Nonparametric analysis of nonexponential and multidimensional kinetics. I. Quantifying rate dispersion, rate heterogeneity, and exchange dynamics

The quantification of nonexponential (dispersed) kinetics has relied on empirical functions, which yield parameters that are neither unique nor easily related to the underlying mechanism. Multidimensional kinetics provide more information on dispersed processes, but a good approach to their analysis is even less clear than for standard, one-dimensional kinetics. This paper is the first in a series that analyzes kinetic data in one or many dimensions with a scheme that is nonparametric: it quantifies nonexponential decays without relying on a specific functional form. The quantities obtained are directly related to properties of the mechanism causing the rate dispersion. Log-moments of decays, which parallel the standard moments of distributions (mean, standard deviation, etc.), are introduced for both one- and multi-dimensional decays. Kinetic spectra are defined to visualize the data. The utility of this approach is demonstrated on a simple, but general, model of dispersed kinetics—a nonexponential homogeneous decay combined with slowly exchanging rate heterogeneity. The first log-moments give a geometric-mean relaxation time. Second log-moments quantify the magnitude of rate dispersion, the fraction of the dispersion due to heterogeneity, and the dynamics of exchange between different rate subensembles. A suitable combination of these moments isolates exchange dynamics from three-dimensional kinetics without contamination by the rate-filtering effects that were identified in a recent paper [M. A. Berg and J. R. Darvin, J. Chem. Phys. 145, 054119 (2016)].