Ian Jauslin

Analysis of a non-linear, non-local 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 non-linearity and non-locality of the self-convolution 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 non-linearity. 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 Bose-Einstein 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 many-body Bosonic system.

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