deletions | additions       diff --git a/Kinetic model with dynamic disorder.tex b/Kinetic model with dynamic disorder.tex  index 2c33f48..38d62e4 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.  Assuming an overall second order second-order  process with a time depedent rate coefficient the survival function iswritten as  $S_2(t) = 1/\left(1+(\omega tC_A(0))^{\beta}\right)$. Through the time-derivative of the inverse of the survival function the time depedent tC_2(0))^{\beta}\right)$. The time-depedent  rate coefficient characterizing the decay isexpressed as  $k_{2,KWW}(t) = \beta(C_A(0)\omega t)^{\beta}/t$. We see this t)^{\beta}/t$, from the time-derivative of the inverse of the survival function. This  definition of the rate coefficient maintains a dependence depends  on the initial concentration of the reactants which is a unique result when comapared to the first-order result. 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}  \end{equation}  which also has a concentration dependence to cancel the concentration dimensions of the $\omega$. dependence.  To understand the effect of fluctuations on the rate coefficient we then integrate over the square, square $k(t)$,  determining the divergence. \begin{equation}  \mathcal{J}_{KWW}(\Delta{t}) =  \frac{\Delta tC_A(0)^{2\beta}\beta^{2}\omega^{2\beta}t^{2\beta-1}}{2\beta-1}\bigg|_{t_i}^{t_f} 
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\begin{equation}  \mathcal{J}_{KWW}(\Delta{t}) = \Delta t C_A(0)\beta^2\omega^{2\beta} \ln(t)\big|_{t_i}^{t_f}  \end{equation}  We use of this model as a proof of principle, proof-of-principle,  however any model a time-depedent rate coefficient can be subject to this analysis.