deletions | additions       diff --git a/Connection Betweent the three Formulations.tex b/Connection Betweent the three Formulations.tex  index 695e1e7..31c9653 100644  --- a/Connection Betweent the three Formulations.tex  +++ b/Connection Betweent the three Formulations.tex 
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\begin{equation}  k_{rel}=k_{abs}-\frac{d}{dt}[\ln\sum_i S_i]  \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}  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_i(t)\left(\frac{d}{dt}\ln\gamma_i(t)\right)^2  \end{equation}  \begin{equation}  \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)-\left)  \end{equation}  Therefore we can really write the Fisher Information as follows.  \begin{equation}  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}\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)\left(-\omega+\frac{\omega e^{-\omega t}+\omega'e^{-\omega't}}{e^{-\omega 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 $0$ which is consistent with the variance interpretation.  \begin{equation}  I(t) = \gamma(t)\left(-u+\frac{u e^{-u t}+ue^{-ut}}{e^{-u 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}\left[\ln\sum_i S_i\right]   \end{equation}