of two exponentials $$g^{(1s)}(q,t) = A_{1} \exp(-D_{1} q^{2} t) + A_{2} \exp(-D_{2} q^{2} t)
      \label{eq:doubleexponential}$$ where $A_{i}$ is the scattering amplitude of species $i$ and $D_{i}$ is the diffusion coefficient of species $i$. Correspondingly, the displacement distribution function for all the particles in solution is a weighted sum of two Gaussians, one Gaussian for each sphere size.

Suppose one were used eq \[eq:g1swrong\] to determine $\langle (\Delta x)^{2} \rangle$ of the sphere mixture. According to eq \[eq:g1swrong\] $$\langle (\Delta x)^{2} \rangle = - \ln( g^{(1s)}(q,t))/q^{2}.
      \label{eq:meansquare2}$$ At short times, $g^{(1s)}(q,t)$ contains contributions from both species, and the $\langle (\Delta x)^{2} \rangle$ computed from eq \[eq:meansquare2\] is the weighted average of the mean-square displacements of the two species. At long times, in eq \[eq:doubleexponential\] the exponential corresponding to the more rapidly-moving species has decayed to zero, so the nominal mean-square displacement from eq \[eq:meansquare2\] is determined by scattering from the larger, more slowly-moving spheres. For this simple bidisperse sphere suspension, eq \[eq:meansquare2\] asserts that the slope of $\langle (\Delta x)^{2}\rangle$ depends on time, $\langle (\Delta x)^{2}\rangle$ increasing more rapidly at short times and more slowly at long times. The assertion is incorrect. In this system $\langle (\Delta x)^{2}\rangle$ increases linearly with 