diff options
Diffstat (limited to 'Jauslin_SIAM_2020.tex')
| -rw-r--r-- | Jauslin_SIAM_2020.tex | 222 |
1 files changed, 222 insertions, 0 deletions
diff --git a/Jauslin_SIAM_2020.tex b/Jauslin_SIAM_2020.tex new file mode 100644 index 0000000..1cef79f --- /dev/null +++ b/Jauslin_SIAM_2020.tex @@ -0,0 +1,222 @@ +\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 gasses\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 \href{https://arxiv.org/abs/1912.04987}{1912.04987}} +\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\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: + $$ + H_N:= + -\frac12\sum_{i=1}^N\Delta_i + +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|) + $$ + \item Ground state: $\psi_0$, energy $E_0$. + \item Observables in the {\color{highlight}thermodynamic limit}: ground state energy per particle and condensate fraction: + $$ + e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N + ,\quad + \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\left<\psi_0\right|\sum_{i=1}^N\int\frac{dx_i}V\left|\psi_0\right> + . + $$ +\end{itemize} +\vfill +\eject + +\title{Effective theories} +\begin{itemize} + \item {\color{highlight}Bogolubov} theory: {\color{highlight}approximation scheme} that makes $H$ {\color{highlight}integrable}. + \item Low density 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} + \item At high density: {\color{highlight}Hartree mean-field theory}. +\end{itemize} +\vfill +\eject + +\title{Mathematical results} +\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://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]} + + \item Condensate fraction: {\color{highlight}still open} in the thermodynamic limit. + + \item At high density, {\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{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{{\color{ipurple}Big equation}} +\begin{itemize} +\item Solve for +\begin{equation} + u(x_2-x_1):=1-\frac{\int \frac{dx_3}V\cdots\frac{dx_N}V\ \psi_0(x_1,\cdots,x_N)}{\int \frac{dx_1}V\cdots\frac{dx_N}V\ \psi_0(x_1,\cdots,x_N)} +\end{equation} +\item {\color{ipurple}``Big'' equation}: + $${\color{ipurple} + -\Delta u(x) + = + (1-u(x))\left(v(x)-2\rho u\ast S(x)+\rho^2 u\ast u\ast S(x)\right) + }$$ + $$ + S(y):=(1-u(y))v(y) + $$ +\end{itemize} +\vfill +\eject + +\title{{\color{iblue}Simple equation}} +\vskip-10pt +\begin{itemize} + \item Further approximate $S(x)\approx\frac{2e}\rho\delta(x)$ and $u\ll 1$. + \item {\color{iblue}Simple equation}: + $$ + {\color{iblue}-\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)} + $$ + $$ + e=\frac\rho2\int dx\ (1-u(x))v(x) + $$ + \item {\bf Theorem 1}: + If $v(x)\geqslant 0$ and $v\in L_1\cap L_2(\mathbb R^3)$, then the {\color{iblue}simple equation} has an {\color{highlight}integrable solution} (proved constructively), with $0\leqslant u\leqslant 1$. +\end{itemize} +\vfill +\eject + +\title{Energy for the {\color{iblue}simple equation}} +\vskip-10pt +\begin{itemize} + \item {\bf Theorem 2}: + $$ + \frac{e}{\rho}\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x) + $$ + (note that there is no condition that $\hat v\geqslant 0$). + This coincides with the {\color{highlight}Hartree energy}. + \item {\bf Theorem 3}: + $$ + e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right) + $$ + This coincides with the {\color{highlight}Lee-Huang-Yang prediction}. +\end{itemize} +\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{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}Bogolubov'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{Limitations of the simple and big equations} +\begin{itemize} + \item Only works at high densities for {\color{highlight}$\hat v\geqslant 0$}. + \item Less accurate for {\color{highlight}large potentials}: for $v(x)=16e^{-|x|}$, + + \hfil\includegraphics[width=5.5cm]{energy16.pdf} + \hfil\includegraphics[width=5.5cm]{condensate16.pdf} +\end{itemize} +\vfill +\eject + +\title{Conclusions and outlooks} +\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 Bogolubov 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} +\begin{itemize} + \item Analysis of the {\color{iblue}simple equation}: {\color{highlight}Monotonicity} of $e(\rho)$, and {\color{highlight}convexity} of $\rho e(\rho)$. (So far, we have proofs for small and large $\rho$.). Similarly, prove that $0\leqslant\eta\leqslant 1$. (We have a proof for small $\rho$.) + + \item Analysis of the {\color{ipurple}big equation}: everything is still open. + + \item Relate these equations back to the {\color{highlight}many-body Bose gas}. +\end{itemize} + +\end{document} |