deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  new file mode 100644  index 0000000..8a1e6c0  --- /dev/null  +++ b/Introduction.tex 
...
\section{Introduction}  Rate coefficients define a paramater capable of almost universally characterizing any kinetic process. There are many instances where the traditional kinetics model does not sufficiently describe a population decaying over time [insert citation]. These cases are more accurately described by either a distribution of rate coefficients, or a time dependent rate coefficient, defined as static or dynamic disorder respectively [insert citation]. Therefore the ability to quantify these types of potentially inclusive disorders is essential information for understanding your system. In our previous work a theory was developed for analyzing first order irreversible decay kinetics through an inequality[insert citation]. The convenience of this inequality is through its ability to quantify disorder, with the unique property of becoming an equality only when the system is disorder free, and therefore described by chemical kinetics in its classical formulation. The next problem that should be addressed is that of highter oreder kinetics, what if the physical systems one wishes to understand are more complex kinetic schemes, they would require a modified theoretical framework for analysis, but should and can be addressed. To motivate this type of development systems such as...... are all known to proceed through higher ordered kinetics, and all of these systems possess unique and interesting applications, therefore a more complete kinetics description of them should be pursued[insert citations].   For this work we propose a generalization of our previous first order irreversible decay kinetics to higher orders, with complete framework analyzing any $n^{th}$ order system with this description.  In this work we propose a method for studying these more complex cases in chemical kinetics propopsinmg theory to analyze disorder in $n^{th}$ order kinetics and provide detailed proof of principle analyses for second order kinetics and mixed order kinetics for irreversible decay phenominium. We then connect this theory to previously accepted work on first order kinetics showing how the model simplifies in a consistent manner when working with first order models.  An inequality between two important quantities known as the statistical length and Fisher divergence is able to quantitatively measure the amount of static and dynamic disorder of a rate coefficient over a period of time, and is able to determine when traditional kinetics is truly valid[insert citation]. When the inequality is minimized, the statistical description of rate coefficients is also minimized, which can help one determine the best data set to use during a kinetic analysis. This inequality measures the temporal and spatial variation in the rate coefficient. When there is no static or dynamic disorder, this inequality is reduced to an equality and it is the only time where traditional kinetics is truly valid. This inequality has been interpreted as a measure of how constant a rate coefficient can be. It captures the fluctuations associated with the rate coefficient for first order irreversible decay processes. In this work we expand on this idea providing more generality and utility to the theory by gernealizing to higher order kinetics.   The most important piece in extending current theory 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 are both functions of the time dependent rate coefficient.  \begin{equation}  \mathcal{L}(\Delta{t})^2 = \left[\int_{t_i}^{t_f}k(t)dt\right]^2  \end{equation}  \begin{equation}  \frac{\mathcal{J}(\Delta{t})}{\Delta{t}} = \int_{t_i}^{t_f}k(t)^{2}dt  \end{equation}  The 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  \end{equation}  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 is only dependent on one rate coefficient, $\omega$, and the mechanism of the reaction. The rate law of a first order reaction of A irreversibly decaying into B is   \begin{equation}  \frac{-dC_A(t)}{dt}=\omega C_A  \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 citation]  \begin{equation}  \frac{-d\ln S(t)}{dt} = k(t)  \end{equation}  In order to determine the time dependent rate coefficient in higher order reactions, it is useful to use the survival function, but 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 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})  \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 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 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)  \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)  \end{equation}  And from that definition of k(t), again, we can arrive at the inequality.