= Microrheology of a Maxwell fluid
{title2=$\langle\Delta r^2(t)\rangle=\frac{k_BT}{\pi aG_0}(1+t/\tau),\quad\tau=\eta/G_0$}
A <Maxwell fluid> has a spring and dashpot in series, so its step-stress <creep compliance> is $J(t)=1/G_0+t/\eta$. The <passive microrheology> relation then gives the displayed three-dimensional <mean squared displacement>. The plateau scale $k_BT/(\pi aG_0)$ measures elasticity, while the long-time slope $k_BT/(\pi a\eta)$ measures viscosity. An entropic network with mesh size $\xi$ has the estimate $G_0\sim k_BT/\xi^3$, giving plateau scale $\xi^3/(\pi a)$.
The nonzero zero-lag extrapolation belongs to the idealized massless spring-dashpot response. Actual displacement variance starts at zero; fast solvent dynamics or inertia regularize the short-time behavior. Consequently the plateau formula applies after those transients and before the Maxwell relaxation time, if those scales are well separated. A sampling interval comparable with $\tau$ gives little access to a genuine plateau, even when spatial localization is sufficiently precise.
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