Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit

TL;DR

This paper derives the Bogoliubov time evolution for a large volume mean-field limit of an interacting Bose gas, showing convergence of microscopic dynamics to effective descriptions including pair correlations.

Contribution

It extends mean-field results by incorporating pair correlations and analyzing the dynamics in the large volume and density limit with technical estimates on particle number moments.

Findings

Difference between N-body and Bogoliubov description is small in $L^2$ as density increases.

Microscopic dynamics of localized excitations converges to free evolution with Bogoliubov dispersion.

Established estimates for moments of particles outside the condensate in large volume.

Abstract

The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both spectral properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically lead to convergence for the reduced density matrix only, one obtains norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In this article we consider an interacting Bose gas in the case where both the volume and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (non-zero) speed of sound in the gas. In our first main result we show that the difference between the N-body and the Bogoliubov description is small in $L^2$ as the density of the gas tends to infinity and the volume does not grow too fast. This describes the dynamics of delocalized excitations of the order of the volume. In our second main result we consider an interacting Bose gas near the ground state with a macroscopic localized excitation of order of the density. We prove that the microscopic dynamics of the excitation coming from the N-body Schr\"odinger equation converges to an effective dynamics which is free evolution with the Bogoliubov dispersion relation. The main technical novelty are estimates for all moments of the number of particles outside the condensate for large volume, and in particular control of the tails of their distribution.