deletions | additions       diff --git a/Nonlinear irreversible kinetics.tex b/Nonlinear irreversible kinetics.tex  index 32a8649..f420fab 100644  --- a/Nonlinear irreversible kinetics.tex  +++ b/Nonlinear irreversible kinetics.tex 
...
\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 $k_i(t)\to\omega$. This is straightforward to show for the case of an $i^{th}$-order reaction, 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 t.  \end{equation} 
...
\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 to be  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 $k(t)$. Theinequality between the  statistical length and divergence can also be derived for these irreversible decay reactions. The time dependent time-dependent  rate coefficient is \begin{equation}  k_i(t)  \equiv \frac{d}{dt}\frac{1}{S(t)^{i-1}} \frac{d}{dt}\frac{1}{S_i(t)^{i-1}}  = (i-1)\omega C_A(0)^{i-1} C_i(0)^{i-1}  \end{equation}  As shown in equation 4, the statistical length $\mathcal{L}$ $\mathcal{L}_i$  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 \mathcal{L}_i(\Delta  t)^2 = \left[\int_{t_i}^{t_f}(n-1)\omega([A_0]^{n-1})dt\right]^2 \left[\int_{t_i}^{t_f}(i-1)\omega(C_i^{i-1})dt\right]^2  \end{equation}  Following length, and  theFisher divergence is the integral of the cumulative time dependent rate coefficient squared over a period of time $\Delta{t}$. The Fisher  divergence is \begin{equation}  \frac{\mathcal{J}_n(\Delta \frac{\mathcal{J}_i(\Delta  t)}{\Delta t} = \int_{t_i}^{t_f}{(n-1)^2\omega^2}([A_0]^{n-1})^{2} dt \int_{t_i}^{t_f}{(i-1)^2\omega^2}\left(C_i^{i-1}\right)^{2} dt.  \end{equation}  Both the length squared and the divergence are $(n-1)^2\omega^2([A_0]^{n-1})^2\Delta $(i-1)^2\omega^2([A_0]^{n-1})^2\Delta  t^2$: the bound holds when there is no static or dynamic disorder, and a single rate coefficient is sufficient for the to characterize  irreversible decay process. 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 $\mathcal{L}$.