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Jonathan Nichols added Kinetic Model With Static Disorder.tex
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The difference between the Fisher Divergence and statistical length can also serve as a way of measuring the amount of static disorder in a second order irreversible decay processes. The same model used to show that this is true in first order irreversible decay can be examined again in second order. This model examines two experimentally indistinguishable states A and A', which inter-convert between each other[cite Plonka]. These two states eventually decay irreversibly into B with two time independent rate coefficients $\omega$ and $\omega$' describing the rates of reaction. It can be shown that when a distribution of rate coefficients is necessary ($\omega\neq\omega'$), the decay is bi-exponential in second order. When the rate coefficients $\omega$ and $\omega '$ are equal, the decay follows the second order survival function shown in equation 2. In a general case of any order, the survival function is
\begin{equation}
S(t) = \frac{N_A (t)+N_A' (t)}{N_A (0)+N_A' (0)}
\end{equation}
Assuming the states A and A' which decay into B follow second order irreversible decay, the survival function becomes
\begin{equation}
S(t) = \frac{\frac{N_A(0)}{1+\omega tN_A(0)}+\frac{N_A'(0)}{1+\omega'tN_A'(0)}}{N_A (0)+N_A' (0)}
\end{equation}
To simplify and show that the decay is bi-exponential when $\omega\neq\omega'$, the initial concentrations of A and A' are set to be equal. By setting $N_A (0)=N_A' (0)$, the survival function is
\begin{equation}
S(t)=\left(\frac{1}{2+2\omega tN_A(0)}+\frac{1}{2+2\omega'tN_A(0)}\right)
\end{equation}
The survival function in equation 26 shows a bi-exponential decay when $\omega\neq\omega'$, which shows that static disorder is present since a distribution of rate coefficients is necessary to describe the system.
When $\omega=\omega'$, the survival function in equation 29 turns into the standard survival function for a second order reaction shown in equation 2.
\begin{equation}
S(t)=\left(\frac{1}{2+2\omega tN_A(0)}+\frac{1}{2+2\omega tN_A(0)}\right)=\frac{2}{2+2\omega t[A_0]}=\frac{1}{1+\omega t[A_0]}
\end{equation}
\section{Plonka Plots $\frac{1}{S(t)}$ vs time. with changing w/w'}
The inequality between $\mathcal{L}^2(\Delta{t})$ and $\mathcal{J}(\Delta{t})$ measures not only the amount of static and dynamic order in a first order irreversible decay process, but also in second order, mixed second order, and $n^{th}$ order irreversible decay reactions. All of these inequalities rely on two functions of the Fisher information, the statistical length and divergence. The inequality between the statistical length and Fisher divergence measures the amount of static and dynamic disorder in the rate coefficient. A single rate coefficient is sufficient only when $\mathcal{L}^2(\Delta{t})$=$\mathcal{J}(\Delta{t})$, and is when classical kinetics truly works. In the future this work may be useful at looking at other kinetic theories such as Michaelis-Menten kinetics.
