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Microrheology infers local mechanical response from microscopic probes embedded in a material. Active methods apply a calibrated force and measure displacement; passive microrheology uses thermal fluctuations. Probe size, boundary conditions at the probe, temporal resolution and material heterogeneity determine which continuum response can be inferred.

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 $k_{B}T/(\pi a{G}_0)$ measures elasticity, while the long-time slope $k_{B}T/(\pi a\eta )$ measures viscosity. An entropic network with mesh size $\xi$ has the estimate $G_0\sim k_{B}T/{\xi }^3$, 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.

For a no-slip spherical tracer of radius $a$ in an isotropic incompressible elastic continuum, a small translation produces restoring force $6\pi aG\mathbf{x}$. ThusCombining this with the equilibrium three-dimensional positional variance givesEquivalently, if $P_{\infty }$ is the stationary three-dimensional displacement-MSD plateau, $G=k_{B}T/(\pi a{P}_{\infty })$. For a single measured coordinate, $⟨x_i^2⟩=k_{B}T/k$ and $G=k_{B}T/(6\pi a⟨x_i^2⟩)$.

This is passive microrheology. Calibrate bead radius, position sensitivity, temperature and localization noise; subtract drift and test that a plateau is resolved. The continuum must be meaningful on the bead scale, the bead must couple without slip, and thermal equilibrium must hold. Boundaries, compressibility, active motion, probe attachment and unresolved relaxation can change the inferred response. An imposed optical trap adds stiffness and must be removed or independently calibrated rather than mistaken for material elasticity.

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,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 $⟨\Delta r^2⟩=k_{B}Tt/(\pi \eta a)=6Dt$, exactly the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

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.