Measuring a hidden coordinate: Rate-exchange kinetics from 3D correlation functions.

Nonexponential kinetics imply the existence of at least one slow variable other than the observable, that is, the system has a "hidden" coordinate. We develop a simple, but general, model that allows multidimensional correlation functions to be calculated for these systems. Homogeneous and heterogeneous mechanisms are both included, and slow exchange of the rates is allowed. This model shows that 2D and 3D correlation functions of the observable measure the distribution and kinetics of the hidden coordinate controlling the rate exchange. Both the mean exchange time and the shape of the exchange relaxation are measurable. However, complications arise because higher correlation functions are sums of multiple "pathways," each of which measures different dynamics. Only one 3D pathway involves exchange dynamics. Care must be used to extract exchange dynamics without contamination from other processes.