deletions | additions       diff --git a/Kinetic model with dynamic disorder.tex b/Kinetic model with dynamic disorder.tex  index 38d62e4..ad9bda2 100644  --- a/Kinetic model with dynamic disorder.tex  +++ b/Kinetic model with dynamic disorder.tex 
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To study the effect of dynamical disorder on higher-order kinetics, we adapt the Kohlrausch-Williams-Watts (KWW) model for stretched exponential decay. In first-order kinetics, stretched exponential decay involves a two-parameter survival function $\exp(-\omega t)^\beta$ and associated time-depedent rate coefficient, $k_{1,KWW}(t) = \beta(\omega t)^{\beta}/t$. The parameter $\omega$ is a characteristic rate or inverse time scale and the parameter $\beta$ is a measure of the degree of stretching. We showed in Reference~[citation] that stretching the exponential with $\beta$ increases the degree of dynamic disorder nonlinearly.  In general we can write the $n^{th}$ order rate law for the KWW model using the new formulations for the nonlinear differential KWW and survival functions.  \begin{equation}  S_{n,KWW}(t)=(\frac{1}{1+(n-1)(\omega t C_A(0)^{n-1})^\beta})^{\frac{1}{n-1}}=\frac{1}{1+(n-1)(z^\beta t^\beta})^{\frac{1}{n-1}}  \end{equation}  Assuming an overall second-order process with a time depedent rate coefficient the survival function is $S_2(t) = 1/\left(1+(\omega tC_2(0))^{\beta}\right)$. The time-depedent rate coefficient characterizing the decay is $k_{2,KWW}(t) = \beta(C_A(0)\omega t)^{\beta}/t$, from the time-derivative of the inverse of the survival function. This definition of the rate coefficient depends on the initial concentration of the reactants which is consistent with units of rate constants in traditional kinetics. Integrating the time-depedent rate coefficient gives the statistical length  \begin{equation}  \mathcal{L}_{KWW}(\Delta{t}) = \left(C_A(0)\omega t\right)^{\beta}\big|_{t_i}^{t_f}