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