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\subsection{Bound for constant rate coefficients}  These forms of $k(t)$ satisfy the bound $\mathcal{J}-\mathcal{L}^2 = 0$ in the absence of disorder, when $k_i(t)\to\omega$. $k_i(t)\to\omega_i$.  This is straightforward to show for the case of an $i^{th}$-order reaction ($i\geq 2$), 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 \frac{1}{C_i(0)^{i-1}}+(i-1)\omega_i  t. \end{equation}  and associated survival function  \begin{equation}  S_i(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega \sqrt[i-1]{\frac{1}{1+(i-1)\omega_i  tC_i(0)^{i-1}}}. \end{equation}  In traditional kinetics, the rate coefficient of irreversible decay is assumed to be constant, in which case $k(t)\to\omega$, $k(t)\to\omega_i$,  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 $k(t)$. The statistical length and divergence can also be derived for these irreversible decay reactions. The time-dependent rate coefficient is  \begin{equation}  k_i(t)  \equiv \frac{d}{dt}\frac{1}{S_i(t)^{i-1}}  = (i-1)\omega (i-1)\omega_i  C_i(0)^{i-1} \end{equation}  The statistical length $\mathcal{L}_i$ is the integral of the cumulative time-dependent rate coefficient over a period of time $\Delta{t}$, and the divergence is the cumulative square of the rate coefficient, multiplied by the time interval. %The statistical length is   %\begin{equation}  % \mathcal{L}_i(\Delta t)^2 = \left[\int_{t_i}^{t_f}(i-1)\omega \left[\int_{t_i}^{t_f}(i-1)\omega_i  C_i^{i-1}(0)\,dt\right]^2 %\end{equation}  %and the divergence is  %\begin{equation}  % \frac{\mathcal{J}_i(\Delta t)}{\Delta t} = \int_{t_i}^{t_f}{(i-1)^2\omega^2}\left(C_i^{i-1}(0)\right)^{2} \int_{t_i}^{t_f}{(i-1)^2\omega_i^2}\left(C_i^{i-1}(0)\right)^{2}  dt. %\end{equation}  For the equations governing traditional kinetics, both the statistical length squared and the divergence are $(i-1)^2\omega^2\left(C_i^{i-1}(0)\right)^2\Delta $(i-1)^2\omega_i^2\left(C_i^{i-1}(0)\right)^2\Delta  t^2$: the bound holds when there is no static or dynamic disorder, and a single rate coefficient is sufficient to characterize irreversible decay. The nonlinearity of the rate law leads to solutions that depend on concentration. This concentration dependence is also present in both $\mathcal{J}$ and $\mathcal{L}$.