Analysis of a nonlinear, nonlocal PDE to study Bose gases at all densities
Rutgers University, New Brunswick, New Jersey, USA
December 16, 2020
In this talk, I will discuss a partial differential equation that arises in the study of interacting systems of Bosonic particles. This equation arises in the study of interacting systems of Bosonic particles. The nonlinearity and nonlocality of the selfconvolution term and of the multiplication by e make this equation rather difficult to study. I will first introduce a set of tools that have allowed us to prove a number of properties of this equation, which we have shown to be relevant to our understanding of the interacting Bose gas. In particular, the solution to this equation is shown to make accurate predictions for various physical observables of the Bose gas both when the particle density is low, and when it is high. Until now, the only means to accomplish this was to carry out difficult and computationally intensive numerical computations.
I will then discuss the next steps in this project. The equation presented above is actually an approximation of a larger equation with a more complicated nonlinearity. We have studied this larger equation numerically, and found remarkable quantitative agreement with physical predictions for the interacting Bose gas, for all particle densities. However, the tools introduced to study the simpler equation cannot be used for the larger one, and new techniques will need to be developed. Another point of interest stems from the fact that interacting Bose gases are widely expected to exhibit a quantum phase of matter called a BoseEinstein condensate (BEC), but this has not yet been proved mathematically. We can show that the simpler equation exhibits a BEC, and it may provide new ideas to prove the emergence of a BEC in the manybody Bosonic system.
Slides
PDF:
LaTeX source:
 tarball: 20rutgers1.0.tar.gz
 git repository: 20rutgersgit (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)
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)
pdf, source