%$Id: notes.tex,v 1.102 2022/04/05 19:25:44 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
\usepackage{graphicx,natbib,bm,url,color}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{}
\author{}
\date{\today,~ $ $Revision: 1.102 $ $}
\begin{document}
%\maketitle

\section{New runs}

%\begin{enumerate}\itemsep-0.2mm
%\item
%\end{enumerate}

All runs have $\mu_{50}=10^6$, $\tilde{\lambda}^{1/2}=7\times10^{11}$,
and $\nu=\eta=D=5\times10^{-8}$.
Thus, we have
\begin{equation}
v_\lambda=\mu_{50}/\lambda^{1/2}=1.4\times10^{-6},
\end{equation}
and
\begin{equation}
v_\mu=\mu_{50}\eta=5\times10^{-2},
\end{equation}
corresponding to regime~II.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_D512_1e5_1e6_49e22_5em8a4}
\end{center}\caption[]{
\url{pcomp_D512_1e5_1e6_49e22_5em8a4}
}\label{pcomp_D512_1e5_1e6_49e22_5em8a4}\end{figure}

%error bars, E* value, 2 assumptions for alpha
%do the same intermediate vlam
%do plot for spectra, together with extrapolation
%/home/brandenb/data/axel/GW/CME
\begin{table}[htb]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{cccl}
$k_1$ & $\EEM^{\max}$ & $\EEGW^{\rm sat}$ & $\EEGW^*$ & color \\
\hline
$5\times10^3$ & $4.7\times10^{-13}$ & $1.9\times10^{-37}$ & $1.0\times10^{38}$ & black \\
       $10^4$ & $4.9\times10^{-13}$ & $1.7\times10^{-38}$ & & red \\
$2\times10^4$ & $4.9\times10^{-13}$ & $3.4\times10^{-39}$ & & orange \\
       $10^5$ & $5.0\times10^{-13}$ & $1.0\times10^{-41}$ & & blue \\
\label{Tsummary}\end{tabular}}\end{table}

The difference between the four runs is the minimum wavenumber, $k_1$,
available in the simulations (ideally, it should be $\to0$).
\Fig{pcomp_D512_1e5_1e6_49e22_5em8a4} shows the time traces for those runs.
They are all reasonably similar, except for $\hrms$ and $\EEGW^{\rm sat}$,
which drop significantly for large $k_1$.

% data/axel/GW/CME/D512_1e5_1e6_49e22_5em8a4_k5e3

%   k1      mu0      lam      eta      vmu      vlam     etak2    EEM        EEGW     run
%5.0e+03  1.0e+06  4.9e+23  5.0e-08  5.0e-02  1.4e-06  2.5e-04  4.695e-13  1.928e-37  D512_1e5_1e6_49e22_5em8a4_k5e3
%1.0e+04  1.0e+06  4.9e+23  5.0e-08  5.0e-02  1.4e-06  5.0e-04  4.918e-13  1.735e-38  D512_1e5_1e6_49e22_5em8a4_k1e4
%2.0e+04  1.0e+06  4.9e+23  5.0e-08  5.0e-02  1.4e-06  1.0e-03  4.931e-13  3.407e-39  D512_1e5_1e6_49e22_5em8a4_k2e4
%1.0e+05  1.0e+06  4.9e+23  5.0e-08  5.0e-02  1.4e-06  5.0e-03  4.952e-13  1.036e-41  D512_1e5_1e6_49e22_5em8a4

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{EEGW_vs_EEKM}
\end{center}\caption[]{
\url{EEGW_vs_EEKM}
}\label{EEGW_vs_EEKM}\end{figure}

The chiral chemical potential drops rapidly when the linear phase of the
chiral plasma instability is over ($t\approx1.001$).
It then levels off near $10^5$, which corresponds to the wavenumber
where the magnetic energy spectra peak; see below.

\Fig{EEGW_vs_EEKM} compares $\EEM^{\max}$ and $\EEGW^{\rm sat}$ (black
filled symbols) with those of earlier runs (gray filled symbols and
gray lines).
The efficiency of GW production is evidently very low.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pspec_sat_D512_1e5_1e6_49e22_5em8a4_k5e3}
\end{center}\caption[]{
\url{pspec_sat_D512_1e5_1e6_49e22_5em8a4_k5e3}
}\label{pspec_sat_D512_1e5_1e6_49e22_5em8a4_k5e3}\end{figure}

\Fig{pspec_sat_D512_1e5_1e6_49e22_5em8a4_k5e3} compares magnetic and GW
energy spectra.
As found in our 2021 paper, the GW energy spectra don't capture the
lowest wavenumber and therefore the GW energies are not converged.

\section{Next?}

It would be useful to have a model of how $\mu_5$ builds up.

\end{document}
