Author: Ngọc Nhi Nguyễn

TBA

TBA,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

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SPEAKER: Annalisa Panati (Université de Toulon, Centre de Physique Théorique)

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Entropic fluctuations in quantum two-time measurement framework

Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, since the ground-breaking formulation of the transient and steady entropic Fluctuation Relations (FR) in the early nineties.
The extension of these results to the quantum setting has turned out to be surprisingly challenging and it is still an ongoing effort. Kurchan and Hal Tasaki’s seminal works (2000) showed quantum formulation of the transient version of FR is possible by introducing the two-time measurement framework.
In this talk, we present some results in a recent series of papers, where we attempt to introduce a quantum equivalent of steady entropic functional and compare it to the transient version for open quantum system. We consider both the case of idealised direct measurement on the reservoirs and experimentally accessible indirect measurement through coupling with an ancilla. We analyse in particular stability with respect to the initial state. In order to deal with the thermodynamic limit and to have general results, we use methods of $C^*$- algebras and modular theory.

(Joint work with T. Benoist, L. Bruneau, V. Jakšić, C.A. Pillet)

Thursday, April 3, 2025 - 11:00

Aula dottorato,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

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SPEAKER: Antoine Borie (Université de Rennes)

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Scattering for the positive density Hartree equation

In this talk, we explore the long-time behavior of solutions to of the positive density Hartree equation, which models the evolution of a homogeneous quantum gas. Our focus is the stability of certain stationary states, extending the original result introduced by Mathieu Lewin and Julien Sabin to higher dimensions and more singular interaction potentials. Using tools from dispersive partial differential equations, such as Strichartz estimates and fractional Leibniz rules, we develop a new approach tailored to density matrices.

This talk is based on joint work with Julien Sabin (Rennes University ) and Sonae Hadama (Kyoto University).

Wednesday, December 6, 2024 - 14:00

Aula Dottorato,
Dipartimento di Matematica,
Università degli Studi di Milano,
Via Cesare Saldini 50.

Disentangling pulse-coupled oscillators in the mean-field regime through the pseudo-inverse in a dilated timescale

Systems of pulse-coupled oscillators model synchronization through singular interactions occurring at discrete times, when particles reach a specific firing phase. They have numerous applications in physics, biology and engineering, for example to cardiac cells, neurons and fireflies. In the mean-field limit, the probability density in phase satisfies a singular continuity equation prone to finite-time blow-up, for which very few theoretical results are available. With José A. Carrillo, Xu’an Dou and Zhennan Zhou, we have introduced a reformulation of the mean-field system based on the inverse distribution function seen in a dilated timescale. It allows to show a hidden contraction/expansion mechanism and to propose simple and rigorous proofs of the long-time behaviour, the existence of steady states, the rates of convergence and the occurence of finite time blow-up for a large class of monotone phase response functions.

Wednesday, June 5, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Umberto Morellini (Université Paris Dauphine, CEREMADE, PSL) ________________________________________________________________________

The free energy of Dirac’s vacuum in purely magnetic fields

The Dirac vacuum is a non-linear polarisable medium rather than an empty space. This non-linear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at non-zero temperature needs to be better understood.
In this talk, we will present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of quantum vacuum in a classical magnetic field. After introducing our model, we will properly define the free energy functional using the Pauli-Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well known issue of the theory. The study of the properties of this functional will be addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg
formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in 1979. The talk is based on the work available at arXiv:2404.12733.

Monday, April 22, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Antoine Prouff (Université Paris-Saclay, Laboratoire de Mathématique d’Orsay) _________________________________________________________________________

Egorov’s theorem in the Weyl-Hörmander calculus and application to the control of PDEs

It is known that geometric optics can be derived as the high-frequency limit of the wave equation, from both experimental and theoretical perspectives. This fact can be regarded as an instance of a “quantum-classical correspondence principle”, made rigorous by Egorov’s theorem, which relates the evolution of a linear PDE (e.g. the wave equation) to the natural underlying classical dynamics (e.g. the geodesic flow).

We will present a version of Egorov’s theorem in the Euclidean space, in the setting of the “Weyl-Hörmander calculus”. This general framework of microlocal analysis involves Riemannian metrics on the phase space adapted to the dynamics under consideration, and allows for a fairly large range of applications (study of Schrödinger, wave and transport equations).

If time allows, we will discuss in more detail an application to the observability of the Schrödinger equation with a confining potential in the Euclidean space.