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roadmap for Applications chapter

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src/hops_tweak.tex
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∫_0^t\dd{s}A(s)\qty(α(s)-α_0(s)) =\\ ∫_0^t\dd{s}A(s)\qty(α(s)-α_0(s)) =\\
∑_{m,n}\frac{A_nG_m}{C_n+W_m}\qty(1-\eu^{-(C_n+W_m)t}) - ∑_{m,n}\frac{A_nU_m}{C_n+Q_m}\qty(1-\eu^{-(C_n+Q_m)t}). ∑_{m,n}\frac{A_nG_m}{C_n+W_m}\qty(1-\eu^{-(C_n+W_m)t}) - ∑_{m,n}\frac{A_nU_m}{C_n+Q_m}\qty(1-\eu^{-(C_n+Q_m)t}).
\end{multline} \end{multline}
\subsection{Technical Notes on the Code}
\label{sec:code}

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src/num_results.tex
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\chapter{Numerical Results} \chapter{Applications}
\label{chap:numres} \label{chap:numres}
\begin{itemize} After developing the tools for obtaining information about bath
\item all the theory developed in \cref{chap:flow,chap:analytsol} related observables in \cref{chap:flow} and the means for their
wants some applications verification in \cref{chap:analytsol}, we are now in a position to
\item roadmap: first verify numerics and systematics using apply those results.
\cref{chap:analytsol} in \cref{sec:hopsvsanalyt} and then using
energy conservation \cref{sec:prec_sim}. Although not the focus: The roadmap is the following. Using \cref{chap:analytsol} we will
also some discussion of observations and the role of stronger verify the results of \cref{chap:flow} in \cref{sec:hopsvsanalyt}. A
coupling and non-markovianity striking phenomenon will be noticed and explained in a brief detour
\item especially short time behaviour in \cref{sec:pure_deph} \cref{sec:pure_deph}.
\item after systematics interlude about basic quantum-thermo
\cref{sec:basic_thermo} In the generic case where no analytic solution we nevertheless are
\item finally short applications: single bath \cref{sec:singlemod} and able to obtain consistent results as is demonstrated in
a simple otto cycle \cref{sec:otto} \cref{sec:prec_sim}.
\end{itemize}
In this chapter some application of the results described in These results will strengthen the confidence in
\cref{chap:flow,chap:analytsol} are presented. In the method so that we can turn to more complicated applications.
\cref{sec:hopsvsanalyt}, we begin by considering the bath energy for First a brief overview of interesting features of quantum
an analytically solvable model and contrasting the analytical results thermodynamics is given in \cref{sec:basic_thermo}. Subsequently we
with the results obtained by hops. will turn to two applications to demonstrate these features in
\cref{sec:singlemod,sec:otto}.
An overview and explanation of the codes used in this chapter can be found
in \cref{sec:code}.
\section{Some Remarks on the Methods} \section{Some Remarks on the Methods}
\label{sec:meth} \label{sec:meth}