sphere probes in hydroxypropylcellulose: water, this author and Lacroix[@lacroix1997a] found stretched exponentials in time $$g^{(1s)}(q,t) =  a \exp(- \theta t^{\beta}) \equiv a \exp(- (t/\tau)^{\beta}).
     \label{eq:gisqeext}$$ Here $\beta$ is a scaling exponent while $\theta$ and $\tau$ are prefactors. A series of papers by Streletzky and collaborators on the same chemical system (most recently, ref. ) established by viewing a much wider range of delay times that $g^{(1s)}(q,t)$ is in fact a sum of two stretched exponentials in time.

Finally, I note a very simple model system in which eqs \[eq:gaussianform\] and \[eq:g1swrong\] fail. The system is a dilute aqueous dispersion of polystyrene spheres, in which the spheres are of two different sizes. There are no sphere-sphere interactions. Each sphere individually performs Brownian motion as described by the Langevin equation. Therefore, for each sphere in the mixture, $P(\Delta x, t)$ is a Gaussian in $\Delta x$ and $\langle (\Delta x)^{2} \rangle$ increases linearly in time. For the mixture as a whole, the mean-square displacement averaged over all the particles must therefore also increase linearly with time, at a rate determined by a weighted average of the diffusion coefficients of the two sphere species.

However, the mixture’s field correlation function is a sum 