Distinguishing disorder from order in irreversible decay processes

Jonathan W. Nichols, Shane W. Flynn, William E. Fatherley, Jason R. Green
Department of Chemistry, University of Massachusetts Boston


Fluctuating rate coefficients are necessary when modeling disordered kinetic processes with mass-action rate equations. However, measuring the fluctuations of rate coefficients is a challenge, particularly for nonlinear rate equations. Here we present a measure of the total disorder in irreversible decay \(i\,A\to \textrm{products}\), \(i=1,2,3,\ldots n\) governed by (non)linear rate equations – the inequality between the time-integrated square of the rate coefficient (multiplied by the time interval of interest) and the square of the time-integrated rate coefficient. We apply the inequality to empirical models for statically and dynamically disordered kinetics with \(i\geq 2\). These models serve to demonstrate that the inequality quantifies the cumulative variations in a rate coefficient, and the equality is a bound only satisfied when the rate coefficients are constant in time.


Rates are a way to infer the mechanism of kinetic processes, such as chemical reactions. They typically obey the empirical mass-action rate laws when the reaction system is homogeneous, with uniform concentration (s) throughout. Deviations from traditional rate laws are possible when the system is heterogeneous and there are fluctuations in structure, energetics, or concentrations. When traditional kinetic descriptions break down [insert citation], the process is statically and/or dynamically disordered [insert Zwanzig citation], and it is necessary to replace the rate constant in the rate equation with a time-dependent rate coefficient. Measuring the variation of time-dependent rate coefficients is a means of quantifying the fidelity of a rate coefficient and rate law.

In our previous work a theory was developed for analyzing first-order irreversible decay kinetics through an inequality[insert citation]. The usefulness 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 higher order 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].

Static and dynamic disorder lead to an observed rate coefficient that depends on time \(k(t)\). The main result here, and in Reference[cite], is an inequality \[\mathcal{L}(\Delta{t})^2 \leq \mathcal{J}(\Delta{t})\] between the statistical length (squared) \[\mathcal{L}(\Delta{t})^2 \equiv \left[\int_{t_i}^{t_f}k(t)dt\right]^2\] and the divergence \[\frac{\mathcal{J}(\Delta{t})}{\Delta{t}} \equiv \int_{t_i}^{t_f}k(t)^{2}dt\] over a time interval \(\Delta t = t_f - t_i\). Both \(\mathcal{L}\) and \(\mathcal{J}\) are functions of a possibly time-dependent rate coefficient, originally motivated by an adapted form of the Fisher information[cite]. Reference 1 showed that 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 result to irreversible decay processes with “order” higher than one. We show \(\mathcal{J}-\mathcal{L}^2=0\) is a condition for a constant rate coefficient for any \(i\). Accomplishing this end requires reformulating the definition of the time-dependent rate coefficient.

In this work we extend the application of this inequality to measure disorder in irreversible decay kinetics with nonlinear rate laws (i.e., kinetics with total “order” greater thane unity). We illustrate this framework with proof-of-principle analyses for second-order kinetics for irreversible decay phenomena. We also connect this theory to previous work on first-order kinetics showing how the model simplifies in a consistent manner when working with first order models.