this is for holding javascript data
Jason R. Green deleted file $n^{th}$ Order Irreversible Decay.tex
over 9 years ago
Commit id: 6e19a6abbdc764fb92c212196feca403bd868517
deletions | additions
diff --git a/$n^{th}$ Order Irreversible Decay.tex b/$n^{th}$ Order Irreversible Decay.tex
deleted file mode 100644
index 78fa971..0000000
--- a/$n^{th}$ Order Irreversible Decay.tex
+++ /dev/null
...
We now shift our attention to $n^{th}$ order reactions. These $n^{th}$ order reactions are of the form of one reactant turning into product. The inequality between the statistical distance and Fisher divergence can also be derived for these irreversible decay reactions.
The time dependent rate coefficient is
\begin{equation}
k(t)\equiv(\frac{d\frac{1}{(S(t)^{n-1)}}}{dt})\equiv(n-1)\omega[A_0]^{n-1}
\end{equation}
As shown in equation 4, the statistical length 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}(\Delta t)^2=\left[\int_{t_i}^{t_f}(n-1)\omega([A_0]^{n-1})dt\right]^2
\end{equation}
Following length, the Fisher 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}( \Delta t)}{\Delta t}=\int_{t_i}^{t_f}{(n-1)^2\omega^2}([A_0]^{n-1})^{2} dt
\end{equation}
The inequality becomes
\begin{equation}
(n-1)^2\omega^2([A_0]^{n-1})^2\Delta t^2-[(n-1)\omega[A_0]^{n-1}\Delta t]^2\geq0
\end{equation}
Again, we see that when the bound $(n-1)^2\omega^2([A_0]^{n-1})^{2}\Delta t^2=[(n-1)\omega[A_0]^{n-1}\Delta t]^2$ holds, there is no static or dynamic disorder, and a single rate coefficient can be defined for the irreversible decay process.
