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Derivation of Effective Dynamics from Microscopic Evolution Equations

Periodic Reporting for period 1 - DEDMEE (Derivation of Effective Dynamics from Microscopic Evolution Equations)

Reporting period: 2015-06-01 to 2017-05-31

Problem:
We are interested in the derivation of effective equations from microscopic considerations. The particular interest of our approach relies on the combination of rigorous methods of Quantum Field Theory and Quantum Statistical Mechanics, which is both innovative and creative. We paid particular attention to the macroscopic properties of interacting quantum systems governed by a microscopic dynamics. The research project yielded significant outcomes in this field, while developing important knowledge transfers, locally between Breteaux, Bru, and Ratsimanetrimanana, and globally between BCAM and the TU Braunschweig, University of Paris Nord, ETH Zurich and university of São Paulo, Université de Rennes 1, LMU Münich, and université de Metz.

For interacting systems such issues are notoriously difficult problems at the interface between Mathematics and Physics and we focus on the mathematical derivation of two types of effective (macroscopic) theories:
I. Hartree-type equations
II. Kinetic equations

Importance for Society:
We disseminate the knowledge gained by our research
- in Scientific Journals indexed related to Mathematics,
- by the organization of workshops and seminars,
- through courses, workshops, our website.

Objectives :
I.a Derive the Hartree-Fock equation from first principle of Quantum Mechanics for fermionic systems with Coulomb two body interaction potential in the semiclassical scaling.
I.b Derive the Hartree-Fock-Bogoliubov equation, and use it to study the thermal effects in a Bose-Einstein condensate, and their effects on the stability of such a condensate.
II.a Prove that the Wigner measure associated with a family of states satisfies the linear Boltzmann equation, for general initial data and a poissonian random field.
II.b Clarify the relation between kinetic equations and transport property for interacting systems.
"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"
"We list the progresses beyond the state of the art for our different objectives.

I. Hartree-type equations :

a.
We derive the Hartree-Fock equation for a class of potential including the Coulomb potential, which is very relevant physically. We are still working on the semiclassical version of such bounds. Here is a quote the report of the referee:
""There are two main ingredients in the proof. The first one is to introduce a quantity designed to control the number of particles which do not follow the Hartree-Fock dynamics. [...] The second main ingredient of the proof is the use of a formula due to Fefferman and de la Llave that allows to write the Coulomb potential as a sum of multiplication operators by pure tensor products. [...] The combination of these two ingredients leads to a very elegant proof of the main estimate of the article.
In conclusion, 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.""

b.
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 for sufficiently regular pair interaction potentials, 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. Kinetic equations :

II.a
Our project about Wigner measures changed shape as it evolved, and 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. We linked of mean-field asymptotics and second microlocalized semiclassical measures. We illustrated the phase space geometric description by various examples, like Gibbs states.

II.b About kinetic equations and transport property for interacting systems, with J.-B. Bru - my supervisor at BCAM -, W. de Siqueira Pedra - my supervisor at the host institution Universidad de Sao Paolo - and R. Yamagati, we are in the process of writing an article. I went to Sao Paulo for 7 days only, but W. De Siqueira Pedra joint the BCAM for a year, while Rafael Yamagati recently stays at BCAM during more than two months."
Poster Workshop MMBTA