From 3c35cef0ee24d369d6793ce3ebcd002deab7f9b2 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Wed, 10 Mar 2021 17:14:51 -0500
Subject: As presented at IAS on 2021-03-10

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+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{array}
+\usepackage{xcolor}
+
+
+\definecolor{ipurple}{HTML}{4B0082}
+\definecolor{iyellow}{HTML}{DAA520}
+\definecolor{igreen}{HTML}{32CD32}
+\definecolor{iblue}{HTML}{4169E1}
+\definecolor{ired}{HTML}{DC143C}
+
+\definecolor{highlight}{HTML}{328932}
+\definecolor{highlight}{HTML}{981414}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Many interacting quantum particles:\par
+\medskip
+\hfil\large open problems, and a new point of view on an old problem
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\rm
+\hfil collaborators: {\bf E.A.\-~Carlen, E.H.\-~Lieb, M.\-~Holzmann, M.P.\-~Loss}\par
+\vfil
+arXiv:{\tt\ \parbox[b]{3cm}{
+  \href{https://arxiv.org/abs/1912.04987}{1912.04987}\par
+  \href{https://arxiv.org/abs/2010.13882}{2010.13882}\par
+  \href{https://arxiv.org/abs/2011.10869}{2011.10869}
+}}
+\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Fermions/Bosons}
+\bigskip
+\hfil\includegraphics[height=5.5cm]{bosons-weinersmith.png}
+
+\hfil{\tiny Zach Weinersmith, \href{https://creativecommons.org/licenses/by-nc/3.0/legalcode}{CC-BY-NC 3.0}}
+\vfill
+\eject
+
+\title{Bose-Einstein condensation}
+\begin{itemize}
+\item System of Bosons, e.g. {\color{highlight}Helium} atoms, {\color{highlight}Rubidium} atoms, etc...
+\item At low temperatures, {\color{highlight}superfluidity} (flow with zero viscocity) and {\color{highlight}superconductivity} (currents with zero resistance).
+\item {\color{highlight}Bose-Einstein condensate}: most particles are in the same quantum state.
+\item Predicted theoretically in {\color{highlight}1924-1925}, experimentally observed in {\color{highlight}1995}.
+\item Mathematical understanding: still {\color{highlight}no proof} of the existence of a condensate (at finite density, in the presence of interactions and in the continuum).
+\end{itemize}
+\vfill
+\eject
+
+\title{Repulsive Bose gas}
+\begin{itemize}
+  \item {\color{highlight}$N$-particle} quantum state in a volume $V$:
+  $$
+    \psi(x_1,\cdots,x_N)\in L^2_{\mathrm{symmetric}}((V\mathbb T^3)^N)
+  $$
+
+  \item $|\psi|^2$: probability distribution on the positions of the $N$ particles.
+
+  \item Hamiltonian operator acting on $\psi$:
+  $$
+    H_N\psi:=
+    -\frac12\sum_{i=1}^N\Delta_i\psi
+    +\sum_{1\leqslant i<j\leqslant N}{\color{highlight}v(|x_i-x_j|)}\psi
+  $$
+  {\color{highlight}$v(r)\geqslant 0$}.
+\vfill
+\eject
+
+\title{Energy and condensate fraction}
+\vskip-10pt
+  \item {\color{highlight}Zero-temperature:} Ground state: $\psi_0$, energy $E_0=\inf\mathrm{spec}H_N$:
+  $$
+    H_N\psi_0={\color{highlight}E_0}\psi_0
+  $$
+\vskip-10pt
+  \item In the {\color{highlight}thermodynamic limit}:
+  $$
+    e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N
+    .
+  $$
+\vskip-5pt
+  \item Condensate fraction: proportion of particles in the Bose-Einstein condensate: in the {\color{highlight}thermodynamic limit}: ($P_i$: projection onto condensate wavefunction)
+  $$
+    \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right>
+    .
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Ground state energy}
+\begin{itemize}
+  \item At low density: {\color{highlight}Bogolyubov theory}: [Bogolyubov, 1947], \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
+  $$
+    e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)
+  $$
+  \item {\color{highlight} Proved}:
+  \href{https://doi.org/10.1103/PhysRevLett.80.2504}{[Lieb, Yngvason, 1998]},
+  \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]},
+  (\href{https://arxiv.org/abs/2101.06222}{[Basti, Cenatiempo, Schlein, 2021]}),
+  \href{https://doi.org/10.4007/annals.2020.192.3.5}{[Fournais, Solovej, 2020]}.
+
+  \item At high density: {\color{highlight}Hartree theory}: ({\color{highlight}Proved} in \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}).
+  $$
+    {\color{highlight}e_0\sim\frac\rho2\int v}
+    .
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Condensate fraction}
+\begin{itemize}
+  \item At low density: {\color{highlight}Bogolyubov theory}: [Bogolyubov, 1947], \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}:
+  $$
+    1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}
+  $$
+
+  \item {\color{highlight}Still open} in the thermodynamic limit. (No proof of Bose-Einstein condensation, in the continuum, at finite density.)
+
+  \item At high density: {\color{highlight}Hartree theory}: ({\color{highlight}open})
+  $$
+    \eta_0\to1
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Effective equations}
+\begin{itemize}
+  \item {\color{highlight}Boltzmann equation}: $N$ classical hard particles with an infinitely small radius (dilute limit)
+  [Lanford, 1976].
+  \item {\color{highlight}Thomas-Fermi theory}: $Z$ electrons orbiting a nucleus in the $Z\to\infty$ limit
+  \href{https://doi.org/10.1103/PhysRevLett.31.681}{[Lieb, Simon, 1973]}.
+  \item{\color{highlight}Hartree-Fock equation}: dynamics of many Fermions in the weakly-interacting limit
+  \href{https://doi.org/10.1142/9789814618144_0011}{[Benedikter, Porta, Schlein, 2015]}.
+  \item{\color{highlight}Hartree-Fock-Bogolyubov equation}: dynamics of many Bosons in the weakly-interacting limit
+  \href{https://arxiv.org/abs/1602.05171}{[Bach, Breteaux, Chen, Fr\"ohlich, Sigal, 2016]}.
+\end{itemize}
+\vfill
+\eject
+
+\title{{\color{iblue}Simple equation}}
+\begin{itemize}
+  \item {\color{iblue}Simple equation}
+  $$
+    -\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)
+  $$
+  $$
+    e=\frac\rho2\int dx\ (1-u(x))v(x)
+  $$
+  \item $\rho>0$, $v(x)\geqslant 0$, $v\in L_1(\mathbb R^3)$.
+  \item {\color{highlight}Non-linear} and {\color{highlight}non-local} partial differential equation.
+  \item {\color{highlight}Effective equation} for the ground state of a Bose gas.
+  \item Main idea: think of {\color{highlight}$\psi$ as a probability distribution} instead of $|\psi|^2$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Energy as a function of density for the {\color{iblue}Simple equation}}
+For $v(x)=e^{-|x|}$:
+
+\hfil\includegraphics[height=5.5cm]{erho_lowhigh.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Energy as a function of density for the {\color{iblue}Simple equation}}
+For $v(x)=e^{-|x|}$:
+
+\hfil\includegraphics[height=5.5cm]{erho_effective.pdf}
+\vfill
+\eject
+
+\title{Energy}
+$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}
+
+\hfil\includegraphics[height=5.5cm]{erho_fulleq.pdf}
+\vfill
+\eject
+
+\title{{\color{ipurple}Big equation}}
+\begin{itemize}
+\item $x\in\mathbb R^3$,
+  $$
+    -\Delta u(x)
+    =
+    (1-u(x))\left(v(x)-2\rho K(x)+\rho^2 L(x)\right)
+  $$
+  $$
+    K:=
+    u\ast S
+    ,\quad
+    S(y):=(1-u(y))v(y)
+  $$
+  $$
+    L:=
+    u\ast u\ast S
+    -2u\ast(u(u\ast S))
+    .
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Condensate fraction}
+$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo}
+
+\hfil\includegraphics[height=5.5cm]{condensate.pdf}
+\vfill
+\eject
+
+\title{Conclusions}
+\begin{itemize}
+  \item Two {\color{highlight}effective equations}: the {\color{ipurple}big equation} and the {\color{iblue}simple equation}, which are {\color{highlight}non-linear 1-particle equations}.
+  \item Reproduce the known results for both {\color{highlight}small and large densities}.
+  \item Their derivation is {\color{highlight}different from Bogolyubov theory}, so they may give new insights onto studying the Bose gas in these asymptotic regimes.
+  \item The {\color{ipurple}big equation} is {\color{highlight}quantitatively accurate} at intermediate densities.
+  \item This opens up the possibility of studying the physics of the {\color{highlight}Bose gas at intermediate densities}.
+\end{itemize}
+
+\end{document}
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