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diff --git a/Jauslin_Milan_2022.tex b/Jauslin_Milan_2022.tex new file mode 100644 index 0000000..f0f8433 --- /dev/null +++ b/Jauslin_Milan_2022.tex @@ -0,0 +1,430 @@ +\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}\par + \href{https://arxiv.org/abs/2202.07637}{2202.07637} +}} +\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: + $$ + H_N:= + -\frac12\sum_{i=1}^N\Delta_i + +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|) + $$ + \vskip-5pt + \item Ground state: $\psi_0$, energy $E_0$. + \item Observables in the {\color{highlight}thermodynamic limit}: ground state energy per particle and condensate fraction: $P_i$: projection onto condensate state + $$ + e_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac{E_0}N + ,\quad + \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\sum_{i=1}^N\left<\psi_0\right|P_i\left|\psi_0\right> + . + $$ +\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 $H_N\psi_0=E_0\psi_0$: + $$ + \int dx_1\cdots dx_N\ + \left( + -\frac12\sum_{i=1}^N\Delta_i\psi_0 + +\sum_{1\leqslant i<j\leqslant N} v(x_i-x_j)\psi_0 + \right) + =E_0\int dx_1\cdots dx_N\ \psi_0 + $$ + \item Therefore, + $$ + \frac{N(N-1)}2\int dx_1dx_2\ v(x_1-x_2)\frac{\int dx_3\cdots dx_N\ \psi_0}{\int dx_1\cdots dx_N\ \psi_0} + =E_0 + $$ +\end{itemize} +\vfill +\eject + +\title{Derivation of the equation} +\begin{itemize} + \item Thus, + $$ + \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(x)g_2(0,x) + $$ + \item {\color{highlight}$\psi_0\geqslant 0$}, so it can be thought of as a probability distribution. + \item $g_n$: {\color{highlight}correlation functions} of $V^{-N}\psi_0$ + $$ + g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dx_1\cdots dx_N\ \psi_0(x_1,\cdots,x_N)} + $$ +\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 {\color{highlight}assumption}: + $$ + g_3(x_1,x_2,x_3)=g_2(x_1,x_2)g_2(x_1,x_3)g_2(x_2,x_3) + $$ + $$ + g_4(x_1,x_2,x_3,x_4)=\prod_{i<j}(g_2(x_i,x_j)+O(V^{-1})) + $$ +\end{itemize} +\vfill +\eject + +\title{{\color{ipurple}Big equation}} +\begin{itemize} +\item In the thermodynamic limit, if ${\color{highlight}u(x):=1-g_2(0,x)}$, + $$ + -\Delta u(x) + = + (1-u(x))\left(v(x)-2\rho K(x)+\rho^2 L(x)\right) + $$ + $$ + K:= + u\ast S + ,\quad + S(y):=(1-u(y))v(y) + $$ + $$ + L:= + u\ast u\ast S + -2u\ast(u(u\ast S)) + +\frac12 + \int dydz\ u(y)u(z-x)u(z)u(y-x)S(z-y) + . + $$ + + \item {\color{ipurple}``Big'' equation}: + $$ + L\approx + u\ast u\ast S + -2u\ast (u(u\ast S)) + $$ +\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 Simple equation + $$ + {\color{iblue}-\Delta u(x)=(1-u(x))v(x)- 4eu(x)+2e\rho\ u\ast u(x)} + $$ + $$ + {\color{iblue}e=\frac\rho2\int dx\ (1-u(x))v(x)} + $$ + \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{Energy} +$v(x)=e^{-|x|}$, Blue: {\color{iblue}simple equation}; red: {\color{ired}Jastrow}; purple: {\color{ipurple}big equation} + +\hfil\includegraphics[height=5.5cm]{cmp_energy1.pdf} +\vfill +\eject + +\title{Condensate fraction} +\begin{itemize} + \item Add a parameter $\mu$ to the Hamiltonian: + $$ + H_N(\mu):= + -\frac12\sum_{i=1}^N\Delta_i + +\sum_{1\leqslant i<j\leqslant N}v(x_i-x_j) + {\color{highlight}-\mu\sum_{i=1}^NP_i} + $$ + \item Projection onto condensate wavefunction: $P_i$. + \item Condensate fraction: + $$ + {\color{highlight}\eta_0}:=\frac1N\left<\psi_0\right|\sum_{i=1}^NP_i\left|\psi_0\right> + =-\frac1N\partial_\mu \left<\psi_0\right|H_N(\mu)\left|\psi_0\right>|_{\mu_0} + \equiv + {\color{highlight}-\partial_\mu e_0(\mu)|_{\mu=0}} + $$ +\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: + $$ + \psi=\prod_{i<j}e^{-u(x_i-x_j)} + $$ + + \item Why this: $\rho\ll1$, and {\color{highlight}if $\rho\|u\|_1\ll1$}, + $$ + g_2\sim1-u + . + $$ + + \item Again, {\color{highlight}if $\rho\|u\|_1\ll1$}, we would be able to compute the energy of $\psi$ using a cluster expansion! + + \item However, {\color{highlight}$\|u\|_1=\frac1\rho$}! +\end{itemize} +\vfill +\eject + +\title{Existence for the {\color{ipurple}Big Equation}} +\begin{itemize} + \item Numerical method: {\color{highlight}Newton algorithm}. + + \item For the existence of a solution, it would suffice to prove that the Newton algorithm has a {\color{highlight}Basin of attraction}. (Kantorovich-like theorem?) + + \item Such a result, applied to the {\color{iblue}Simple Equation}, would imply the {\color{highlight}uniqueness} of a solution (provided we have convergence in an appropriate norm). +\end{itemize} + +\end{document} |