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Jonathan Nichols added Kinetic Model With Dynamic Disorder.tex
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As demonstrated already in first order kinetics, the Kohlrausch-Williams-Watts (KWW) stretched exponential function can measure the amount of dynamic disorder in a system.[cite] The KWW model has been widely applicable to many fields of science such as the discharge of capacitors[cite]. The KWW model adds an exponential term, $\beta$,to the survival function model. The term $\omega$ represents a time independent rate coefficient and $\beta$ represents dynamic disorder where $0<\beta\leq1$. Applying KWW to second order kinetics, the survival probability is represented as $S(t)=\frac{1}{1+(\omega t[A_0])^{\beta}}$. The time dependent rate coefficient is $k(t)=\frac{\beta([A_0]\omega t)^{\beta}}{t}$. This time dependent rate coefficient gives statistical length and divergences of
\begin{equation}
\mathcal{L}_K_W_W(\Delta{t})=[A_0]^{\beta}\omega^{\beta}t^{\beta}\bigg|_{t_i}^{t_f}
\end{equation}
\begin{equation}
\mathcal{J}_K_W_W(\Delta{t})=\frac{\Delta t[A_0]^{2\beta}\beta^{2}\omega^{2\beta}t^{2\beta-1}}{2\beta-1}\bigg|_{t_i}^{t_f}
\end{equation}
And when $\beta=\frac{1}{2}$,
\begin{equation}
\mathcal{J}_K_W_W(\Delta{t})=\Delta t [A_0]\beta^2\omega^{2\beta} \ln(t)\bigg|_{t_i}^{t_f}
\end{equation}
These results closely match the results of first order irreversible decay, where $\mathcal{L}_K_W_W(\Delta{t})=\omega^{\beta}t^{\beta}\bigg|_{t_i}^{t_f}$, $\mathcal{J}_K_W_W(\Delta{t})=\frac{\Delta t\beta^{2}\omega^{2\beta}t^{2\beta-1}}{2\beta-1}\bigg|_{t_i}^{t_f}$, and when $\beta=\frac{1}{2}$, $\mathcal{J}_K_W_W(\Delta{t})=\Delta t \omega^{2\beta} \ln(t)\bigg|_{t_i}^{t_f}$.
\section{Notes on graphs yet to be done}
Figure 1(a) shows how the inverse of the survival function multiplied by $[A_0]$ versus time depends on the value of $\beta$. There is only a linear dependence on time when $\beta=1$, which corresponds to exponential kinetics and a time independent rate coefficient $k(t)\rightarrow\omega $. For all $\beta$ values higher than 0 and less than 1, the second order s
