Ian Jauslin
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-rw-r--r--Jauslin_LMU_2021.tex398
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diff --git a/Jauslin_LMU_2021.tex b/Jauslin_LMU_2021.tex
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+\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:
+ $$
+ H_N:=
+ -\frac12\sum_{i=1}^N\Delta_i
+ +\sum_{1\leqslant i<j\leqslant N}v(|x_i-x_j|)
+ $$
+ \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{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{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 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}
+\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}.
+
+ \item Other setups: {\color{highlight}trapping potential}.
+\end{itemize}
+
+\end{document}
diff --git a/Makefile b/Makefile
new file mode 100644
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--- /dev/null
+++ b/Makefile
@@ -0,0 +1,45 @@
+PROJECTNAME=$(basename $(wildcard *.tex))
+LIBS=$(notdir $(wildcard libs/*))
+FIGS=$(notdir $(wildcard figs/*.fig))
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME))
+
+all: $(PROJECTNAME)
+
+$(PROJECTNAME): $(LIBS) $(FIGS)
+ pdflatex -file-line-error $@.tex
+ pdflatex -synctex=1 $@.tex
+
+$(SYNCTEXS): $(LIBS) $(FIGS)
+ pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@)
+
+$(LIBS):
+ ln -fs libs/$@ ./
+
+
+$(FIGS):
+ make -C figs/$@
+ for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done
+
+
+clean-aux: clean-figs-aux
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+ rm -f $(addsuffix .out, $(PROJECTNAME))
+
+clean-libs:
+ rm -f $(LIBS)
+
+clean-figs:
+ $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; )
+ rm -f $(notdir $(wildcard figs/*.fig/*.pdf))
+
+clean-figs-aux:
+ $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; )
+
+
+clean-tex:
+ rm -f $(PDFS) $(SYNCTEXS)
+
+clean: clean-aux clean-tex clean-libs clean-figs
diff --git a/README b/README
new file mode 100644
index 0000000..d64b4a7
--- /dev/null
+++ b/README
@@ -0,0 +1,33 @@
+This directory contains the source files to typeset the presentation, and
+generate the figures. This can be accomplished by running
+ make
+
+This document uses a custom class file, located in the 'libs' directory, which
+defines a number of commands.
+
+
+* Dependencies:
+
+ pdflatex
+ TeXlive packages:
+ amsfonts
+ array
+ graphics
+ hyperref
+ latex
+ pgf
+ standalone
+ GNU make
+ gnuplot
+
+* Files:
+
+ Jauslin_LMU_2021.tex:
+ main LaTeX file
+
+ libs:
+ custom LaTeX class file
+
+ figs:
+ source code for the figures
+
diff --git a/figs/condensate.fig/Makefile b/figs/condensate.fig/Makefile
new file mode 100644
index 0000000..c45de65
--- /dev/null
+++ b/figs/condensate.fig/Makefile
@@ -0,0 +1,34 @@
+PROJECTNAME=condensate
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+DATS= simpleq.dat bigeq.dat
+
+all: $(PDFS)
+
+$(PDFS): $(DATS)
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+ pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+simpleq.dat:
+ julia $(SIMPLEQ)/main.jl -p "tolerance=1e-11;order=100;maxiter=100;v_a=1" -M simpleq condensate_fraction > $@
+bigeq.dat:
+ julia -p 8 $(SIMPLEQ)/main.jl -p "eq=bigeq;N=12;P=8;v_a=1" -M medeq condensate_fraction > $@
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+$(LIBS):
+ ln -fs libs/$@ ./
+
+clean-libs:
+ rm -f $(LIBS)
+
+clean-aux:
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+ rm -f $(addsuffix .tex, $(PROJECTNAME))
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-libs clean-aux clean-tex
diff --git a/figs/condensate.fig/bigeq.dat b/figs/condensate.fig/bigeq.dat
new file mode 100644
index 0000000..e48cbc3
--- /dev/null
+++ b/figs/condensate.fig/bigeq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-06 1.942109602248065e-03 1.702970665674168e-12
+ 1.202264434617413e-06 2.144142093189251e-03 1.178808810520995e-13
+ 1.445439770745928e-06 2.370859341242408e-03 1.192336354552407e-12
+ 1.737800828749376e-06 2.623985546948715e-03 4.076932518598860e-12
+ 2.089296130854041e-06 2.904135074720890e-03 3.065737407737518e-12
+ 2.511886431509582e-06 3.210803051150611e-03 6.684762465649123e-13
+ 3.019951720402019e-06 3.543752141002869e-03 4.093999456324028e-14
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diff --git a/figs/condensate.fig/condensate.gnuplot b/figs/condensate.fig/condensate.gnuplot
new file mode 100644
index 0000000..f1c689d
--- /dev/null
+++ b/figs/condensate.fig/condensate.gnuplot
@@ -0,0 +1,36 @@
+set ylabel "$\\eta$" norotate
+set xlabel "$\\rho$"
+
+set yrange [0.9:1]
+set ytics 0.91, 0.03
+set mytics 3
+
+set xrange [1e-6:100]
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 2
+set style line 2 linetype rgbcolor "#DC143C" linewidth 2
+set style line 3 linetype rgbcolor "#32CD32" linewidth 2
+set style line 4 linetype rgbcolor "#4B0082" linewidth 2
+set style line 5 linetype rgbcolor "#DAA520" linewidth 2
+
+set pointsize 0.6
+
+set label at 0.000005,0.91 "Bogolyubov" textcolor "#32CD32"
+
+set logscale x
+
+a=1.25435641059
+
+plot "simpleq.dat" u 1:(1-$2) w l ls 1, \
+ "bigeq.dat" u 1:(1-$2) w l ls 4 , \
+ 1-8/(3*sqrt(pi))*sqrt(x*a**3) linestyle 3 dashtype "." ,\
+ "holzmann_2019-12-25.dat" u 1:(1-$3) ls 2
diff --git a/figs/condensate.fig/holzmann_2019-12-25.dat b/figs/condensate.fig/holzmann_2019-12-25.dat
new file mode 100644
index 0000000..96689cb
--- /dev/null
+++ b/figs/condensate.fig/holzmann_2019-12-25.dat
@@ -0,0 +1,11 @@
+## data from M. Holzmann, 2019-09-22
+# rho E0 n0
+1e-6 7.902e-6
+1e-4 8.3441e-4 0.0171
+1e-3 9.13384e-3 0.0481
+1e-2 1.061073e-1 0.0871
+1e-1 1.19182e+0 0.0851
+1e-0 1.24302e+1 0.0587
+1e+1 1.25442e+2 0.0319
+5e+1 6.28032e+2 0.021
+
diff --git a/figs/condensate.fig/simpleq.dat b/figs/condensate.fig/simpleq.dat
new file mode 100644
index 0000000..348121c
--- /dev/null
+++ b/figs/condensate.fig/simpleq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-06 2.095081711925015e-03
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+ 1.584893192461114e+01 1.530130079086820e-04
+ 1.905460717963248e+01 1.273233330370514e-04
+ 2.290867652767775e+01 1.059394293104648e-04
+ 2.754228703338169e+01 8.814186420712528e-05
+ 3.311311214825908e+01 7.333072418003886e-05
+ 3.981071705534969e+01 6.100596972384632e-05
+ 4.786300923226381e+01 5.075095479639026e-05
+ 5.754399373371567e+01 4.221862079919067e-05
+ 6.918309709189363e+01 3.511994492291246e-05
+ 8.317637711026708e+01 2.921428437169262e-05
diff --git a/figs/condensate16.fig/.energy16.gnuplot.swp b/figs/condensate16.fig/.energy16.gnuplot.swp
new file mode 100644
index 0000000..d23db08
--- /dev/null
+++ b/figs/condensate16.fig/.energy16.gnuplot.swp
Binary files differ
diff --git a/figs/condensate16.fig/2020-05-22+16.dat b/figs/condensate16.fig/2020-05-22+16.dat
new file mode 100644
index 0000000..c28285d
--- /dev/null
+++ b/figs/condensate16.fig/2020-05-22+16.dat
@@ -0,0 +1,7 @@
+# rho E0 n0
+ 1e-3 0.05195 0.61
+ 5e-3 0.449 0.40
+ 1e-2 1.1335 0.34
+ 2e-2 2.748 0.33
+ 1e-1 17.797 0.43
+ 1e+0 197.35 0.70
diff --git a/figs/condensate16.fig/Makefile b/figs/condensate16.fig/Makefile
new file mode 100644
index 0000000..d44f73c
--- /dev/null
+++ b/figs/condensate16.fig/Makefile
@@ -0,0 +1,35 @@
+PROJECTNAME=condensate16
+
+SIMPLEQ=~/Work/Research/2018+bose_gas/cmp/simpleq
+
+DATS=simpleq16.dat medeq16.dat
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(DATS)
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+ pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+simpleq16.dat:
+ julia $(SIMPLEQ)/main.jl -p "tolerance=1e-11;order=100;maxiter=100;v_a=16" -M simpleq condensate_fraction > $@
+medeq16.dat:
+ julia -p 8 $(SIMPLEQ)/main.jl -p "aK=1;aL1=1;aL2=1;aL3=0;bK=1;bL1=1;bL2=1;gK=1;gL2=1;gL3=0;N=12;P=8;v_a=16" -M medeq condensate_fraction > $@
+
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+clean-aux:
+ rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-dat:
+ rm -f $(DATS)
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-aux clean-tex
diff --git a/figs/condensate16.fig/condensate16.gnuplot b/figs/condensate16.fig/condensate16.gnuplot
new file mode 100644
index 0000000..e94eb1a
--- /dev/null
+++ b/figs/condensate16.fig/condensate16.gnuplot
@@ -0,0 +1,34 @@
+set ylabel "$\\eta$" norotate offset -1,0
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+#set ytics 0.6, 0.1
+set mytics 2
+
+set xrange [0.000001:100]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 2
+set style line 2 linetype rgbcolor "#DC143C" linewidth 2
+set style line 3 linetype rgbcolor "#32CD32" linewidth 2
+set style line 4 linetype rgbcolor "#4B0082" linewidth 2
+set style line 5 linetype rgbcolor "#DAA520" linewidth 2
+
+set pointsize 1
+
+set logscale x
+
+plot \
+ "simpleq16.dat" using 1:($3<1e-5 ? 1-$2 : 1/0) with lines ls 1 ,\
+ "medeq16.dat" using 1:($3<1e-5 ? 1-$2 : 1/0) with lines ls 4 ,\
+ "2020-05-22+16.dat" using 1:3 with points ls 2
+
diff --git a/figs/condensate16.fig/medeq16.dat b/figs/condensate16.fig/medeq16.dat
new file mode 100644
index 0000000..16ef72d
--- /dev/null
+++ b/figs/condensate16.fig/medeq16.dat
@@ -0,0 +1,100 @@
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diff --git a/figs/condensate16.fig/simpleq16.dat b/figs/condensate16.fig/simpleq16.dat
new file mode 100644
index 0000000..c19b5fd
--- /dev/null
+++ b/figs/condensate16.fig/simpleq16.dat
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diff --git a/figs/correlation.fig/2020-08-27+correlation.dat b/figs/correlation.fig/2020-08-27+correlation.dat
new file mode 100644
index 0000000..c436b89
--- /dev/null
+++ b/figs/correlation.fig/2020-08-27+correlation.dat
@@ -0,0 +1,102 @@
+# Potential: 16 e^{-|x|}, rho=0.02
+# |x| 2-pt correlation
+ 0.04641589 0.00532330 0.00040199 0.00532330 0.00040199 0.00000000 0.00000000 0.00000000 0.00000000
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diff --git a/figs/correlation.fig/Makefile b/figs/correlation.fig/Makefile
new file mode 100644
index 0000000..fd4b6c3
--- /dev/null
+++ b/figs/correlation.fig/Makefile
@@ -0,0 +1,36 @@
+PROJECTNAME=correlation
+
+SIMPLEQ=~/Work/Research/2018+bose_gas/cmp/simpleq
+
+DATS=bigeq.dat mueq.dat simpleq.dat
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(DATS)
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+ pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+simpleq.dat:
+ julia $(SIMPLEQ)/main.jl -p "rho=2e-2;minlrho=-6;nlrho=60;eq=simpleq;N=12;P=8;xmax=10" -M medeq 2pt-simpleq > $@
+mueq.dat:
+ julia $(SIMPLEQ)/main.jl -p "rho=2e-2;minlrho=-6;nlrho=60;eq=mueq;N=12;P=8;xmax=10" -M medeq 2pt > $@
+bigeq.dat:
+ julia $(SIMPLEQ)/main.jl -p "rho=2e-2;minlrho=-6;nlrho=60;v_a=16;eq=bigeq;N=12;P=8;xmax=10" -M medeq 2pt > $@
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+clean-aux:
+ rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-dat:
+ rm -f $(DATS)
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-aux clean-tex
diff --git a/figs/correlation.fig/bigeq.dat b/figs/correlation.fig/bigeq.dat
new file mode 100644
index 0000000..07b50b3
--- /dev/null
+++ b/figs/correlation.fig/bigeq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-01 -7.492541163311106e-06
+ 2.000000000000000e-01 -8.016902978356378e-06
+ 3.000000000000000e-01 -8.679554787819034e-06
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+ 5.000000000000000e-01 -1.015424596606049e-05
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diff --git a/figs/correlation.fig/correlation.gnuplot b/figs/correlation.fig/correlation.gnuplot
new file mode 100644
index 0000000..9be226f
--- /dev/null
+++ b/figs/correlation.fig/correlation.gnuplot
@@ -0,0 +1,26 @@
+set ylabel "$\\frac{C_2}{\\rho^2}$" norotate offset -1,0
+set xlabel "$|\\mathbf x|$"
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+unset key
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 2
+set style line 2 linetype rgbcolor "#DC143C" linewidth 1
+set style line 3 linetype rgbcolor "#32CD32" linewidth 2
+set style line 4 linetype rgbcolor "#4B0082" linewidth 2
+set style line 5 linetype rgbcolor "#DAA520" linewidth 2
+set style line 6 linetype rgbcolor "#555500" linewidth 2
+
+set pointsize 0.5
+
+rho=2e-2
+
+plot \
+ "simpleq.dat" using 1:($2/rho**2) with lines ls 1,\
+ "bigeq.dat" using 1:($2/rho**2) with lines ls 4,\
+ "2020-08-27+correlation.dat" using 1:2 ls 2
+
diff --git a/figs/correlation.fig/mueq.dat b/figs/correlation.fig/mueq.dat
new file mode 100644
index 0000000..4662058
--- /dev/null
+++ b/figs/correlation.fig/mueq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-01 2.130418403340800e-04
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diff --git a/figs/correlation.fig/simpleq.dat b/figs/correlation.fig/simpleq.dat
new file mode 100644
index 0000000..21074ac
--- /dev/null
+++ b/figs/correlation.fig/simpleq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-01 2.576832012945544e-04
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diff --git a/figs/energy.fig/Makefile b/figs/energy.fig/Makefile
new file mode 100644
index 0000000..e8d3f12
--- /dev/null
+++ b/figs/energy.fig/Makefile
@@ -0,0 +1,28 @@
+PROJECTNAME=erho_effective erho_fulleq erho_lowhigh
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS):
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tex, $@)
+ pdflatex -file-line-error $(patsubst %.pdf, %.tex, $@)
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+$(LIBS):
+ ln -fs libs/$@ ./
+
+clean-libs:
+ rm -f $(LIBS)
+
+clean-aux:
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+ rm -f $(addsuffix .tex, $(PROJECTNAME))
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-libs clean-aux clean-tex
diff --git a/figs/energy.fig/erho.dat b/figs/energy.fig/erho.dat
new file mode 100644
index 0000000..837439f
--- /dev/null
+++ b/figs/energy.fig/erho.dat
@@ -0,0 +1,104 @@
+ # v(x)=exp(-|x|), order=100, tolerance=1e-14
+ #
+ # rho energy
+ #
+ 1.00000000e-06 7.93445832e-06
+ 1.20226443e-06 9.54545091e-06
+ 1.44543977e-06 1.14842358e-05
+ 1.73780083e-06 1.38177301e-05
+ 2.08929613e-06 1.66265809e-05
+ 2.51188643e-06 2.00080034e-05
+ 3.01995172e-06 2.40792133e-05
+ 3.63078055e-06 2.89815779e-05
+ 4.36515832e-06 3.48856405e-05
+ 5.24807460e-06 4.19972045e-05
+ 6.30957344e-06 5.05647046e-05
+ 7.58577575e-06 6.08881446e-05
+ 9.12010839e-06 7.33299394e-05
+ 1.09647820e-05 8.83280780e-05
+ 1.31825674e-05 1.06412115e-04
+ 1.58489319e-05 1.28222614e-04
+ 1.90546072e-05 1.54534805e-04
+ 2.29086765e-05 1.86287394e-04
+ 2.75422870e-05 2.24617677e-04
+ 3.31131121e-05 2.70904370e-04
+ 3.98107171e-05 3.26819905e-04
+ 4.78630092e-05 3.94394317e-04
+ 5.75439937e-05 4.76093386e-04
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+ 8.31763771e-05 6.94502595e-04
+ 1.00000000e-04 8.39296065e-04
+ 1.20226443e-04 1.01470042e-03
+ 1.44543977e-04 1.22730542e-03
+ 1.73780083e-04 1.48514988e-03
+ 2.08929613e-04 1.79804690e-03
+ 2.51188643e-04 2.17798310e-03
+ 3.01995172e-04 2.63960836e-03
+ 3.63078055e-04 3.20083623e-03
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+ 9.12010839e-02 1.11585830e+00
+ 1.09647820e-01 1.34683126e+00
+ 1.31825674e-01 1.62477387e+00
+ 1.58489319e-01 1.95916654e+00
+ 1.90546072e-01 2.36140673e+00
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+ 2.75422870e-01 3.42701297e+00
+ 3.31131121e-01 4.12666196e+00
+ 3.98107171e-01 4.96795983e+00
+ 4.78630092e-01 5.97954056e+00
+ 5.75439937e-01 7.19583118e+00
+ 6.91830971e-01 8.65822353e+00
+ 8.31763771e-01 1.04164830e+01
+ 1.00000000e+00 1.25304420e+01
+ 1.20226443e+00 1.50720365e+01
+ 1.44543977e+00 1.81277534e+01
+ 1.73780083e+00 2.18015742e+01
+ 2.08929613e+00 2.62185132e+01
+ 2.51188643e+00 3.15288712e+01
+ 3.01995172e+00 3.79133506e+01
+ 3.63078055e+00 4.55892039e+01
+ 4.36515832e+00 5.48176270e+01
+ 5.24807460e+00 6.59126465e+01
+ 6.30957344e+00 7.92518063e+01
+ 7.58577575e+00 9.52890139e+01
+ 9.12010839e+00 1.14569987e+02
+ 1.09647820e+01 1.37750822e+02
+ 1.31825674e+01 1.65620322e+02
+ 1.58489319e+01 1.99126836e+02
+ 1.90546072e+01 2.39410530e+02
+ 2.29086765e+01 2.87842186e+02
+ 2.75422870e+01 3.46069846e+02
+ 3.31131121e+01 4.16074894e+02
+ 3.98107171e+01 5.00239474e+02
+ 4.78630092e+01 6.01427558e+02
+ 5.75439937e+01 7.23082394e+02
+ 6.91830971e+01 8.69343678e+02
+ 8.31763771e+01 1.04518842e+03
diff --git a/figs/energy.fig/erho_effective.gnuplot b/figs/energy.fig/erho_effective.gnuplot
new file mode 100644
index 0000000..f26252e
--- /dev/null
+++ b/figs/energy.fig/erho_effective.gnuplot
@@ -0,0 +1,38 @@
+set ylabel "$\\displaystyle\\frac{e}{\\rho}$" norotate offset -1,0
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+set ytics 8,1
+set mytics 2
+
+set xrange [0.000001:100]
+set yrange [7.5:13.5]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 3
+set style line 2 linetype rgbcolor "#DC143C" linewidth 3
+set style line 3 linetype rgbcolor "#32CD32" linewidth 3
+set style line 4 linetype rgbcolor "#4B0082" linewidth 3
+set style line 5 linetype rgbcolor "#DAA520" linewidth 3
+
+set pointsize 1
+
+set logscale x
+
+a=1.25435641059
+
+set label at 0.0001,2*pi*a*1.03 "LHY" textcolor "#DAA520"
+set label at 0.1,4.1*pi "Hartree" textcolor "#32CD32"
+
+plot "erho.dat" using 1:($2/$1) with lines linestyle 1 ,\
+ 2*pi*a*(1+128/15/sqrt(pi)*sqrt(x*a**3)) linestyle 5 dashtype "." ,\
+ 4*pi linestyle 3 dashtype "."
diff --git a/figs/energy.fig/erho_fulleq.dat b/figs/energy.fig/erho_fulleq.dat
new file mode 100644
index 0000000..351d1b9
--- /dev/null
+++ b/figs/energy.fig/erho_fulleq.dat
@@ -0,0 +1,100 @@
+ 1.000000000000000e-06 7.943042518656766e-06 0
+ 1.202264434617413e-06 9.550219866068997e-06 0
+ 1.445439770745928e-06 1.148608224097378e-05 1
+ 1.737800828749376e-06 1.381647407426905e-05 1
+ 2.089296130854041e-06 1.662160583174429e-05 0
+ 2.511886431509582e-06 2.000065135391551e-05 0
+ 3.019951720402019e-06 2.406839054942614e-05 0
+ 3.630780547701017e-06 2.896662133595565e-05 1
+ 4.365158322401657e-06 3.486570260772683e-05 1
+ 5.248074602497723e-06 4.197087478032896e-05 1
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diff --git a/figs/energy.fig/erho_fulleq.gnuplot b/figs/energy.fig/erho_fulleq.gnuplot
new file mode 100644
index 0000000..2927ead
--- /dev/null
+++ b/figs/energy.fig/erho_fulleq.gnuplot
@@ -0,0 +1,37 @@
+set ylabel "$\\displaystyle\\frac{e}{\\rho}$" norotate offset -1,0
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+set ytics 8,1
+set mytics 2
+
+set xrange [0.000001:100]
+set yrange [7.5:13.5]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 3
+set style line 2 linetype rgbcolor "#DC143C" linewidth 3
+set style line 3 linetype rgbcolor "#32CD32" linewidth 3
+set style line 4 linetype rgbcolor "#4B0082" linewidth 3
+set style line 5 linetype rgbcolor "#DAA520" linewidth 3
+
+set pointsize 1
+
+set logscale x
+
+a=1.25435641059
+
+plot "erho.dat" using 1:($2/$1) with lines linestyle 1 ,\
+ "erho_fulleq.dat" using 1:($2/$1) with lines linestyle 4 ,\
+ 2*pi*a*(1+128/15/sqrt(pi)*sqrt(x*a**3)) linestyle 5 dashtype "." ,\
+ 4*pi linestyle 3 dashtype "." ,\
+ "holzmann_2019-12-25.dat" using 1:($2/$1) with points linestyle 2
diff --git a/figs/energy.fig/erho_lowhigh.gnuplot b/figs/energy.fig/erho_lowhigh.gnuplot
new file mode 100644
index 0000000..9d2a590
--- /dev/null
+++ b/figs/energy.fig/erho_lowhigh.gnuplot
@@ -0,0 +1,37 @@
+set ylabel "$\\displaystyle\\frac{e}{\\rho}$" norotate offset -1,0
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+set ytics 8,1
+set mytics 2
+
+set xrange [0.000001:100]
+set yrange [7.5:13.5]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 3
+set style line 2 linetype rgbcolor "#DC143C" linewidth 3
+set style line 3 linetype rgbcolor "#32CD32" linewidth 3
+set style line 4 linetype rgbcolor "#4B0082" linewidth 3
+set style line 5 linetype rgbcolor "#DAA520" linewidth 3
+
+set pointsize 1
+
+set logscale x
+
+a=1.25435641059
+
+set label at 0.0001,2*pi*a*1.03 "LHY" textcolor "#DAA520"
+set label at 0.1,4.1*pi "Hartree" textcolor "#32CD32"
+
+plot 2*pi*a*(1+128/15/sqrt(pi)*sqrt(x*a**3)) linestyle 5 dashtype "." ,\
+ 4*pi linestyle 3 dashtype "."
diff --git a/figs/energy.fig/holzmann_2019-12-25.dat b/figs/energy.fig/holzmann_2019-12-25.dat
new file mode 100644
index 0000000..96689cb
--- /dev/null
+++ b/figs/energy.fig/holzmann_2019-12-25.dat
@@ -0,0 +1,11 @@
+## data from M. Holzmann, 2019-09-22
+# rho E0 n0
+1e-6 7.902e-6
+1e-4 8.3441e-4 0.0171
+1e-3 9.13384e-3 0.0481
+1e-2 1.061073e-1 0.0871
+1e-1 1.19182e+0 0.0851
+1e-0 1.24302e+1 0.0587
+1e+1 1.25442e+2 0.0319
+5e+1 6.28032e+2 0.021
+
diff --git a/figs/energy16.fig/2020-05-22+16.dat b/figs/energy16.fig/2020-05-22+16.dat
new file mode 100644
index 0000000..c28285d
--- /dev/null
+++ b/figs/energy16.fig/2020-05-22+16.dat
@@ -0,0 +1,7 @@
+# rho E0 n0
+ 1e-3 0.05195 0.61
+ 5e-3 0.449 0.40
+ 1e-2 1.1335 0.34
+ 2e-2 2.748 0.33
+ 1e-1 17.797 0.43
+ 1e+0 197.35 0.70
diff --git a/figs/energy16.fig/Makefile b/figs/energy16.fig/Makefile
new file mode 100644
index 0000000..964a8c5
--- /dev/null
+++ b/figs/energy16.fig/Makefile
@@ -0,0 +1,35 @@
+PROJECTNAME=energy16
+
+SIMPLEQ=~/Work/Research/2018+bose_gas/cmp/simpleq
+
+DATS=simpleq16.dat medeq16.dat
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(DATS)
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+ pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+simpleq16.dat:
+ julia $(SIMPLEQ)/main.jl -p "tolerance=1e-11;order=100;maxiter=100;v_a=16" -M simpleq energy_rho > $@
+medeq16.dat:
+ julia -p 8 $(SIMPLEQ)/main.jl -p "aK=1;aL1=1;aL2=1;aL3=0;bK=1;bL1=1;bL2=1;gK=1;gL2=1;gL3=0;N=12;P=8;v_a=16" -M medeq energy_rho > $@
+
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+clean-aux:
+ rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-dat:
+ rm -f $(DATS)
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-aux clean-tex
diff --git a/figs/energy16.fig/energy16.gnuplot b/figs/energy16.fig/energy16.gnuplot
new file mode 100644
index 0000000..dc996c5
--- /dev/null
+++ b/figs/energy16.fig/energy16.gnuplot
@@ -0,0 +1,34 @@
+set ylabel "$\\displaystyle\\frac{e}{\\rho}$" norotate offset -1,0
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+
+#set ytics 0.6, 0.1
+set mytics 2
+
+set xrange [0.000001:100]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+set key off
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 2
+set style line 2 linetype rgbcolor "#DC143C" linewidth 2
+set style line 3 linetype rgbcolor "#32CD32" linewidth 2
+set style line 4 linetype rgbcolor "#4B0082" linewidth 2
+set style line 5 linetype rgbcolor "#DAA520" linewidth 2
+
+set pointsize 1
+
+set logscale x
+
+plot \
+ "simpleq16.dat" using 1:($3<1e-5 ? ($2/$1) : 1/0) with lines ls 1 ,\
+ "medeq16.dat" using 1:($3<1e-5 ? ($2/$1) : 1/0) with lines ls 4 ,\
+ "2020-05-22+16.dat" using 1:($2/$1) with points ls 2
+
diff --git a/figs/energy16.fig/medeq16.dat b/figs/energy16.fig/medeq16.dat
new file mode 100644
index 0000000..b670ef5
--- /dev/null
+++ b/figs/energy16.fig/medeq16.dat
@@ -0,0 +1,100 @@
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diff --git a/figs/energy16.fig/medeq4.dat b/figs/energy16.fig/medeq4.dat
new file mode 100644
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diff --git a/figs/energy_relative_error.fig/2020-10-15+energy.dat b/figs/energy_relative_error.fig/2020-10-15+energy.dat
new file mode 100644
index 0000000..8a6566c
--- /dev/null
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+1e-0 1.2430e+1 1.2431e+1 0.058 0.059 1.245e+1 1.246e+1 0.05 0.06
+1e+1 1.2544e+2 1.2545e+2 0.031 0.032
+5e+1 6.2803e+2 6.2804e+2 0.02 0.03
+
diff --git a/figs/energy_relative_error.fig/Makefile b/figs/energy_relative_error.fig/Makefile
new file mode 100644
index 0000000..93e03f5
--- /dev/null
+++ b/figs/energy_relative_error.fig/Makefile
@@ -0,0 +1,51 @@
+PROJECTNAME=cmp_energy1
+
+SIMPLEQ=~/Work/Research/2018+bose_gas/cmp/simpleq
+
+DATS= simpleq1.dat mueq1.dat bigeq1.dat
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
+
+all: $(PDFS)
+
+$(PDFS): $(DATS)
+ gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
+ pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
+
+bigeq1.dat:
+ for rho in 1e-6 1e-4 1e-3 5e-3 1e-2 1.5e-2 2e-2 5e-2 1e-1 1.0 1e1 5e1; do \
+ echo -n $$rho " " >> $@-tmp ;\
+ julia $(SIMPLEQ)/main.jl -p "N=12;P=8;J=10;maxiter=100;v_a=1;rho=$$rho;minlrho=-6;nlrho=50;eq=bigeq" -M medeq energy >> $@-tmp ;\
+ done
+ paste $@-tmp 2020-10-15+energy.dat > $@
+ rm $@-tmp
+simpleq1.dat:
+ for lrho in -6 -4 -3 -2.3010299956639813 -2 -1.8239087409443189 -1.6989700043360187 -1.3010299956639813 -1 0 1 1.6989700043360187; do\
+ julia $(SIMPLEQ)/main.jl -p "tolerance=1e-11;order=100;maxiter=100;v_a=1;minlrho=$$lrho;nlrho=1" -M simpleq energy_rho >> $@-tmp ;\
+ done
+ paste $@-tmp 2020-10-15+energy.dat > $@
+ rm $@-tmp
+mueq1.dat:
+ for lrho in -6 -4 -3 -2.3010299956639813 -2 -1.8239087409443189 -1.6989700043360187 -1.3010299956639813 -1 0 1 1.6989700043360187; do\
+ julia $(SIMPLEQ)/main.jl -p "tolerance=1e-11;order=100;maxiter=100;v_a=1;minlrho=$$lrho;nlrho=1" -M mueq energy_rho >> $@-tmp ;\
+ done
+ paste $@-tmp 2020-10-15+energy.dat > $@
+ rm $@-tmp
+
+
+install: $(PDFS)
+ cp $^ $(INSTALLDIR)/
+
+clean-aux:
+ rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
+ rm -f $(addsuffix .aux, $(PROJECTNAME))
+ rm -f $(addsuffix .log, $(PROJECTNAME))
+
+clean-dat:
+ rm -f $(DATS)
+ rm -f Abar.dat
+
+clean-tex:
+ rm -f $(PDFS)
+
+clean: clean-aux clean-tex
diff --git a/figs/energy_relative_error.fig/bigeq1.dat b/figs/energy_relative_error.fig/bigeq1.dat
new file mode 100644
index 0000000..ccab7cc
--- /dev/null
+++ b/figs/energy_relative_error.fig/bigeq1.dat
@@ -0,0 +1,13 @@
+1e-6 7.933345669932108e-06 1.702110068633820e-12 1e-6 7.90e-6 7.91e-6
+1e-4 8.349333596092273e-04 1.112527655484278e-13 1e-4 8.344e-4 8.345e-4 0.017 0.018 8.355e-4 8.356e-4 0.015 0.016
+1e-3 9.146316224214820e-03 2.077370190827862e-12 1e-3 9.1338e-3 9.1339e-3 0.048 0.049 9.142e-3 9.143e-3 0.045 0.046
+5e-3 5.063551121414726e-02 8.398015568932763e-12 5e-3 5.0613e-2 5.0614e-2 0.080 0.081 5.0648e-2 5.0649e-2 0.070 0.071
+1e-2 1.061166635113156e-01 5.017547334745932e-15 1e-2 1.06107e-1 1.06108e-1 0.087 0.088 1.0627e-1 1.0628e-1 0.073 0.074
+1.5e-2 1.633545333357252e-01 2.212732350724188e-15 1.5e-2 1.6337e-1 1.6338e-1 0.090 0.091 1.6342e-1 1.6343e-1 0.088 0.089
+2e-2 2.216006926766234e-01 4.554921896809050e-15 2e-2 2.2162e-1 2.2163e-1 0.093 0.094 2.2169e-1 2.2170e-1 0.090 0.091
+5e-2 5.805467427278628e-01 3.333614946727334e-15 5e-2 5.8044e-1 5.8045e-1 0.091 0.092 5.8046e-1 5.8047e-1 0.091 0.092
+1e-1 1.191928562890024e+00 2.450444415773880e-15 1e-1 1.1918e+0 1.1919e+0 0.088 0.089 1.1919e+0 1.1920e+0 0.085 0.086
+1.0 1.243051761633436e+01 3.791416729017179e-12 1e-0 1.2430e+1 1.2431e+1 0.058 0.059 1.245e+1 1.246e+1 0.05 0.06
+1e1 1.254476421597336e+02 2.677873467933947e-13 1e+1 1.2544e+2 1.2545e+2 0.031 0.032
+5e1 nan nan 5e+1 6.2803e+2 6.2804e+2 0.02 0.03
+
diff --git a/figs/energy_relative_error.fig/cmp_energy1.gnuplot b/figs/energy_relative_error.fig/cmp_energy1.gnuplot
new file mode 100644
index 0000000..c2b1afd
--- /dev/null
+++ b/figs/energy_relative_error.fig/cmp_energy1.gnuplot
@@ -0,0 +1,37 @@
+set title "$v(\\mathbf x)=e^{-|\\mathbf x|}$"
+set ylabel "energy relative error" offset +3
+set xlabel "$\\rho$"
+
+set xtics 1e-6, 100, 100
+set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
+unset mxtics
+set xrange [1e-6:1e2]
+
+set ytics 1e-5, 10, 1e-1
+set ytics add ("$10^{-5}$" 1e-5, "$10^{-4}$" 1e-4, "$10^{-3}$" 1e-3, "$10^{-2}$" 1e-2, "$10^{-1}$" 1e-1)
+set yrange [1e-5:1e-1]
+
+# default output canvas size: 12.5cm x 8.75cm
+set term lua tikz size 8,6 standalone
+
+unset key
+
+
+# set linestyle
+set style line 1 linetype rgbcolor "#4169E1" linewidth 2
+set style line 2 linetype rgbcolor "#DC143C" linewidth 2
+set style line 3 linetype rgbcolor "#32CD32" linewidth 2
+set style line 4 linetype rgbcolor "#4B0082" linewidth 2
+set style line 5 linetype rgbcolor "#DAA520" linewidth 2
+
+set pointsize 1
+
+max(x,y)=(x>y ? x : y)
+
+set logscale xy
+
+plot \
+ "simpleq1.dat" using 1:($3<1e-5 ? max(abs($2-$5)/$5,abs($2-$6)/$6) : 1/0) every ::1 with lp ls 1 ,\
+ "bigeq1.dat" using 1:($3<1e-5 ? max(abs($2-$5)/$5,abs($2-$6)/$6) : 1/0) every ::1 with lp ls 4 ,\
+ "2020-10-15+energy.dat" using 1:(max(max(abs($6-$2)/$2,abs($6-$3)/$3),max(abs($7-$2)/$2,abs($7-$3)/$3))) every ::1 with lp ls 2
+
diff --git a/figs/energy_relative_error.fig/mueq1.dat b/figs/energy_relative_error.fig/mueq1.dat
new file mode 100644
index 0000000..6680ed3
--- /dev/null
+++ b/figs/energy_relative_error.fig/mueq1.dat
@@ -0,0 +1,13 @@
+ 1.000000000000000e-06 7.934286257047026e-06 3.235575358862730e-14 1e-6 7.90e-6 7.91e-6
+ 1.000000000000000e-04 8.377752370367244e-04 3.201262322514326e-14 1e-4 8.344e-4 8.345e-4 0.017 0.018 8.355e-4 8.356e-4 0.015 0.016
+ 1.000000000000000e-03 9.227516472314044e-03 5.926882960328585e-12 1e-3 9.1338e-3 9.1339e-3 0.048 0.049 9.142e-3 9.143e-3 0.045 0.046
+ 4.999999999999999e-03 5.110614907540598e-02 1.069784208067329e-14 5e-3 5.0613e-2 5.0614e-2 0.080 0.081 5.0648e-2 5.0649e-2 0.070 0.071
+ 1.000000000000000e-02 1.069255387977072e-01 1.572627931557844e-14 1e-2 1.06107e-1 1.06108e-1 0.087 0.088 1.0627e-1 1.0628e-1 0.073 0.074
+ 1.500000000000000e-02 1.643925118385577e-01 2.494863657028789e-14 1.5e-2 1.6337e-1 1.6338e-1 0.090 0.091 1.6342e-1 1.6343e-1 0.088 0.089
+ 2.000000000000000e-02 2.228019992986293e-01 1.448625206361080e-14 2e-2 2.2162e-1 2.2163e-1 0.093 0.094 2.2169e-1 2.2170e-1 0.090 0.091
+ 5.000000000000000e-02 5.821600306289594e-01 2.429130993459621e-14 5e-2 5.8044e-1 5.8045e-1 0.091 0.092 5.8046e-1 5.8047e-1 0.091 0.092
+ 1.000000000000000e-01 1.193625606724180e+00 9.289627651289616e-14 1e-1 1.1918e+0 1.1919e+0 0.088 0.089 1.1919e+0 1.1920e+0 0.085 0.086
+ 1.000000000000000e+00 1.243092200540192e+01 3.956891809887943e-12 1e-0 1.2430e+1 1.2431e+1 0.058 0.059 1.245e+1 1.246e+1 0.05 0.06
+ 1.000000000000000e+01 1.254464788052155e+02 1.136222076562384e-14 1e+1 1.2544e+2 1.2545e+2 0.031 0.032
+ 4.999999999999999e+01 6.280479158155819e+02 4.085062088184932e-15 5e+1 6.2803e+2 6.2804e+2 0.02 0.03
+
diff --git a/figs/energy_relative_error.fig/simpleq1.dat b/figs/energy_relative_error.fig/simpleq1.dat
new file mode 100644
index 0000000..0a9946c
--- /dev/null
+++ b/figs/energy_relative_error.fig/simpleq1.dat
@@ -0,0 +1,13 @@
+ 1.000000000000000e-06 7.934458317554763e-06 1.386474846476237e-14 1e-6 7.90e-6 7.91e-6
+ 1.000000000000000e-04 8.392960650211980e-04 4.992772141681879e-14 1e-4 8.344e-4 8.345e-4 0.017 0.018 8.355e-4 8.356e-4 0.015 0.016
+ 1.000000000000000e-03 9.335885024335848e-03 9.476600935679491e-12 1e-3 9.1338e-3 9.1339e-3 0.048 0.049 9.142e-3 9.143e-3 0.045 0.046
+ 4.999999999999999e-03 5.253256909806337e-02 2.904203225041083e-14 5e-3 5.0613e-2 5.0614e-2 0.080 0.081 5.0648e-2 5.0649e-2 0.070 0.071
+ 1.000000000000000e-02 1.105504827786785e-01 3.695967730970493e-14 1e-2 1.06107e-1 1.06108e-1 0.087 0.088 1.0627e-1 1.0628e-1 0.073 0.074
+ 1.500000000000000e-02 1.702888708763101e-01 2.044316796426063e-14 1.5e-2 1.6337e-1 1.6338e-1 0.090 0.091 1.6342e-1 1.6343e-1 0.088 0.089
+ 2.000000000000000e-02 2.309054339970606e-01 1.984104771646051e-14 2e-2 2.2162e-1 2.2163e-1 0.093 0.094 2.2169e-1 2.2170e-1 0.090 0.091
+ 5.000000000000000e-02 6.014214358150070e-01 1.045253287889987e-14 5e-2 5.8044e-1 5.8045e-1 0.091 0.092 5.8046e-1 5.8047e-1 0.091 0.092
+ 1.000000000000000e-01 1.226000446492040e+00 3.630493925691041e-15 1e-1 1.1918e+0 1.1919e+0 0.088 0.089 1.1919e+0 1.1920e+0 0.085 0.086
+ 1.000000000000000e+00 1.253044200066420e+01 1.230015282154983e-13 1e-0 1.2430e+1 1.2431e+1 0.058 0.059 1.245e+1 1.246e+1 0.05 0.06
+ 1.000000000000000e+01 1.256270224354739e+02 6.086298884133178e-15 1e+1 1.2544e+2 1.2545e+2 0.031 0.032
+ 4.999999999999999e+01 6.282817756461592e+02 1.506690451671515e-15 5e+1 6.2803e+2 6.2804e+2 0.02 0.03
+
diff --git a/libs/ian-presentation.cls b/libs/ian-presentation.cls
new file mode 100644
index 0000000..91bd487
--- /dev/null
+++ b/libs/ian-presentation.cls
@@ -0,0 +1,187 @@
+%%
+%% Ian's presentation class
+%%
+
+%% TeX format
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+
+%% class name
+\ProvidesClass{ian-presentation}[2017/09/29]
+
+\def\ian@defaultoptions{
+ \pagestyle{plain}
+ \RequirePackage{color}
+ \RequirePackage{amssymb}
+}
+
+%% paper dimensions
+\setlength\paperheight{240pt}
+\setlength\paperwidth{427pt}
+
+%% fonts
+\input{size11.clo}
+\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
+\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
+\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
+\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
+\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
+
+%% text dimensions
+\textheight=208pt
+\textwidth=370pt
+\hoffset=-1in
+\voffset=-1in
+\oddsidemargin=24pt
+\evensidemargin=24pt
+\topmargin=8pt
+\headheight=0pt
+\headsep=0pt
+\marginparsep=0pt
+\marginparwidth=0pt
+\footskip=16pt
+
+
+%% remove default skips
+\parindent=0pt
+\parskip=0pt
+\baselineskip=0pt
+
+%% something is wrong with \thepage, redefine it
+\gdef\thepage{\the\c@page}
+
+%% correct vertical alignment at the end of a document
+\AtEndDocument{
+ % save total slide count
+ \immediate\write\@auxout{\noexpand\gdef\noexpand\slidecount{\thepage}}
+ \vfill
+ \eject
+}
+
+
+%% footer
+\def\ps@plain{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{\tiny\hfill\thepage/\safe\slidecount\hfill}
+ \def\@evenfoot{\@oddfoot}
+}
+\def\ps@empty{
+ \def\@oddhead{}
+ \def\@evenhead{\@oddhead}
+ \def\@oddfoot{}
+ \def\@evenfoot{\@oddfoot}
+}
+
+
+%% title of slide
+\def\title#1{
+ \hfil{\bf\large #1}\par
+ \hfil\vrule width0.75\textwidth height0.3pt\par
+ \vskip5pt
+}
+
+
+%% hyperlinks
+% hyperlinkcounter
+\newcounter{lncount}
+% hyperref anchor
+\def\hrefanchor{%
+\stepcounter{lncount}%
+\hypertarget{ln.\thelncount}{}%
+}
+
+%% define a command and write it to aux file
+\def\outdef#1#2{%
+ % define command%
+ \expandafter\xdef\csname #1\endcsname{#2}%
+ % hyperlink number%
+ \expandafter\xdef\csname #1@hl\endcsname{\thelncount}%
+ % write command to aux%
+ \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}%
+ \immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}%
+}
+
+%% can call commands even when they are not defined
+\def\safe#1{%
+ \ifdefined#1%
+ #1%
+ \else%
+ {\color{red}\bf?}%
+ \fi%
+}
+
+
+%% itemize
+\newlength\itemizeskip
+% left margin for items
+\setlength\itemizeskip{20pt}
+\newlength\itemizeseparator
+% space between the item symbol and the text
+\setlength\itemizeseparator{5pt}
+% penalty preceding an itemize
+\def\itemizepenalty{0}
+% counter counting the itemize level
+\newcounter{itemizecount}
+
+% item symbol
+\def\itemizept#1{
+ \ifnum#1=1
+ \textbullet
+ \else
+ $\scriptstyle\blacktriangleright$
+ \fi
+}
+
+\newlength\current@itemizeskip
+\setlength\current@itemizeskip{0pt}
+\def\itemize{
+ \par\penalty\itemizepenalty\medskip\penalty\itemizepenalty
+ \addtocounter{itemizecount}{1}
+ \addtolength\current@itemizeskip{\itemizeskip}
+ \leftskip\current@itemizeskip
+}
+\def\enditemize{
+ \addtocounter{itemizecount}{-1}
+ \addtolength\current@itemizeskip{-\itemizeskip}
+ \par\leftskip\current@itemizeskip
+ \medskip
+}
+\newlength\itempt@total
+\def\item{
+ \settowidth\itempt@total{\itemizept\theitemizecount}
+ \addtolength\itempt@total{\itemizeseparator}
+ \par
+ \medskip
+ \hskip-\itempt@total\itemizept\theitemizecount\hskip\itemizeseparator
+}
+
+%% enumerate
+\newcounter{enumerate@count}
+\def\enumerate{
+ \setcounter{enumerate@count}0
+ \let\olditem\item
+ \let\olditemizept\itemizept
+ \def\item{
+ % counter
+ \stepcounter{enumerate@count}
+ % set header
+ \def\itemizept{\theenumerate@count.}
+ % hyperref anchor
+ \hrefanchor
+ % define tag (for \label)
+ \xdef\tag{\theenumerate@count}
+ \olditem
+ }
+ \itemize
+}
+\def\endenumerate{
+ \enditemize
+ \let\item\olditem
+ \let\itemizept\olditemizept
+}
+
+
+%% end
+\ian@defaultoptions
+
+\endinput