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 

 As is well known, many materials freeze at low temperatures. Microscopically, this means that their molecules form a phase where there is long range order in their positions. Despite their ubiquity, proving that these freezing transitions occur in realistic microscopic models has been a significant challenge, and it remains an open problem in continuum models at positive temperatures. In this talk, I will focus on lattice particle models, in which the positions of particles are discrete, and discuss a general criterion under which crystallization can be proved to occur. The class of models that the criterion applies to are those in which there is no sliding, that is, particles are largely locked in place when the density is large. The tool used in the proof is Pirogov-Sinai theory and cluster expansions. I will present the criterion in its general formulation, and discuss some concrete examples. This is joint work with Qidong He and Joel L. Lebowitz  

In this talk, I present a short proof of BEC for the ground state of $N$ bosons moving in the three-dimensional unit torus and interacting through a pair potential with scattering length of order $N^{\kappa-1}$, for a parameter $kappa \in [0,1/20)$. Such a system is equivalent to a strongly diluted system of interacting bosons with unscaled pair potential. In the talk, I introduce the problem of BEC for dilute Bose gases, summarize the current state of the art and then outline a new proof for suitable ultra-dilute systems which is based on joint work with M. Brooks, C. Caraci and J. Oldenburg. 

 We discuss some existence results for a nonlinear Schröedinger equation with nonlinearity concentrated on a sphere in dimension three.

 One of the main topics in statistical mechanics is the derivation of equations of fluids from microscopic Hamiltonian dynamics. We consider a family of asymmetric/weakly asymmetric exclusions with different velocities, coupled through binary collisions that conserve the total momentum. Assuming special initial distribution given by a small perturbation on the equilibrium state, a law of large numbers holds for the properly rescaled density-velocity field. Depending on the chosen velocity set, the limit field evolves with different PDEs, including the incompressible Navier-Stokes equations. We will review some important existing results and introduce recent progress on this topic. Based on joint works with Patrick van Meurs, Kenkichi Tsunoda and Linjie Zhao. 

 In this talk, I will focus on a two-dimensional lattice gas subject to the conservative Kawasaki dynamics at positive inverse temperature in a large finite box, the volume of which is exponentially large in the inverse temperature. Each pair of neighbouring particles has a negative binding energy, while each particle has a positive activation energy. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. In the metastable regime and in the limit where temperature and density simultaneously vanish, I will describe how and when the system nucleates, i.e., creates a critical droplet somewhere in the domain that subsequently grows by absorbing particles from the surrounding gas.

This talk is based on joint works with A. Gaudillière, F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola. 

 We compute the longitudinal conductivity for a wide class of two-dimensional tight-binding models, whose Hamiltonian displays conical intersections of the Bloch bands at the Fermi level. Our setting allows us to consider generic transitions between integer quantum Hall phases. We obtain an explicit expression for the longitudinal conductivity, completely determined by the number of conical intersections and by the shape of the cones. In particular, the formula reproduces the known quantized values obtained for graphene and for the critical Haldane model. This talk is on a joint work with L. Pigozzi and M. Porta: https://arxiv.org/abs/2503.01381 . 

 In this talk, we discuss how chaos -- in the sense of near independence of the relevant random variables -- can be studied via the smallness of relevant joint cumulants. Focusing on the stochastic Kac model model for velocity exchange of a system of $N$ particles, we quantify the notion of chaos in terms of weighted norms on the space of joint cumulants of the energy variables. The time-evolution of the joint cumulants in such spaces demonstrate how chaos is generated or propagated in the system. Known spectral gap results imply that typical initial densities converge to uniform distribution on the energy sphere at a time which has order of $N$ expected collisions. We prove that the finite order cumulants become small and converge to their small stationary values much faster, already at a time scale of order 1 or log(N) collisions. The proof relies on stability analysis of the closed, nonlinear hierarchy of energy cumulants around the fixed point formed by their values in the stationary spherical distribution. In addition, we describe the accuracy of the Boltzmann--Kac kinetic equation after the onset of chaos in the system.

Based on joint work with Jani Lukkarinen.

 The hydrodynamic limit of a stochastic epidemiological model, where two infection scenarios alternate, namely 

a) infections in separated groups of finite size 

b) and infections at meeting places of finite capacity, where individuals meet randomly, 

results in a McKendrick system with a polynomial infection force. 

For this system of kinetic equations, invariants can be determined 

which uniquely determine the outcome of the model epidemics. 

Such kinetic equations allow to link global data of an epidemic with not so easily observable local rate dependencies.

(Joint work with Stephan Luckhaus, Universität Leipzig) 

 We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology,  can be derived from an interacting particle model and its discretized version   As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing. We present some numerical results comparing the model to a known exact solution. For different choices of the parameters, the numerical algorithm can be connected to  a Bence–Merriman–Osher scheme for surface evolution or a convolutional neural network.