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diff --git a/Jauslin_UCDavis_2020.tex b/Jauslin_UCDavis_2020.tex new file mode 100644 index 0000000..1899f15 --- /dev/null +++ b/Jauslin_UCDavis_2020.tex @@ -0,0 +1,304 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{array} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil A 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 +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{Lieb's simple equation} +\vskip-10pt +\begin{itemize} + \item \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}: $x\in\mathbb R^3$ + $$ + (-\Delta+v(x)+4e)u(x)=v(x)+2e\rho\ u\ast u(x) + $$ + $$ + e=\frac\rho2\int dx\ (1-u(x))v(x) + $$ + \item with + $$ + \rho>0 + ,\quad + v(x)\geqslant 0 + ,\quad + v\in L_1\cap L_2(\mathbb R^3) + $$ + \item and + $$ + u\in L_1(\mathbb R^3) + ,\quad + u\ast u(x):=\int dy\ u(x-y)u(y) + $$ +\end{itemize} +\vfill +\eject + + +\title{Interacting Bose gas} +\vskip-10pt +\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 Probability distribution: $|\psi(x_1,\cdots,x_N)|^2$ + \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 $v(x-y)\geqslant 0$ and $v\in L_1\cap L_{\frac d2+\epsilon}(\mathbb R^d)$. +\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 $H_N\psi_0=E_0\psi_0$: + $$ + \frac{E_0}N=\frac{N-1}{2V}\int dx\ v(x)g_2(0,x) + $$ + \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]}: + $$ + (-\Delta+v(x)+4e)u(x)=v(x)+2e\rho\ u\ast u(x) + $$ + $$ + e=\frac\rho2\int dx\ (1-u(x))v(x) + $$ + \item with $\rho:=\frac NV$ + $$ + g_2(x,y)=1-u(x-y) + $$ +\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 for the Bose gas} +\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 ($a$: scattering length) + $$ + \frac{e_0}{\rho}\mathop{\longrightarrow}_{\rho\to0}2\pi a + $$ + \href{https://doi.org/10.1103/PhysRev.106.1135}{[Lee, Huang, Yang, 1957]}, \href{https://doi.org/10.1007/s10955-009-9792-3}{[Yau, Yin, 2009]}, \href{https://arxiv.org/abs/1904.06164}{[Fournais, Solovej, 2019]}: + $$ + e_0=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right) + $$ +\end{itemize} +\vfill +\eject + +\title{Comparison with Bose gas (Monte Carlo)} +\vskip-10pt +\hfil\includegraphics[width=8cm]{erho_holzmann.pdf} + +\hfil{\footnotesize Monte Carlo computation courtesy of M. Holzmann} +\vfill +\eject + +\title{Main Theorem} +\vskip-5pt +\begin{itemize} + \item If $v(x)\geqslant 0$ and $v\in L_1\cap L_{\frac d2+\epsilon}(\mathbb R^d)$, then Lieb's simple equation + $$ + (-\Delta+4e+v)u=v+2e\rho u\ast u + ,\quad + e=\frac\rho2\int dx\ (1-u(x))v(x) + $$ + has an integrable solution (proved constructively), with $0\leqslant u\leqslant 1$. + + \item In 3 dimensions, + $$ + e=2\pi\rho a\left(1+\frac{128}{15\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt\rho)\right) + ,\quad + \frac{e}\rho\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x) + $$ + + \item If $v(x)\equiv v(|x|)$ is radially symmetric and decays exponentially, + $$ + u(|x|)\mathop\sim_{|x|\to\infty}\frac\alpha{|x|^4} + $$ +\end{itemize} +\vfill +\eject + +\title{Existence of a solution (sketch)} +\begin{itemize} + \item Change the point of view: fix $e>0$, and compute $\rho$ and $u$. + + \item Iteration: $u_0=0$, + $$ + (-\Delta+4e+v)u_n=v+2e\rho_{n-1}u_{n-1}\ast u_{n-1} + ,\quad + \rho_n:=\frac{2e}{\int dx\ (1-u_n(x))v(x)} + . + $$ + + \item Prove by induction that $u_n(x)$ is an increasing sequence, and is bounded $u_n(x)\leqslant 1$. It therefore converges to a function $u$, which is the unique integrable solution of the equation with $e$ fixed. + + \item In addition, we prove that $e\mapsto\rho(e)$ is continuous, and $\rho(0)=0$ and $\rho(\infty)=\infty$, which allows us to compute solutions for the problem at fixed $\rho$. This does not imply the uniqueness of the solution. +\end{itemize} +\vfill +\eject + +\title{Asymptotics (sketch)} +\vskip-10pt +\begin{itemize} + \item When $\rho$ is small, $e$ is small as well, so the solution $u$ is {\it not too far from} the solution of the scattering equation + $$ + (-\Delta+v)\varphi=v + . + $$ + + \item The energy of $\varphi$ is + $$ + \frac\rho 2\int dx\ (1-\varphi(x))v(x)=2\pi\rho a + $$ + which yields the first term in the expansion. + + \item The second term comes from approximating + $$ + (1-u(x))v(x)\approx\frac{2e}\rho\delta(x) + $$ + and solving the equation in Fourier space. +\end{itemize} +\vfill +\eject + +\title{Decay (sketch)} +$$ + (-\Delta+4e+v)u=v+2e\rho u\ast u + ,\quad + e=\frac\rho2\int dx\ (1-u(x))v(x) +$$ +\begin{itemize} + \item $u$ and $u\ast u$ have to decay at the same rate. This is a property of algebraically decaying functions. + + \item (Remark: if $f(x)\geqslant f\ast f(x)$ and $\int f=\frac12$, then (morally) $f\sim|x|^{d+1}$.) + + \item (Remark: $u_n(x)$ decays exponentially). + + \item Proof is based on the Fourier transform and complex analysis. + + \item Remark: The truncated two-point correlation function of the Bose gas is also conjectured to decay like $|x|^{-4}$. +\end{itemize} +\vfill +\eject + +\title{Conclusion} +\vfill +\begin{itemize} + \item Simple equation: correct asymptotics for the ground state energy at both high and low densities. + + \item Good approximation for intermediate densities (relative error of 5\%). + + \item Intriguing non-linear PDE. + + \item Proved existence, asymptotics, and decay rate. +\end{itemize} +\vfill +\eject + +\title{Open problems and conjectures} +\begin{itemize} + \item Monotonicity of $e\mapsto\rho(e)$, and concavity of $e\mapsto\frac1{\rho(e)}$ (would imply uniqueness). + + \item Other observables? Condensate fraction? (in progress) + + \item Crystallization? + + \item {\it Lieb's simple equation} is actually a simplified version of a more complicated one: {\it Lieb's full equation}. Can it improve on the simple one? (in progress) +\end{itemize} + +\end{document} |