As presented at UC Davis on 2020-02-06HEADv1.0master

11 files changed, 818 insertions, 0 deletions

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}
diff --git a/Makefile b/Makefile
new file mode 100644
index 0000000..4f30ca1
--- /dev/null
+++ b/Makefile
@@ -0,0 +1,49 @@
+PROJECTNAME=$(basename $(wildcard *.tex))
+LIBS=$(notdir $(wildcard libs/*))
+FIGS=$(notdir $(wildcard figs/*.fig))
+PNGS=$(notdir $(wildcard figs/*.png))
+
+PDFS=$(addsuffix .pdf, $(PROJECTNAME))
+SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME))
+
+all: $(PROJECTNAME)
+
+$(PROJECTNAME): $(LIBS) $(FIGS) $(PNGS)
+	pdflatex -file-line-error $@.tex
+	pdflatex -synctex=1 $@.tex
+
+$(SYNCTEXS): $(LIBS) $(FIGS) $(PNGS)
+	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
+
+$(PNGS):
+	ln -fs figs/$@ ./
+
+
+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) $(SYNCTEXS)
+
+clean: clean-aux clean-tex clean-libs clean-figs
diff --git a/README b/README
new file mode 100644
index 0000000..add2525
--- /dev/null
+++ b/README
@@ -0,0 +1,33 @@
+This directory contains the source files to typeset the presentation, and
+generate the figures. This can be accomplished by running
+  make
+
+This document uses a custom class file, located in the 'libs' directory, which
+defines a number of commands.
+
+
+* Dependencies:
+
+  pdflatex
+  TeXlive packages:
+    amsfonts
+    array
+    graphics
+    hyperref
+    latex
+    pgf
+    standalone
+  GNU make
+  gnuplot
+
+* Files:
+
+  Jauslin_UCDavis_2020.tex:
+    main LaTeX file
+
+  libs:
+    custom LaTeX class file
+
+  figs:
+    source code for the figures
+
diff --git a/figs/1d.png b/figs/1d.png
new file mode 100644
index 0000000..e64f960
--- /dev/null
+++ b/figs/1d.png
diff --git a/figs/plots.fig/Makefile b/figs/plots.fig/Makefile
new file mode 100644
index 0000000..39d1a0b
--- /dev/null
+++ b/figs/plots.fig/Makefile
@@ -0,0 +1,28 @@
+PROJECTNAME=erho_bare erho erho_holzmann
+
+PDFS=$(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/plots.fig/erho.dat b/figs/plots.fig/erho.dat
new file mode 100644
index 0000000..837439f
--- /dev/null
+++ b/figs/plots.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
+ 6.91830971e-05  5.74914264e-04
+ 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
+ 4.36515832e-04  3.88357954e-03
+ 5.24807460e-04  4.71465030e-03
+ 6.30957344e-04  5.72685888e-03
+ 7.58577575e-04  6.96035420e-03
+ 9.12010839e-04  8.46425370e-03
+ 1.09647820e-03  1.02986208e-02
+ 1.31825674e-03  1.25368572e-02
+ 1.58489319e-03  1.52685874e-02
+ 1.90546072e-03  1.86031268e-02
+ 2.29086765e-03  2.26736380e-02
+ 2.75422870e-03  2.76420977e-02
+ 3.31131121e-03  3.37052167e-02
+ 3.98107171e-03  4.11014850e-02
+ 4.78630092e-03  5.01195435e-02
+ 5.75439937e-03  6.11081308e-02
+ 6.91830971e-03  7.44879092e-02
+ 8.31763771e-03  9.07655413e-02
+ 1.00000000e-02  1.10550483e-01
+ 1.20226443e-02  1.34575061e-01
+ 1.44543977e-02  1.63718539e-01
+ 1.73780083e-02  1.99036030e-01
+ 2.08929613e-02  2.41793284e-01
+ 2.51188643e-02  2.93508614e-01
+ 3.01995172e-02  3.56003446e-01
+ 3.63078055e-02  4.31463277e-01
+ 4.36515832e-02  5.22511186e-01
+ 5.24807460e-02  6.32296416e-01
+ 6.30957344e-02  7.64601077e-01
+ 7.58577575e-02  9.23968602e-01
+ 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
+ 2.29086765e-01  2.84519724e+00
+ 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/plots.fig/erho.gnuplot b/figs/plots.fig/erho.gnuplot
new file mode 100644
index 0000000..f2f04bf
--- /dev/null
+++ b/figs/plots.fig/erho.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=1.2544
+set arrow from 0.01,4*pi to 100,4*pi linewidth 2 dashtype "." linecolor "#DC143C" nohead
+set label at 0.00001,4*pi "$\\displaystyle\\frac12\\int dx\\ v(x)$" textcolor "#DC143C"
+set arrow from 0.000001,2*pi*a to 0.01,2*pi*a linewidth 2 dashtype "." linecolor "#DC143C" nohead
+set label at 0.01,2*pi*a "$2\\pi a$" textcolor "#DC143C"
+
+plot "erho.dat" using 1:($2/$1) with lines linestyle 1
diff --git a/figs/plots.fig/erho_bare.gnuplot b/figs/plots.fig/erho_bare.gnuplot
new file mode 100644
index 0000000..381bf8f
--- /dev/null
+++ b/figs/plots.fig/erho_bare.gnuplot
@@ -0,0 +1,31 @@
+set ylabel "$\\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
+
+plot "erho.dat" using 1:($2/$1) with lines linestyle 1
diff --git a/figs/plots.fig/erho_holzmann.gnuplot b/figs/plots.fig/erho_holzmann.gnuplot
new file mode 100644
index 0000000..94af8f5
--- /dev/null
+++ b/figs/plots.fig/erho_holzmann.gnuplot
@@ -0,0 +1,34 @@
+set ylabel "$\\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 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
+
+plot "erho.dat" using 1:(0.95*$2/$1):(1.05*$2/$1) with filledcurves linetype rgbcolor "#DDDDDD", \
+ "erho.dat" using 1:($2/$1) with lines linestyle 1, \
+ "holzmann_2019-12-25.dat" using 1:($2/$1) with points linestyle 2
+
diff --git a/figs/plots.fig/holzmann_2019-12-25.dat b/figs/plots.fig/holzmann_2019-12-25.dat
new file mode 100644
index 0000000..96689cb
--- /dev/null
+++ b/figs/plots.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/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