= Solution
<Passive microrheology> uses thermal bead motion as a probe of mechanical compliance. The <fluctuation-dissipation theorem> relates equilibrium displacement fluctuations to the same linear response measured by a small applied force. The stated thermal <stress> scale has units of <energy> per bead volume; dividing the mean-square displacement by $a^2$ provides a dimensionless effective <strain>. With that prescribed analogy,
$$
J(t)=\frac{\langle\Delta r^2(t)\rangle/a^2}{k_BT/(\pi a^3)},\qquad
\boxed{\langle\Delta r^2(t)\rangle=\frac{k_BT}{\pi a}J(t).}
$$
The displacement is relative to the starting position, and the variance here is the sum over three spatial coordinates. For a Newtonian fluid, substitution gives $\langle\Delta r^2\rangle=k_BTt/(\pi\eta a)=6Dt$, exactly the <Stokes–Einstein relation> $D=k_BT/(6\pi\eta a)$.
The continuum interpretation assumes a dilute no-slip spherical probe, thermal equilibrium, linear response and a homogeneous incompressible medium on the bead scale. The stated <stress> estimate is an effective scale for the analogy; a literal instantaneous <stress> variance would also require a microscopic or temporal averaging prescription.
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