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\section{Theory}
The most important piece in extending current theory Central to
higher order irreversible decay is determining the time dependent rate coefficient, which is dependent on the
order of reaction. The statistical length and divergence kinetic theory in Reference[cite] are
both two functions of
the time dependent a possibly time-dependent rate
coefficient. coefficient: the statistical length (squared)
\begin{equation}
\mathcal{L}(\Delta{t})^2 = \left[\int_{t_i}^{t_f}k(t)dt\right]^2
\end{equation}
and the Fisher divergence
\begin{equation}
\frac{\mathcal{J}(\Delta{t})}{\Delta{t}} =
\int_{t_i}^{t_f}k(t)^{2}dt \int_{t_i}^{t_f}k(t)^{2}dt.
\end{equation}
The They generally satisfy an inequality
between the statistical length and divergence can be shown in two different ways
\begin{equation}
\mathcal{L}(\Delta{t})^2\leq
\mathcal{J}(\Delta{t})
\end{equation}
\begin{equation}
\mathcal{J}(\Delta{t})-\mathcal{L}(\Delta{t})^2\geq 0 \mathcal{J}(\Delta{t}).
\end{equation}
To define the length and divergence, and arrive at this inequality, we adapted the Fisher information[cite]. We showed how the difference $\mathcal{J}(\Delta t)-\mathcal{L}(\Delta t)^2$ is a measure of the variation in the rate coefficient, due to static or dynamic disorder, for decay kinetics with a first-order rate law. The lower bound holds only when the rate coefficient is constant in first-order irreversible decay. Here we extend this inequality to disorder in irreversible decay proecesses with order higher than one. We show $\mathcal{J}-\mathcal{L}^2$ is a condition for constant rate coefficients. Accomplishing this end requires reformulating the definition of the time-dependent rate coefficient.
The
second form of the inequality is the most useful, as the difference in $\mathcal{J}(\Delta t)$ and $\mathcal{L}(\Delta t)^2$ measures the variation in the rate coefficient. When the difference between these two quantities is zero, the rate coefficient is constant.
This inequality is derived from the first order rate law and survival
function. In traditional kinetics, irreversible decay function is
only dependent on one rate coefficient, $\omega$, and the mechanism a measure of the
reaction. The rate law concentration of
species at a
first order reaction of A irreversibly decaying into B is time $t$ compared to the initial concentration
\begin{equation}
\frac{-dC_A(t)}{dt}=\omega C_A S(t) = \frac{C_A(t)}{C_A(0)}.
\end{equation}
The time dependent rate coefficient, k(t), is determined by integrating the rate law of the reaction and forming a survival function from the integrated rate law. The survival function is simply a measure of a the cocentration of species at some time compared to its initital concentration.
\begin{equation}
S(t) = \frac{C_A(t)}{C_A(0)}
\end{equation} From the survival function, the time dependent rate coefficient is determined by taking various time derivatives of the survival function, depending on the total order of reaction. For first order irreversible decay reactions, the time dependent rate coefficient
is the negative time derivative of the natural log of the survival function[insert is[insert citation]
\begin{equation}
\frac{-d\ln S(t)}{dt} = k(t)
\end{equation}
In traditional kinetics, irreversible decay is only dependent on one rate coefficient, $\omega$. The rate law of a first-order reaction in which $A$ irreversibly decays to B is
\begin{equation}
\frac{-dS(t)}{dt} = \omega S(t)
\end{equation}
The time-dependent rate coefficient, $k(t)$, is determined by integrating the rate law of the reaction and forming a survival function from the integrated rate law.
In order to determine
the time dependent rate coefficient $k(t)$ in higher order reactions,
it is useful to we use the survival function, but
it is not necessary. The survival function of any reaction involving any number of the same molecule can be derived using the integrated rate laws of reactions. For example, the $n^{th}$ order integrated rate law is
\begin{equation}
\frac{1}{C_A(t)]^{n-1}}=\frac{1}{C_A(0)]^{n-1}}+(n-1)\omega \frac{1}{[c_A(t)]^{n-1}} = \frac{1}{[c_A(0)]^{n-1}}+(n-1)\omega t
\end{equation}
From the integrated rate law, we get the survival function
\begin{equation}
S(t)=\frac{C_A(t)}{C_A(0)}=\sqrt[n-1]{\frac{1}{1+(n-1)\omega tC_A(0)^{n-1}}}
\end{equation}
From the survival function, the time dependent rate coefficient is
\begin{equation}
k(t)=(\frac{d\frac{1}{(S(t)^{n-1)}}}{dt}) k(t) = (\frac{d\frac{1}{(S(t)^{n-1)}}}{dt})
\end{equation}
Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to to second order irreversible decay. The integrated rate law in second order is
\begin{equation}
C_A(t)=\frac{C_A(0)}{1+\omega C_A(t) = \frac{C_A(0)}{1+\omega tC_A(0)}
\end{equation}
The second order survival function is
\begin{equation}
S(t)=\frac{C_A(t)}{C_A(0)}=\frac{1}{1+\omega S(t) = \frac{C_A(t)}{C_A(0)}=\frac{1}{1+\omega tC_A(0)}
\end{equation}
From this survival function, taking the inverse of the survival function and then the time derivative yields the time dependent rate coefficient.
\begin{equation}
k(t)=\left(\frac{dS(t)^{-1}}{dt}\right) k(t) = \left(\frac{dS(t)^{-1}}{dt}\right)
\end{equation}
When it comes to mixed second order reactions, the rate law does not allow a survival function to be obtained for the process because the rate law only allows the survival function of one reactant to be looked at. The rate law of a mixed second order reaction is
\begin{equation}
\int\frac{dx}{(C_A(0)]-x)(C_B(0)-x)}=k(t)
\end{equation}
Taking the time derivative of the left hand side gives
\begin{equation}
\int\frac{\frac{dx}{dt}}{{(C_A(0)-x)(C_B(0)-x)}}dt=k(t) \int\frac{\frac{dx}{dt}}{{(C_A(0)-x)(C_B(0)-x)}}dt = k(t)
\end{equation}
And from that definition of k(t), again, we can arrive at the inequality.
