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Order and disorder in irreversible decay processes

Dynamical disorder motivates fluctuating rate coefficients in phenomenological, mass-action rate equations. The reaction order in these rate equations is the fixed exponent controlling the dependence of the rate on the number of species. Here we clarify the relationship between these notions of (dis)order in irreversible decay, $n\,A\to B$, $n=1,2,3,\ldots$, by extending a theoretical measure of fluctuations in the rate coefficient. The measure, $\mathcal{J}_n-\mathcal{L}_n^2\geq 0$, is the magnitude of the inequality between $\mathcal{J}_n$, the time-integrated square of the rate coefficient multiplied by the time interval of interest, and $\mathcal{L}_n^2$, the square of the time-integrated rate coefficient. Applying the inequality to empirical models for non-exponential relaxation, we demonstrate that it quantifies the cumulative deviation in a rate coefficient from a constant, and so the degree of dynamical disorder. The equality is a bound satisfied by traditional kinetics where a single rate constant is sufficient. For these models, we show how increasing the reaction order can increase or decrease dynamical disorder and how, in either case, the inequality $\mathcal{J}_n-\mathcal{L}_n^2\geq 0$ can indicate the ability to deduce the reaction order in dynamically disordered kinetics.