deletions | additions       diff --git a/Second-order decay.tex b/Second-order decay.tex  index 2dc5f74..99e3b6e 100644  --- a/Second-order decay.tex  +++ b/Second-order decay.tex 
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\section{Second-order decay, $A+A\to\textrm{products}$}  These forms of $k(t)$ satisfy the bound $\mathcal{J}-\mathcal{L}^2 = 0$ in the absence of disorder, when $k_i(t)\to\omega$. This is straightforward to show for the case of an $i^{th}$-order reaction, with the traditional integrated rate law  \begin{equation}  \frac{1}{C_i(t)^{i-1}} = \frac{1}{C_i(0)^{i-1}}+(i-1)\omega t.  \end{equation}  and associated survival function  \begin{equation}  S_i(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega tC_i(0)^{i-1}}}.  \end{equation}  In traditional kinetics, the rate coefficient of irreversible decay is assumed constant, in which case $k(t)\to\omega$, but this will not be the case when the kinetics are statically or dynamically disordered. In these cases, we will use the above definitions of $k(t)$.  Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to to second order irreversible decay. The integrated second-order rate law gives the survival function  %\begin{equation}  % C_A(t) = \frac{C_A(0)}{1+\omega tC_A(0)}