"We describe the results we obtained about the objectives we recalled above.
I.a Solved
Article published, 30 pages, by Bach, Breteaux, Petrat, Pickl, Tzaneteas in a first decile journal
Citation from the referee report:
""This article provides an important contribution to the derivation of time-dependent Hartree-Fock type equations, both at the level of the nature of the results and the type of proofs.""
We studied the time evolution of a system of N spinless fermions in ℝ3 which interact through a pair potential. We compared the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we gave an estimate for the accuracy of this approximation in terms of the initial total energy of the system.
We are still working on the semiclassical scaling, and we did some good progresses during my stays in Braunschweig and LMU Münich.
I.b Solved
Article submitted for publication, 36 pages, by Bach, Breteaux, Chen, Fröhlich, Sigal
We used quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate. We proved global well-posedness for the HFB equations, and established key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system.
II.a
Modified strategy, but a first step has been solved.
Article submitted for publication, 42 pages
Ammari, Breteaux, Nier
Our project about Wigner measures has evolved: we decided to study more generally Quantum mean field asymptotics and multiscale analysis.
More precisely we study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples, like Gibbs states.
II.b
Partially solved, ongoing project.
Article in preparation.
About kinetic equations and transport property for interacting systems, with Bru - my supervisor at BCAM -, Pedra - my supervisor at the host institution Universidad de Sao Paolo - and Yamagati, we are in the process of writing an article. I went to Sao Paulo for 7 days only, but De Siqueira Pedra joint the BCAM for a year, while Yamagati recently stays at BCAM during more than two months.
I travelled 62 days at secondment or hosting institutions, and 23 days in LMU Münich and Rennes, to work with Pickl, Petrat, Ammari and Nier on objectives I.a and II.a.
Communication and dissemination:
- Publication of the results in Scientific Journals
- Organization of 2 workshops
- Presentations at 9 international events and 6 seminars
- Dissemination to a wider public
- Invitation of 16 guests
- 4 BCAM courses
- Website"
