A new approach to the Mathematics of the Bose gas
Texas A&M, College Station, Texas, USA
November 30, 2020
I will discuss an effective equation, which is used to study the ground state of an interacting system of Bosonic particles. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. This system has been the focus of much research in the past twenty years, and yet some of the fundamental questions about it are still open. For instance, there still is no proof of the emergence of a Bose-Einstein condensate at finite density. A very successful, but approximate approach to solving this problem is Bogolubov theory, in which a series of approximations are made after which the analysis reduces to a one-particle problem, which incorporates the many-body correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem, in the form of a non-linear and non-local partial differential equation in three dimensions. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and high densities. I will discuss some recent results which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. In other words, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov theory can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas, about which there still are many important open questions, such us Bose-Einstein Condensation. In addition, it opens an avenue to understand the properties of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.
- tarball: 20tamu-b-1.0.tar.gz
- git repository: 20tamu-b-git (the git repository contains detailed information about the changes in the slides as well as the source code for all previous versions).
This presentation is based on
[CJL19]: Analysis of a simple equation for the ground state energy of the Bose gas
Eric Carlen, Ian Jauslin, Elliott H. Lieb, 2019
(published in Pure and Applied Analysis, volume 2, issue 3, pages 659-684)
[CJL20]: Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction
Eric A. Carlen, Ian Jauslin, Elliott H. Lieb, 2020
(published in SIAM Journal on Mathematical Analysis, Volume 53, Number 5, pages 5322-5360, 2021)
[CHJL20]: A simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy
Eric A. Carlen, Markus Holzmann, Ian Jauslin, Elliott H. Lieb, 2020
(published in Physical Review A, volume 103, number 053309)