deletions | additions       diff --git a/Nonlinear irreversible kinetics.tex b/Nonlinear irreversible kinetics.tex  index 700d28c..96318af 100644  --- a/Nonlinear irreversible kinetics.tex  +++ b/Nonlinear irreversible kinetics.tex 
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\end{cases}  \end{equation}  \section{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 $k_i(t)\to\omega$. 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. 
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The inequality between the statistical length and divergence can also be derived for these irreversible decay reactions. The time dependent rate coefficient is  \begin{equation}  k_n(t) k_i(t)  \equiv \frac{d}{dt}\frac{1}{S(t)^{n-1}} \frac{d}{dt}\frac{1}{S(t)^{i-1}}  = (n-1)\omega C_A(0)^{n-1} (i-1)\omega C_A(0)^{i-1}  \end{equation}  As shown in equation 4, the statistical length $\mathcal{L}$  is the integral of the cumulative time dependent rate coefficient over a period of time $\Delta{t}$. The statistical length is \begin{equation}  \mathcal{L}_n(\Delta t)^2 = \left[\int_{t_i}^{t_f}(n-1)\omega([A_0]^{n-1})dt\right]^2   \end{equation}