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diff --git a/Jauslin_MC_2021.tex b/Jauslin_MC_2021.tex new file mode 100644 index 0000000..ff6d0b7 --- /dev/null +++ b/Jauslin_MC_2021.tex @@ -0,0 +1,235 @@ +\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} |