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 phi(t) in the general form phi(t) = integral g(tau)chi(t/tau)d tau, instead of focusing on the usual special casein which the basis functions chi(t/tau) are exponentials. In practice, phi(t) is often fit by of stretched exponential, phi(t)= exp[-(t/tau)(beta)]. Here we analyze the properties of the probability density g(tau) for the case in which phi(t) 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, chi(t/tau). Various degrees of nonexponentiality intrinsic in each basis function translate into different values for the time-dependent variance sigma(2)(t) of the stochastic quantity chi(t/tau), in which tau is considered to be a spatially varying characteristic time scale. We slate a simple but exact solution for sigma(2)(t), and assess its relation to experimental data on the inhomogeneous optical linewidth sigma(inh)(t), measured in the course of solvation processes in a supercooled liquid.