\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 An effective equation to study Bose gases\par \smallskip \hfil at all densities\par \vfil \large \hfil Ian Jauslin \normalsize \vfil \hfil\rm joint with {\bf Eric A. Carlen}, {\bf Markus Holzmann}, {\bf Elliott H. Lieb}\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{Bose-Einstein condensation} \begin{itemize} \item System of many Bosons, e.g. {\color{highlight}Helium} atoms, {\color{highlight}Rubidium} atoms, etc... \item {\color{highlight}Bose-Einstein condensate}: most particles are in the same quantum state. \item Related to the phenomena of {\color{highlight}superfluidity} (flow with zero viscocity) and {\color{highlight}superconductivity} (currents with zero resistance). \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 Potential: {\color{highlight}$v(r)\geqslant 0$} and {\color{highlight}$v\in L_1(\mathbb R^3)$}, Hamiltonian: $$ H_N:= -\frac12\sum_{i=1}^N\Delta_i +\sum_{1\leqslant i . $$ \end{itemize} \vfill \eject \title{Low density} \begin{itemize} \item Bogolyubov theory: {\color{highlight}approximation scheme} that reduces the problem to an effective {\color{highlight}1-particle problem}. \item Predictions \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}: \begin{itemize} \item Energy: $$ {\color{highlight}e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right)} $$ \vskip-10pt \item Condensate fraction: $$ {\color{highlight}1-\eta_0\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi}} $$ \end{itemize} \end{itemize} \vfill \eject \title{Low density} \begin{itemize} \item Energy asymptotics: {\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://doi.org/10.4007/annals.2020.192.3.5}{[Fournais, Solovej, 2020]}. \item Condensate fraction: {\color{highlight}still open} in the theormodynamic limit, but there are proofs of condensation in the Gross-Pitaevskii regime (ultra-dilute): \href{https://doi.org/10.1103/PhysRevLett.88.170409}{[Lieb, Seiringer, 2002]}, \href{https://doi.org/10.1007/s00220-017-3016-5}{[Boccato, Brennecke, Cenatiempo, Schlein, 2018]}. \end{itemize} \vfill \eject \title{High density} \begin{itemize} \item [Bogolyubov, 1947]: if $\hat v\geqslant 0$. $$ {\color{highlight}e_0\sim\frac\rho2\int v} $$ {\color{highlight}Hartree} (mean field) energy. \item {\color{highlight}Proved} in \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}. \item Condensate fraction $$ \eta\to1 $$ {\color{highlight}open}. \end{itemize} \vfill \eject \title{Energy as a function of density} 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 $v(x)=e^{-|x|}$: \hfil\includegraphics[height=5.5cm]{erho_effective.pdf} \vfill \eject \title{Derivation of the equation} \begin{itemize} \item \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}. \item Integrate $H_N\psi_0=E_0\psi_0$: $$ \int dx_1\cdots dx_N\ \left( -\frac12\sum_{i=1}^N\Delta_i\psi_0 +\sum_{1\leqslant i\displaystyle l} -\frac12(\Delta_x+\Delta_y) g_2(x,y) +\frac{N-2}V\int dz\ (v(x-z)+v(y-z))g_3(x,y,z) \\[0.5cm]\hfill +v(x-y)g_2(x,y) +\frac{(N-2)(N-3)}{2V^2}\int dzdt\ v(z-t)g_4(x,y,z,t) =E_0g_2(x,y) \end{array} $$ \item Factorization {\color{highlight}assumption}: $$ g_3(x_1,x_2,x_3)=g_2(x_1,x_2)g_2(x_1,x_3)g_2(x_2,x_3) $$ $$ g_4(x_1,x_2,x_3,x_4)=\prod_{i =-\frac1N\partial_\mu \left<\psi_0\right|H_N(\mu)\left|\psi_0\right>|_{\mu_0} \equiv {\color{highlight}-\partial_\mu e_0(\mu)|_{\mu=0}} $$ \end{itemize} \vfill \eject \title{Condensate fraction} \begin{itemize} \item {\bf Theorem 4}: For the {\color{iblue}simple equation}, as $\rho\to0$ $$ 1-\eta\sim\frac{8\sqrt{\rho a^3}}{3\sqrt\pi} $$ which coincides with {\color{highlight}Bogolyubov's prediction}. \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{Two point correlation function} $v(x)=16e^{-|x|}$, Blue: {\color{iblue}simple equation}; purple: {\color{ipurple}big equation}; red: {\color{ired}Monte Carlo} \hfil\includegraphics[height=5.5cm]{correlation.pdf} \vfill \eject \title{Summary and outlook} \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} \vfill \eject \title{Open problems} \vfill \begin{itemize} \item {\color{highlight}Uniqueness} of the solution of the {\color{iblue}simple equation} (done for small and large $\rho$). \vfill \item LHY as an {\color{highlight}upper bound} at low density using the {\color{iblue}simple equation} to construct an Ansatz. \vfill \item {\color{highlight}Existence} (and uniqueness) of the solution of the {\color{ipurple}big equation}. \end{itemize} \vfill \eject \title{The uniqueness problem} $$ -\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x) ,\quad e=\frac\rho2\int dx\ (1-u(x))v(x) $$ \begin{itemize} \item Change the point of view: {\color{highlight}fix $e>0$}, and compute $\rho$ and $u$. \item {\color{highlight}Iteration}: $u_0=0$, $$ (-\Delta+4e+v)u_n=v+2e\rho_{n-1}u_{n-1}\ast u_{n-1} ,\quad \rho_n:=\frac{2e}{\int dx\ (1-u_n(x))v(x)} . $$ \end{itemize} \vfill \eject \title{The uniqueness problem} \begin{itemize} \item {\bf Lemma}: $u_n(x)$ is an {\color{highlight}increasing} sequence, and is {\color{highlight}bounded} $u_n(x)\leqslant 1$. It converges to a function $u$, which is the {\color{highlight}unique} integrable solution of the equation {\color{highlight}with $e$ fixed}. \item {\bf Lemma}: $e\mapsto\rho(e)$ is {\color{highlight}continuous}, and $\rho(0)=0$ and $\rho(\infty)=\infty$, which allows us to compute solutions for the problem at fixed $\rho$. \item We thus have a {\color{highlight}restricted} notion of uniqueness. The full uniqueness would follow from a proof that $e\mapsto\rho(r)$ is {\color{highlight}monotone increasing} (which must be true for the physics to make sense). \end{itemize} \vfill \eject \title{Upper bound at low density} \begin{itemize} \item \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]}: proof for {\color{highlight}weak, smooth}, rapidly decaying potentials. \item \href{https://arxiv.org/abs/2101.06222}{[Basti, Cenatiempo, Schlein, 2021]}: extended for {\color{highlight}$L_3$} (and compactly supported) potentials (excludes hard-core interactions). \item {\color{iblue}Simple Equation}: our analysis holds for the hard-core, so if one could find a good {\color{highlight}Ansatz} from it, one might get an upper bound for the energy in this case. \item Idea for Ansatz? (Jastrow, Dyson-Jastrow?). \end{itemize} \vfill \eject \title{Upper bound at low density: Jastrow function} \begin{itemize} \item Idea: $$ \psi=\prod_{i