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\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Analysis of a non-linear, non-local PDE\par
\hfil to study Bose gases at all densities
\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{{\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.
\end{itemize}
\vfill
\eject

\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 {\color{highlight}$N$-particle} quantum state in a volume $V$:
  $$
    \psi_N(x_1,\cdots,x_N)\in L^2_{\mathrm{symmetric}}((V\mathbb T^3)^N)
  $$

  \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}$v(r)\geqslant 0$}, {\color{highlight}$\hat v(k)\geqslant 0$} and {\color{highlight}$v\in L_1(\mathbb R^3)$}.
\vfill
\eject

\title{Energy and condensate fraction}
\vskip-10pt
  \item Ground state: $\psi_0$, energy $E_0$:
  $$
    H_N\psi_0={\color{highlight}E_0}\psi_0
  $$
\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}:
  $$
    \eta_0:=\lim_{\displaystyle\mathop{\scriptstyle V,N\to\infty}_{{\color{highlight}\frac NV=\rho}}}\frac1N\left<\psi_0\right|\sum_{i=1}^N\int\frac{dx_i}V\left|\psi_0\right>
    .
  $$
\end{itemize}
\vfill
\eject

\title{Low density conjectures}
\begin{itemize}
  \item Bogolyubov theory: {\color{highlight}approximation scheme} [Bogolyubov, 1947].
  \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 conjectures}
\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 thermodynamic limit. (No proof of Bose-Einstein condensation.)

  \item There are proofs of condensation in the ultra-dilute (Gross-Pitaevskii) regime:
  \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]}.

  \item There is also a proof of condensation for a {\color{highlight}lattice} Bose gas
  \href{https://doi.org/10.1007/BF01023854}{[Kennedy, Lieb, Shastry, 1988]}.
\end{itemize}
\vfill
\eject

\title{High density conjectures}
\begin{itemize}
  \item [Bogolyubov, 1947]:
  $$
    {\color{highlight}e_0\sim\frac\rho2\int v}
  $$
  \item {\color{highlight}Proved} in \href{https://doi.org/10.1103/PhysRev.130.2518}{[Lieb, 1963]}.

  \item Condensate fraction: mean field regime: $\eta_0\to 1$. (No proof of Bose-Einstein condensation at any density.)
\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{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{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 $\psi_0$
  $$
    g_n(x_1,\cdots,x_n):=\frac{V^n\int dx_{n+1}\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}{\int dx_1\cdots dx_N\ \psi_0(x_1,\cdots,x_N)}
  $$
\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}:
  $$
    L\approx
    u\ast u\ast S.
  $$
\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}:
  $$
    \frac{e}{\rho}\mathop{\longrightarrow}_{\rho\to\infty}\frac12\int dx\ v(x)
    .
  $$
  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 Add a parameter $\mu$ to the Hamiltonian:
  $$
    H_N(\mu):=
    -\frac12\sum_{i=1}^N\Delta_i
    +\sum_{1\leqslant i<j\leqslant N}v(x_i-x_j)
    {\color{highlight}-\mu\sum_{i=1}^N\int\frac{dx_i}V\ \cdot}
  $$
  \item Projection onto condensate wavefunction: $\sum_i\int\frac{dx_i}V$.
  \item Condensate fraction:
  $$
    {\color{highlight}\eta_0}:=\frac1N\left<\psi_0\right|\sum_{i=1}^N\int\frac{dx_i}V\left|\psi_0\right>
    =-\frac1N\partial_\mu \left<\psi_0\right|H_N(\mu)\left|\psi_0\right>|_{\mu_0}
    \equiv
    {\color{highlight}-\partial_\mu e_0(\mu)|_{\mu=0}}
  $$
\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}.

  \item In particular {\color{highlight}there is Bose-Einstein condensation} for the simple equation.
\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 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 and next steps}
\vskip-10pt
\begin{itemize}
  \item Analysis of the {\color{ipurple}big equation}: everything is still open.
\vskip-10pt
  \begin{itemize}
    \item Main tool: {\color{highlight}Newton algorithm}, which works numerically.
    \item There is a family of {\color{highlight}intermediate equations} that extrapolate between the {\color{iblue} simple} and {\color{ipurple}big} equations.
  \end{itemize}

  \item Relate these equations back to the {\color{highlight}many-body Bose gas}.
  \vskip-10pt
  \begin{itemize}
    \item {\color{highlight}Upper bound} for the ground state energy, using a {\color{highlight}Bijl function} as a test function.
    \item {\color{highlight}Lee-Huang Yang formula} by studying the low-density properties of the {\color{highlight}Bijl function}.
    \item Extend the proof to the {\color{highlight}condensate fraction}.
  \end{itemize}
\end{itemize}

\end{document}