Poster

Florian Haberberger (LMU Munich)

Title: The free energy of dilute Bose gases at low temperatures

Abstract: We consider a system of N bosons confined in a box of sidelength L. The interaction between the bosons is described by a Hamiltonian with a repulsive potential. At low densities (the density is given by N/L^3), we obtain a Lee-Huang-Yang type formula for the free energy of the system. Moreover, we give some idea of the proof for both lower and upper bound.

This is joint work with: C.Hainzl, P.T.Nam, B.Schlein, R.Seiringer, A.Triay.

Matthias Herdzik (TU Braunschweig)

Title: On the Bogolubov-Hartree-Fock Energy of the Pauli-Fierz Hamiltonian 

Abstract: Following recent analyses of the Pauli-Fierz model we investigate the corresponding Bogolubov-Hartree-Fock variational problem. Bach, Breteaux and Tzaneteas (2013) showed that the minimum of the energy on quasifree states coincides with the minimum of the energy on pure quasifree states. Starting from this fact, we utilize techniques of Bach and Hach (2022) to study positivity properties of the minimizing Bogolubov transformation. Further symmetries of the minimizer are investigated with the aim of simplifying the energy functional. Finally, we use a parametrization introduced by Bach and Hach in order to write the simplified energy functional as a sum of polynomials and resolvents of the variables. Euler-Lagrange equations with respect to this new variable are derived.

Kiyeon Lee (KAIST)

Title: Scattering results for the (1+4) dimensional massive Maxwell-Dirac system under Lorenz gauge condition

Abstract: In this poster, we investigate the \emph{massive} Maxwell-Dirac system under the Lorenz gauge condition in (4+1) dimensional Minkowski space. The focus is on establishing global existence and scattering results for small solutions on the weighted Sobolev class. The imposition of the Lorenz gauge condition transforms the Maxwell-Dirac system into a set of Dirac equations coupled with an electromagnetic potential derived from five quadratic wave equations. To comprehensively understand the global solution and its behavior, we employ various energy estimates based on the space-time resonance argument. This involves addressing diverse resonance functions stemming from the free Dirac and wave propagators. Additionally, we identify the space-time resonant sets associated with the \emph{massive} Maxwell-Dirac system.

Yoonjung Lee (Yonsei University)

Title : On the global well-posedness of the Euler-Reisz systems.

Abstract : We consider the Euler-Riesz system which governs the motion of an inviscid compressible fluid with an interaction force satisfying the Riesz equation. The existence of solutions to such system can be expected to be local-in-time due to the hyperbolic conservation laws though, a fascinating series of works of D. Serre and M. Grassin showed that the compressible Euler system admits a global smooth solution under a suitable dispersive spectral condition on the initial velocity and a smallness condition on the initial density. Following the works of D. Serre and M. Grassin, we construct the global-in-time unique solution to the Euler–Riesz system and we provide the algebraic time decay rates of convergence for the constructed solution.

Ben Li (Wanli University)

Tital: Believe Damping Revisit

Abstract. According to the Bogoliubov theory the low energy behaviour of the Bose gas at zero temperature can be described by non-interacting bosonic quasiparticles called phonons. In this work the damping rate of phonons at low momenta, the so-called Beliaev damping, is explained and computed with simple arguments involving the Fermi Golden Rule and Bogoliubov’s quasiparticles.

Umberto Morellini (Ceremade, Université Paris Dauphine - PSL)

Title: Relativistic electrons coupled with Newtonian nuclear dynamics

Abstract: In the study of electronic structure of heavy atoms, relativistic effects cannot be neglected anymore and the Dirac operator naturally appears in place of the Schrödinger operator, raising up a number of additional difficulties. The complexity of these systems has been addressed by various approximations. In this poster, we will consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei, where the former is described by the Bogoliubov-Dirac-Fock equation of quantum electrodynamics. A global well-posedness result will be achieved by addressing the Cauchy problem for this model. We think that this system can be seen as a first step in the study of molecular dynamics phenomena in relativistic quantum chemistry.

Andrew Rout (University of Rennes 1)

Title: Microscopic derivation of Gibbs measures for the 1D focusing quintic NLS

Abstract: We obtain a microscopic derivation of Gibbs measures for the quintic focusing nonlinear Schrödinger equation (NLS) on the torus from many-body quantum Gibbs states. We prove results in both the time-independent and time-dependent settings. This corresponds to a three-body interaction on the many-body level. This is a continuation of earlier work deriving cubic measures in the same setting, which corresponded to a two-body interaction in the many-body setting. This is the first such known result in the three-body regime.

To derive a well-defined measure, we truncate in the mass of the classical free field and the rescaled particle number in the quantum setting. Our methods are based on the perturbative expansion previously developed by Fröhlich, Knowles, Schlein, and Sohinger.

Zhiwei Sun (Soochow University)

Title: Derivation of relativistic Vlasov equation in the simultenous semi-classical and mean-field limit using Husimi measure 

Abstract: Our work aims to derive the relativistic Vlasov equation in the simultaneous limit, namely semi-classical and mean-field limit, achieved by setting $\hbar=N^{-1/3}$ and taking $N\to\infty$. Beginning with the many-body semi-relativistic Schrodinger equation, we transform it into the dynamic equation of Husimi measure and undertake a limiting procedure to obtain the relativistic BBGKY hierarchy. Through a unique argument for the relativistic BBGKY hierarchy, we arrive at the solution for the relativistic Vlasov equation.

The primary challenge in this discourse arises from the inconsistency between the relativistic pseudo-differential operator and the Husimi measure. While the pseudo-differential operator commutes with the Wigner transform to some extent, this property does not hold for the Husimi measure.  To address this challenge, we derive the commutator derivative formula for the relativistic pseudo-differential operator.  Additionally, we introduce a novel treatment of the Husimi transform of the interaction energy, relaxing the regularity assumptions on the potential compared to existing literature.