan J. L. Velázquez

(Universität Bonn).

References

[1] Théophile Dolmaire, Juan J. L. Velázquez, “Collapse of inelastic hard spheres in dimension d ≥ 2”, to appear in Journal of Nonlinear Science, preprint arXiv:2402.13803v2 (02/2024).

[2] Théophile Dolmaire, Juan J. L. Velázquez, “Properties of some dynamical systems for

three collapsing inelastic particles”, preprint arXiv:2403.16905 (03/2024).

[3] Tong Zhou, Leo P. Kadanoff, “Inelastic collapse of three particles”, Physical Review E,

54:1, 623–628 (07/1996).

Many systems, taking the scaling limit in a parameter, converge to a deterministic limit equation. Examples are fast-slow system in climate and their limit averaging equation; interacting particle systems and their mean field or hydrodynamic limit; stochastic turbulent transport and their diffusive limit, similar to diffusion limit in homogenization. One could however stop before taking the limit and look for a simplified stochastic model, similar to the limit model but stochastic, which well represents the approximating problem without completely neglecting the fluctuations. This was the viewpoint in particular of Hasselmann proposal in climate modeling, here reviewed in comparison with Dean-Kawasaki approach to fluctuating hydrodynamics, with a detour on stochastic transport.

 To study the statistical properties of dynamical systems exhibiting chaotic behavior, researchers have built on a foundational method introduced by Ruelle in the 1970s. This approach analyzes the evolution of probability densities, rather than individual trajectories, to gain insights into the system. The evolution of these densities is governed by an operator known as the transfer operator, whose spectral properties in suitable Banach spaces relate directly to key statistical features of the system, such as the existence of invariant measures, ergodicity, mixing and CLT. In this talk, I will present the core aspects of this powerful method and explore recent and prospective advancements in the field, including applications to interacting dynamical systems coupled by mean-field type forces. 

 I will present some recent results on large deviations for  binary collision stochastic models. The paradigmatic model is the Kac’s walk, described, in the kinetic limit, by the homogeneous Boltzmann equation. Although the dynamics preserves the energy, atypical paths that violate energy conservation may occur with exponentially small probability in the number of particles. I will exhibit energy increasing paths, condensation-like phenomena and a phase transition for the asymptotics  of the number of collisions. Finally, I will discuss a formulation of the homogeneous Boltzmann equation in terms of an entropy dissipation inequality which involves the large deviation rate function. This is a joint work with L. Bertini, D. Benedetto, E. Caglioti and D. Heydecker. Founded by the European Union - Next Generation EU