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\section{Rates,
Three I(t), Three GFE} three $I(t)$, three GFEs}
\subsection{Rates}
To
utilize use the framework we have previosly developed we need to generalize the theory to handle multiple S(t) functions all contributing to the model. To do this we need to make some useful definitions and see how the connect to both the formulation analyzing chemical kinetics by Ross, and to the work proposed by Fisher himself. First we can define three different forms of rates. The first we will call the relative rate, the all $\gamma$ formulation.
\begin{equation}
k_{rel}\equiv
\frac{d}{dt}[ln\gamma_i]=\frac{d}{dt}[lnS_i-ln\Sigma S_i] \frac{d}{dt}[\ln\gamma_i]=\frac{d}{dt}\left[\ln S_i-\ln\sum_i S_i\right]
\end{equation}
Next we define an absolute rate which is the mixed formulation.
\begin{equation}
k_{abs}\equiv
\frac{d}{dt}[lnS_i(t)] \frac{d}{dt}[\ln S_i(t)]
\end{equation}
Comparing these two definitions one immediately notices that the $k_{abs}$ is actually within the relative formulation. This explains the termonology choice the relative case is understood to this absolute formulation, they are depedent upon eachother.
\begin{equation}
k_{rel}=k_{abs}-\frac{d}{dt}[ln\Sigma k_{rel}=k_{abs}-\frac{d}{dt}[\ln\sum_i S_i]
\end{equation}
\subsection{Fisher Information}
Using these definitions we can then formulate the associated Fisher Information for each of these rates. The relative Fisher Information $I_r$, and the absolute Fisher Information $I_a$.
\begin{equation}
I_r=\Sigma I_r=\sum_i \gamma_i
k_{rel}=\Sigma k_{rel}=\sum_i \gamma_i
\frac{d}{dt}[ln\gamma_i] \frac{d}{dt}[\ln\gamma_i]
\end{equation}
This is where the "all $\gamma$" formulation derives its name.
Likewise we define the
"mixed fomrulation" ``mixed formulation'' which is truly the absolute formulation.
\begin{equation}
I_a=\Sigma I_a=\sum_i \gamma_i
k_{abs}=\Sigma \gamma_i\frac{d}{dt}[lnS_i(t)] k_{abs}=\sum_i\gamma_i\frac{d}{dt}[\ln S_i(t)]
\end{equation}
With these definitions in place the picture would be completed with one final formulation the all S formulation $k_S$ and an associated $I_s$.
\begin{equation}
I_s=\Sigma S_ik_{abs}=\Sigma S_i\frac{d}{dt}[lnS_i(t)] I_s=\sum_i S_ik_{abs}=\sum_i S_i\frac{d}{dt}[\ln S_i(t)]
\end{equation}
%We have three different Fisher Information definitions we should think about graphing each of them and see how the effect of disorder effects them, ultimately seeing the limit of no disorder
\subsection{What is the Fisher
Information} information}
Lets look at $I_r(t)$ in detail, specifically we can evaluate the effect of disorder on the information.
\begin{equation}
I_r(t)\equiv\gamma(t)(\frac{d}{dt}ln\gamma(t))^2 I_r(t) \equiv \gamma_i(t)\left(\frac{d}{dt}\ln\gamma_i(t)\right)^2
\end{equation}
\begin{equation}
\frac{d}{dt}ln\gamma(t)=\frac{d}{dt}ln\frac{S_i}{\Sigma S_i}=\frac{d}{dt}lnS_i -\frac{d}{dt}ln\Sigma \frac{d}{dt}\ln\gamma_i(t)=\frac{d}{dt}\ln\frac{S_i}{\sum_i S_i}=\frac{d}{dt}\ln S_i -\frac{d}{dt}\ln\sum_i S_i
\end{equation}
This form looks just like a variance between the rate coefficient of one specific trajectory and the entire population of trajectories.
\begin{equation}
(k_{r}(t)-) (k_{r}(t)-\left)
\end{equation}
Therefore we can really write the Fisher Information as follows.
\begin{equation}
I_r(t)=\gamma_i(t)(k_{r}(t)-)^2 I_r(t) = \gamma_i(t)(k_{r}(t)-\left)^2
\end{equation}
We can now substitute the Plonka survival functions in to see a specific form of the Fisher Information in this all $\gamma$ formulation.
\begin{equation}
\frac{d}{dt}lne^{-wt} -\frac{d}{dt}ln\Sigma e^{-w't}=\frac{d}{dt}[-\omega t]-\frac{1}{\Sigma \frac{d}{dt}\ln e^{-wt} -\frac{d}{dt}\ln\Sigma e^{-w't}
=\frac{d}{dt}[-\omega t]-\frac{1}{\sum_i S_i}\frac{d}{dt}[e^{-\omega t}+e^{-\omega't}]
\end{equation}
\begin{equation}
I_r(t)=\gamma(t)(-\omega+\frac{\omega I_r(t) = \gamma(t)\left(-\omega+\frac{\omega e^{-\omega t}+\omega'e^{-\omega't}}{e^{-\omega
t}+e^{-\omega't}})^2 t}+e^{-\omega't}}\right)^2
\end{equation}
If we now look at the case where $\omega=\omega'\equiv u$ we can show the Fisher Information becomes
0 $0$ which is consistent with the variance interpretation.
\begin{equation}
I(t)=\gamma(t)(-u+\frac{u I(t) = \gamma(t)\left(-u+\frac{u e^{-u t}+ue^{-ut}}{e^{-u
t}+e^{-ut}})^2=\gamma(t)(0)^2=0 t}+e^{-ut}}\right)^2 = \gamma(t)(0)^2=0
\end{equation}
And with the Fisher Information equal to zero the inequality becomes trivial as both the length and divergence become 0.
From this analysis we can see that the $k_r$ becomes 0 when there is no disorder because the Fisher Information becomes 0. From this simplification we then see that
\begin{equation}
k_{abs}=\frac{d}{dt}[ln\Sigma S_i]
\end{equation}
\subsection{Fisher
Equations} equations}
Finally we can now define three different Fisher Equations one for each of the previously defined Fisher Informations.
\begin{equation}
GFE_r\equiv \frac{d}{dt}[\Sigma
\gamma_ik_{rel}] \gamma_i k_{rel}]
\end{equation}
\begin{equation}
GFE_a\equiv
\frac{d}{dt}[\Sigma\gamma_ik{abs}] \frac{d}{dt}[\Sigma\gamma_i k{abs}]
\end{equation}
\begin{equation}
GFE_s\equiv \frac{d}{dt}[\Sigma
S_ik_{abs}] S_i k_{abs}]
\end{equation}
