deletions | additions       diff --git a/Second Order Decay1.tex b/Second Order Decay1.tex  index 8fc26ae..234f86f 100644  --- a/Second Order Decay1.tex  +++ b/Second Order Decay1.tex 
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\begin{equation}  \frac{dS(t)^{-1}}{dt}=k(t)   \end{equation}  Now consider a population of some species A decaying into B following the reaction $A+A\rightarrow B$. The second order integrated rate law is [cite]  \begin{equation}  [A_t]=\frac{[A_0]}{1+\omega t[A_0]}  \end{equation}  The second order survival function is  \begin{equation}  S(t)=\frac{[A_t]}{[A_0]}=\frac{1}{1+\omega t[A_0]}  \end{equation}  The time dependent rate coefficient for a second order reaction is  \begin{equation}  k(t)=\left(\frac{dS(t)^{-1}}{dt}\right)=\frac{d}{dt}\left[1+\omega t[A_0]\right]=\omega[A_0] 
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\begin{equation}  \mathcal{L}(\Delta{t})=\int_{t_i}^{t_f}k(t)dt  \end{equation}  Which is equal to $\frac{1}{S(t)}\big|_{S_(t_f)}^{S_(t_i)}$, which measures the cumulative rate coefficient, same as in first order irreversible decay. As seen in first order, the statistical length is also dependent on the time interval, with the statistical length being infinite in an infinite time interval.To form an inequality, another quantity called the Fisher divergence is calculated, which is defined as [cite]  \begin{equation}  \frac{\mathcal{J}(\Delta{t})}{\Delta{t}}=\int_{t_i}^{t_f}k(t)^{2}dt  \end{equation}  It has been shown that in first order kinetics the inequality between the statistical length squared and Fisher divergence determines when a rate coefficient is constant, which is only when the inequality turns into an equality.[cite] A similar inequality is found in second order kinetics.The inequality between statistical length squared and Fisher divergence is  \begin{equation}  \mathcal{L}(\Delta{t})^2\leq \mathcal{J}(\Delta{t})  \end{equation}  Which also means that  \begin{equation}  \mathcal{J}(\Delta{t})-\mathcal{L}(\Delta{t})^2\geq 0  \end{equation}  \begin{equation}  \Delta{t}\int_{t_i}^{t_f}k(t)^{2}dt-\left[{\int_{t_i}}^{t_f}k(t)dt\right]^{2}dt\geq0  \end{equation}  Putting in the time dependent rate coefficient for a second order irreversible decay, the inequality becomes  \begin{equation}  \omega^2[A_0]^2\Delta{t}^2-\left(\omega[A_0]\Delta{t}\right)^2\geq0