= Solution
Under a step <stress>, the Maxwell spring extends immediately by $\sigma_0/G_0$, while the dashpot adds $\sigma_0t/\eta$. Hence $J(t)=G_0^{-1}+t/\eta$, and the <microrheology of a Maxwell fluid> relation gives
$$
\boxed{\langle\Delta r^2(t)\rangle=\frac{k_BT}{\pi aG_0}\left(1+\frac t\tau\right),\qquad\tau=\eta/G_0.}
$$
This idealized overdamped relation describes an elastic plateau followed by long-time <diffusion>. Its nonzero extrapolated intercept is not the actual displacement variance at exactly zero lag, which is zero. Inertia or fast solvent relaxation must resolve the initial rise; those processes were omitted from the ideal spring-dashpot model.
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