De Finetti theorems, mean-field limits and Bose-Einstein condensation
Abstract
These notes deal with the mean-field approximation for equilibrium states of N-body systems in classical and quantum statistical mechanics. A general strategy for the justification of effective models based on statistical independence assumptions is presented in details. The main tools are structure theorems {\`a} la de Finetti, describing the large N limits of admissible states for these systems. These rely on the symmetry under exchange of particles, due to their indiscernability. Emphasis is put on quantum aspects, in particular the mean-field approximation for the ground states of large bosonic systems, in relation with the Bose-Einstein condensation phenomenon. Topics covered in details include: the structure of reduced density matrices for large bosonic systems, Fock-space localization methods, derivation of effective energy functionals of Hartree or non-linear Schr{\"o}dinger type, starting from the many-body Schr{\"o}dinger Hamiltonian.
Keywords
Cite
@article{arxiv.1506.05263,
title = {De Finetti theorems, mean-field limits and Bose-Einstein condensation},
author = {Nicolas Rougerie},
journal= {arXiv preprint arXiv:1506.05263},
year = {2020}
}
Comments
Lectures notes from a course at the LMU, Munich. Translated and slightly expanded version of my cours Peccot, hal-01060125v4, arXiv:1409.1182. A wrong proof has been removed from Appendix A