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diff --git a/Jauslin_GLaMP_2019.tex b/Jauslin_GLaMP_2019.tex new file mode 100644 index 0000000..2354d06 --- /dev/null +++ b/Jauslin_GLaMP_2019.tex @@ -0,0 +1,187 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{array} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Lieb's simplified approach to interacting Bose gases\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm joint with {\bf Eric Carlen}, {\bf Elliott H. Lieb}\par +\vfil +\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Bose gas} +\hfil\includegraphics[height=6cm]{BEC.png} +\vfill +\eject + +\title{Interacting Bose gas} +\begin{itemize} + \item State: symmetric wave functions in a finite box of volume $V$ with periodic boundary conditions: + $$ + \psi(x_1,\cdots,x_N) + ,\quad + x_i\in \Lambda_d:=V^{\frac1d}\mathbb T^d + $$ + \item $N$-particle Hamiltonian: + $$ + H_N:= + -\frac12\sum_{i=1}^N\Delta_i + +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|) + $$ + with a repulsive, integrable interaction: $v(|x-y|)\geqslant 0$, $\int dx\ v(x)<\infty$. + +\end{itemize} +\vfill +\eject + +\title{Interacting Bose gas} +\begin{itemize} + \item Ground state: + $$ + H_N\psi_0=E_0\psi_0 + ,\quad + E_0=\min\mathrm{spec}(H_N) + $$ + \item Compute the ground state-energy per particle in the thermodynamic limit: + $$ + e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{\frac NV=\rho}}\frac{E_0}N + $$ +\end{itemize} +\vfill +\eject + +\title{Energy} +\begin{itemize} + \item Integrate $H_N\psi_0=E_0\psi_0$: + $$ + \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(|x-y|)g_2(x,y) + $$ + \item $g_n$: marginal of $\psi_0$ + $$ + g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dx_1\cdots dx_N\ \psi_0(x_1,\cdots,x_N)} + $$ + \item $\psi_0\geqslant 0$, so it can be thought of as a probability distribution +\end{itemize} +\vfill +\eject + +\title{Hierarchy} +\vskip-10pt +\begin{itemize} + \item Equation for $g_2$: integrate $H_N\psi_0=E_0\psi_0$ with respect to $x_3,\cdots,x_N$: + $$ + \begin{array}{>\displaystyle l} + -\frac12(\Delta_x+\Delta_y) g_2(x,y) + +\frac{N-2}V\int dz\ (v(|x-z|)+v(|y-z|))g_3(x,y,z) + \\[0.5cm]\hfill + +v(|x-y|)g_2(x,y) + +\frac{(N-2)(N-3)}{2V^2}\int dzdt\ v(|z-t|)g_4(x,y,z,t) + =E_0g_2(x,y) + \end{array} + $$ + \item Factorization assumption: + $$ + g_3(x_1,x_2,x_3)=g_2(x_1,x_2)g_2(x_1,x_3)g_2(x_2,x_3) + $$ + $$ + g_4(x_1,x_2,x_3,x_4)=\prod_{i<j}(g_2(x_i,x_j)+O(V^{-1})) + $$ +\end{itemize} +\vfill +\eject + +\title{Lieb's simple equation} +\begin{itemize} + \item In the thermodynamic limit, after making a few additional assumptions, \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}: + $$ + (-\Delta+v(|x|)+4e_0)u(|x|)=v(|x|)+2e_0\rho\ u\ast u(|x|) + $$ + $$ + e_0=\frac\rho2\int dx\ (1-u(|x|))v(|x|) + $$ + \item with $\rho:=\frac NV$ + $$ + g_2(x,y)=1-u(|x-y|) + ,\quad + u\ast u(|x|):=\int dy\ u(|x-y|)u(|y|) + $$ +\end{itemize} +\vfill +\eject + +\title{One dimension} +\hfil\includegraphics[height=6cm,angle=0.4]{1d.png} +\vfill +\eject + +\title{Numerical solution for $v(x)=e^{-|x|}$ in 3 dimensions} +\hfil\includegraphics[height=6cm]{erho_bare.pdf} +\vfill +\eject + +\title{Numerical solution for $v(x)=e^{-|x|}$ in 3 dimensions} +\hfil\includegraphics[height=6cm]{erho.pdf} +\vfill +\eject + +\title{Asymptotics} +\vskip-10pt +\begin{itemize} + \item {\bf Theorem} \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}: if $\hat v(k):=\int dx\ e^{ikx}v(x)\geqslant 0$, then + $$ + \frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x) + $$ + \item {\bf Theorem} \href{https://doi.org/10.1103/PhysRevLett.80.2504}{[Lieb, Yngvason, 1998]}: in 3 dimensions + $$ + \frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to0}2\pi a + $$ + \item {\bf Theorem} \href{https://doi.org/10.1023/A:101033721}{[Lieb, Yngvason, 2001]}: in 2 dimensions + $$ + \frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to0}\frac{2\pi}{|\log(\rho a^2)|} + $$ +\end{itemize} +\vfill +\eject + +\title{Lee-Huang-Yang formula} +\begin{itemize} + \item In 3 dimensions, \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]} conjecture: + $$ + e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right) + $$ + \item This was recently proved by \href{https://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]}. +\end{itemize} +\vfill +\eject + +\title{Results} +\begin{itemize} + \item {\bf Theorem}: For Lieb's simplified equation, + $$ + \frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x) + $$ + and, in 3 dimensions, + $$ + e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+O(\rho)\right) + $$ + in 2 dimensions + $$ + \frac{e_0}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac{2\pi}{|\log(\rho a)|} + . + $$ +\end{itemize} + +\end{document} |