Review of a Simplified Approach to study the Bose gas at all densities
In this paper, we will review the results obtained thus far by Eric A. Carlen, Elliott H. Lieb and I on a Simplified Approach to the Bose gas. The Simplified Approach yields a family of effective one-particle equations, which capture some non-trivial physical properties of the Bose gas at both low and high densities, and even some of the behavior at intermediate densities. In particular, the Simplified Approach reproduces Bogolyubov's estimates for the ground state energy and condensate fraction at low density, as well as the mean-field estimates for the energy at high densities. We will also discuss a phase that appears at intermediate densities with liquid-like properties. The simplest of the effective equations in the Simplified Approach can be studied analytically, and we will present several results about it; the others are so far only amenable to numerical analysis, and we will discuss a few numerical results. We will start by reviewing some results and conjectures on the Bose gas, and then introduce the Simplified Approach and its derivation from the Bose gas. We will then discuss the predictions of the Simplified Approach and compare these to results and conjectures about the Bose gas. Finally, we present a few open problems about the Simplified Approach.
- tarball: 22j-0.0.tar.gz
- git repository: 22j-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).
- arXiv preprint: arXiv:2202.07637.
The numerical computations in this paper were carried out using simplesolv, which is available on this website.
[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)
[CJLL20]: On the convolution inequality f>f*f
Eric A. Carlen, Ian Jauslin, Elliott H. Lieb, Michael Loss, 2020
(published in International Mathematics Research Notices, Volume 2021, Issue 24, pages 18604-18612, 2021)
This work has been presented at the following conferences:
[Ga22]: An effective equation to study Bose gases at all densities
Advances in Classical, Quantum and Statistical Mechanics - A celebration of the work and contributions of Giovanni Gallavotti - On the occasion of his 80th birthday, Rome, Italy, May 13 2022
[Mi22]: An effective equation to study Bose gases at all densities
University of Milan, Italy, Feb 21 2022