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\section{Dynamically Disordered System Analysis} \section{Second-order kinetic model with dynamical disorder}  The effects of dynamic disorder on higher ordered kinetics can be investigated through the Kohlrausch-Williams-Watts (KWW) model. This expression takes the expotential decay predicted by classical kinetics and stretches the curve through a time depedent rate coefficient $\omega$ and the coopertavity of decay $\beta$, making it a conventient expression for interpreting dynamic disorder. The use of this model is intended to be a proof of principle, however any system with a time depedent rate coefficient may be analyzed.   Assuming an overall second order process with a time depedent rate coefficient the survival function is written as $S(t)=\frac{1}{1+(\omega t[A_0])^{\beta}}$.Thorugh the time derivitive of the inverse of the survival function the time depedent rate coefficient characterising the deacy is expressed as $k(t)=\frac{\beta([A_0]\omega t)^{\beta}}{t}$. We see this definition of the rate coefficient maintains a depedence on the initial concentration of the reactants which is a unique result when compared to first order kinetics, and propagates thorugh the remainder of the calculations.Next the time depedent rate coefficient is integrated to determine the statistical length.   \begin{equation}  \mathcal{L}_{KWW}(\Delta{t})=[A_0\omega t]^{\beta}\bigg|_{t_i}^{t_f} \end{equation}  To undertand understand  the effect of fluctuations on the rate coefficient we then integrate over the square, determining the divergence. \begin{equation}  \mathcal{J}_{KWW}(\Delta{t})=\frac{\Delta \mathcal{J}_{KWW}(\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}  This form shows the $\frac{1}{2}$ $\beta$ valuse must be evaluated seperately which yields the final result for comparison.   And when $\beta=\frac{1}{2}$,   \begin{equation}  \mathcal{J}_{KWW}(\Delta{t})=\Delta t [A_0]\beta^2\omega^{2\beta} \ln(t)\bigg|_{t_i}^{t_f} \end{equation}