some updates on the figures and add gibbs bound to otto

src/num_results.tex

@@ -1107,10 +1107,10 @@ simplifies the situation.
 To further answer whether a substantial fraction of the maximal
 ergotropy can be extracted from a system we have numerically optimized
 the coupling strengths of the model \cref{eq:one_qubit_model_driven},
-so that after three modulation periods \(τ_{m} = \frac{2 π}{Δ}\) the
+so that over ten modulation periods \(τ_{m} = \frac{2 π}{Δ}\) the
 maximal absolute interaction energy is close to a give value (here
-\(\ev{H_{\inter}}\approx 0.4\) which constitutes quite a strong coupling) for various cutoff
-frequencies \(ω_{c}\).
+\(\ev{H_{\inter}}\approx 0.4\) which constitutes quite a strong
+coupling) for various cutoff frequencies \(ω_{c}\).
 \begin{figure}[h]
   \centering
   \includegraphics{figs/one_bath_mod/omega_interactions}
@@ -1121,6 +1121,10 @@ frequencies \(ω_{c}\).
     the parameters in \cref{tab:plus_omega}.}
 \end{figure}
 
+The resulting couplings can be found in
+\cref{fig:omega_couplings_and_energies}. They correspond roughly to
+demanding \(α_{β}(0)=1.4\), where \(α_{β}\) is the finite temperature
+bath correlation function.
 \Cref{fig:omega_couplings_and_energies} also shows that for the
 simulations with small \(ω_{c}\) the positive parts of the interaction
 energy are especially large.
@@ -1152,14 +1156,14 @@ minimal total energy is achieved earlier and is of greater
 magnitude. Further, we see that a non-trivial amount of energy is
 being extracted relative to ergotropy bound \cref{eq:ergo_mod_model}.
 
-\begin{wrapfigure}[12]{o}{.5\textwidth}
+\begin{figure}
   \centering
   \includegraphics{figs/one_bath_mod/omega_energies_and_powers}
   \caption{\label{fig:omegas_energies_and_powers} One shot power
     (blue) and maximal extracted energy (orange) as a function of
     the bath memory. The grey vertical line marks one
     modulation period.}
-\end{wrapfigure}
+\end{figure}
 In \cref{fig:omegas_energies_and_powers} we can see that longer
 bath\fixme{looks like ``phase transition'' but really isn't} memories,
 defined as the time point at which \(α(τ_{\bath}) = α(0)/10\), lead to
@@ -1177,9 +1181,6 @@ is due to the requirement that power and extracted energy have to be
 finite, and the stroboscopic time view induced by the requirement that
 the interaction energy should be zero.
 
-Note also, that for the two longest bath memories, the relative gain
-in extracted energy is much smaller than the relative gain in power.
-
 In conclusion we can say that in the cases studied here we can extract
 a finite amount of energy from the system as soon as the bath memory
 is somewhat longer than the modulation period. This statement depends
@@ -1423,6 +1424,16 @@ steady state, the system and interaction related quantities are
 constant in the stroboscopic view (the dots in \cref{fig:ottoenergy}),
 while the total energy and the bath energies depend linearly on time.
 
+The Gibbs like inequality \cref{eq:secondlaw_cyclic} derived in
+\cref{sec:operational_thermo} can also be verified in this case.
+We find
+\begin{equation}
+  \label{eq:secondlaw_otto_actual}
+  ∑_iβ_i ΔE_{\bath^i}^\cyc = 0.1096\pm 0.0008 ≥ 0,
+\end{equation}
+which does satisfy the inequality to over 100 standard
+deviations. Minimizing this quantity would maximize the efficiency.
+
 A worthwhile task for future work would be to verify the results
 summarized in \cite{Binder2018} for the Otto cycle. Especially the
 optimization for optimal power which leads to the