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UID:b8179b8e785a35292bb29f2c0df44842
CATEGORIES:Special Colloquium
CREATED:20201214T110200
SUMMARY: Analysis of a non-linear, non-local PDE to study Bose gases at all densities
LOCATION:Zoom
DESCRIPTION:Abstract.\nIn this talk, I will discuss a partial differential equation tha
t arises in the study of interacting systems of Bosonic particles. This equ
ation arises in the study of interacting systems of Bosonic particles. The
non-linearity and non-locality of the self-convolution term and of the mult
iplication 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 properti
es of this equation, which we have shown to be relevant to our understandin
g 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. Un
til now, the only means to accomplish this was to carry out difficult and c
omputationally intensive numerical computations. \nI will then discuss the
next steps in this project. The equation presented above is actually an app
roximation of a larger equation with a more complicated non-linearity. We h
ave studied this larger equation numerically, and found remarkable quantita
tive 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 interacti
ng Bose gases are widely expected to exhibit a quantum phase of matter call
ed a Bose-Einstein condensate (BEC), but this has not yet been proved mathe
matically. 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. \n
X-ALT-DESC;FMTTYPE=text/html:Abstract.

In this talk, I will discuss a partial differential eq
uation that arises in the study of interacting systems of Bosonic particles
. This equation arises in the study of interacting systems of Bosonic parti
cles. The non-linearity and non-locality of the self-convolution term and o
f the multiplication by e make this equation rather difficult to study. I w
ill first introduce a set of tools that have allowed us to prove a number o
f properties of this equation, which we have shown to be relevant to our un
derstanding of the interacting Bose gas. In particular, the solution to thi
s equation is shown to make accurate predictions for various physical obser
vables of the Bose gas both when the particle density is low, and when it i
s high. Until now, the only means to accomplish this was to carry out diffi
cult and computationally intensive numerical computations.

I
will then discuss the next steps in this project. The equation presented ab
ove is actually an approximation of a larger equation with a more complicat
ed non-linearity. We have studied this larger equation numerically, and fou
nd remarkable quantitative agreement with physical predictions for the inte
racting Bose gas, for all particle densities. However, the tools introduced
to study the simpler equation cannot be used for the larger one, and new t
echniques will need to be developed. Another point of interest stems from t
he fact that interacting Bose gases are widely expected to exhibit a quantu
m phase of matter called a Bose-Einstein condensate (BEC), but this has not
yet been proved mathematically. We can show that the simpler equation exhi
bits a BEC, and it may provide new ideas to prove the emergence of a BEC in
the many-body Bosonic system.

CONTACT:Ian Jauslin, Princeton University
DTSTAMP:20240420T022036
DTSTART;TZID=America/New_York:20201216T153000
DTEND;TZID=America/New_York:20201216T163000
SEQUENCE:0
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