Dynamic heterogeneity, spatially distributed stretched-exponential patterns, and transient dispersions in solvation dynamics

In the context of determining the extent of dynamical heterogeneity of relaxation processes, it has proven useful to represent the ensemble-averaged autocorrelation function [Formula Presented] in the general form [Formula Presented] instead of focusing on the usual special case in which the basis functions [Formula Presented] are exponentials. In practice, [Formula Presented] is often fit by a stretched exponential, [Formula Presented] Here we analyze the properties of the probability density [Formula Presented] for the case in which [Formula Presented] is a superposition of stretched exponentials, and is itself a stretched exponential, with a stretching exponent greater than or equal to those of the basis functions, [Formula Presented] Various degrees of nonexponentiality intrinsic in each basis function translate into different values for the time-dependent variance [Formula Presented] of the stochastic quantity [Formula Presented] in which τ is considered to be a spatially varying characteristic time scale. We state a simple but exact solution for [Formula Presented] and assess its relation to experimental data on the inhomogeneous optical linewidth [Formula Presented] measured in the course of solvation processes in a supercooled liquid.