Derivation of the Bogoliubov Time Evolution for a Large Volume\n Mean-field Limit Article Swipe
YOU?
·
· 2017
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1711.01591
· OA: W4298250953
The derivation of mean-field limits for quantum systems at zero temperature\nhas attracted many researchers in the last decades. Recent developments are the\nconsideration of pair correlations in the effective description, which lead to\na much more precise description of both spectral properties and the dynamics of\nthe Bose gas in the weak coupling limit. While mean-field results typically\nlead to convergence for the reduced density matrix only, one obtains norm\nconvergence when considering the pair correlations proposed by Bogoliubov in\nhis seminal 1947 paper. In this article we consider an interacting Bose gas in\nthe case where both the volume and the density of the gas tend to infinity\nsimultaneously. We assume that the coupling constant is such that the\nself-interaction of the fluctuations is of leading order, which leads to a\nfinite (non-zero) speed of sound in the gas. In our first main result we show\nthat the difference between the N-body and the Bogoliubov description is small\nin $L^2$ as the density of the gas tends to infinity and the volume does not\ngrow too fast. This describes the dynamics of delocalized excitations of the\norder of the volume. In our second main result we consider an interacting Bose\ngas near the ground state with a macroscopic localized excitation of order of\nthe density. We prove that the microscopic dynamics of the excitation coming\nfrom the N-body Schr\\"odinger equation converges to an effective dynamics which\nis free evolution with the Bogoliubov dispersion relation. The main technical\nnovelty are estimates for all moments of the number of particles outside the\ncondensate for large volume, and in particular control of the tails of their\ndistribution.\n
