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| -rw-r--r-- | Jauslin_UCDavis_2020.tex | 304 | ||||
| -rw-r--r-- | Makefile | 49 | ||||
| -rw-r--r-- | README | 33 | ||||
| -rw-r--r-- | figs/1d.png | bin | 0 -> 70742 bytes | |||
| -rw-r--r-- | figs/plots.fig/Makefile | 28 | ||||
| -rw-r--r-- | figs/plots.fig/erho.dat | 104 | ||||
| -rw-r--r-- | figs/plots.fig/erho.gnuplot | 37 | ||||
| -rw-r--r-- | figs/plots.fig/erho_bare.gnuplot | 31 | ||||
| -rw-r--r-- | figs/plots.fig/erho_holzmann.gnuplot | 34 | ||||
| -rw-r--r-- | figs/plots.fig/holzmann_2019-12-25.dat | 11 | ||||
| -rw-r--r-- | libs/ian-presentation.cls | 187 |
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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} 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) $(SYNCTEXS) + +clean: clean-aux clean-tex clean-libs clean-figs @@ -0,0 +1,33 @@ +This directory contains the source files to typeset the presentation, and +generate the figures. This can be accomplished by running + make + +This document uses a custom class file, located in the 'libs' directory, which +defines a number of commands. + + +* Dependencies: + + pdflatex + TeXlive packages: + amsfonts + array + graphics + hyperref + latex + pgf + standalone + GNU make + gnuplot + +* Files: + + Jauslin_UCDavis_2020.tex: + main LaTeX file + + libs: + custom LaTeX class file + + figs: + source code for the figures + diff --git a/figs/1d.png b/figs/1d.png Binary files differnew file mode 100644 index 0000000..e64f960 --- /dev/null +++ b/figs/1d.png diff --git a/figs/plots.fig/Makefile b/figs/plots.fig/Makefile new file mode 100644 index 0000000..39d1a0b --- /dev/null +++ b/figs/plots.fig/Makefile @@ -0,0 +1,28 @@ +PROJECTNAME=erho_bare erho erho_holzmann + +PDFS=$(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 |
