= Solution
An entropic network has approximately one thermal <energy> of elastic cost per mesh volume, giving $G_0\sim k_BT/\xi^3$. At $T=300\ \mathrm K$, the supplied mesh size and <viscosity> give
$$
\boxed{G_0\sim0.033\ \mathrm{Pa},\qquad\tau=\eta/G_0\sim0.030\ \mathrm s.}
$$
The intermediate-time plateau in the three-dimensional <mean squared displacement> is
$$
\boxed{\langle\Delta r^2\rangle_{\rm plateau}\sim\frac{\xi^3}{\pi a}=4.0\times10^{-14}\ \mathrm{m^2},\qquad r_{\rm rms}\sim0.20\ \mu\mathrm m.}
$$
After the omitted fast initial transient, the log-log sketch is approximately flat for $t\ll\tau$, crosses over near $\tau$, and grows linearly for $t\gg\tau$. The figure shows the Maxwell compliance prediction over this range.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2008-maxwell-microrheology.png]
{title=Elastic plateau and diffusive growth in the ideal Maxwell microrheology prediction, with the video time scale marked}
An ordinary 25–30 frame-per-second recording samples intervals of about $0.03$–$0.04\ \mathrm s$, comparable to $\tau$. It therefore has little access to a clear plateau at lags well below the crossover, even if its positional precision resolves a displacement of a few tenths of a micrometre. Higher frame rates and calibrated localization noise would improve the modulus measurement. The bead radius is only twice the mesh size, so treating the network as a homogeneous continuum is also an approximation.
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