= Solution
For a thermally elastic network, estimate one elastically active element per mesh <volume>. Each contributes an elastic free-energy scale of order $k_BT$, giving the <entropic network modulus>
$$
\boxed{G_0\sim\frac{k_BT}{m^3}\simeq3.3\times10^{-2}\,\mathrm{Pa}}
$$
at $T=298\,\mathrm K$. Mesh size alone would not determine the modulus of an arbitrary bending-dominated or athermal network; this is the entropic scaling assumption needed for the estimate.
The formal elastic intercept in the given model is consequently
$$
\boxed{\langle r^2\rangle_{\rm plateau}\sim\frac{m^3}{\pi a}
=3.98\times10^{-14}\,\mathrm{m^2}=0.0398\,\mu\mathrm m^2.}
$$
The corresponding three-dimensional root-mean-square <displacement> is about $0.20\,\mu\mathrm m$. The <temperature> cancels from this estimate because the assumed modulus is itself proportional to $T$. This is the extrapolated plateau from the preceding part, not a nonzero exact zero-lag <displacement>. Here $m/a=1/2$, so a continuum estimate is only approximate.
In <passive microrheology>, track dilute inert probe spheres by video microscopy and average squared lag <displacements> over starting <times> and probes. An isotropic two-dimensional image gives $\langle\Delta r^2\rangle_{3d}=(3/2)\langle\Delta r^2\rangle_{2d}$. Alternatively, <dynamic light scattering> probes ensemble <displacement> statistics; the original optical <microrheology> study compared inferred storage and loss moduli with mechanical measurements.<Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids by T. G. Mason and D. A. Weitz>[Physical Review Letters 74 (1995), p. 1250, abstract: dynamic light scattering and comparison with mechanical moduli]{ref}
Remove drift, calibrate spatial scale and frame timing, estimate localization noise using an immobilized-bead control, and account for motion blur. Keep probes away from walls and check that changing probe <radius> or tracer concentration does not change the inferred bulk response unexpectedly. Noise alone can generate a false short-time intercept, while adsorption or confinement can mimic <elasticity>. No numerical long-time slope is fixed because a numerical <viscosity> is not supplied.
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