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
I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce manybody correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a oneparticle problem, which incorporates the manybody 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 oneparticle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the manybody Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture manybody correlations in a oneparticle picture beyond what Bogolubov 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). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.
Slides
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LaTeX source:
 tarball: 22gallavotti801.0.tar.gz
 git repository: 22gallavotti80git (the git repository contains detailed information about the changes in the slides as well as the source code for all previous versions).
References
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 659684, 2020)
pdf, source 
[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 53225360, 2021)
pdf, source 
[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, 2021)
pdf, source 
[Ja22]: Review of a Simplified Approach to study the Bose gas at all densities
Ian Jauslin, 2022
(published in The Physics and Mathematics of Elliott Lieb, The 90th anniversary Volume 1, pages 609635)
pdf, source