De Finetti theorems, mean-field limits and Bose-Einstein condensation

TL;DR

This paper discusses the use of de Finetti theorems to justify mean-field models in classical and quantum systems, focusing on Bose-Einstein condensation and the structure of large bosonic systems.

Contribution

It provides a detailed strategy for deriving effective models from many-body quantum systems using de Finetti theorems and symmetry considerations.

Findings

Derivation of effective energy functionals from many-body Hamiltonians.

Analysis of reduced density matrices in large bosonic systems.

Connection between mean-field approximation 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.