Passive microrheology ($⟨\Delta r^2(t)⟩=k_{B}TJ(t)/(\pi a)$)

For a dilute spherical no-slip probe of radius $a$ in an isotropic incompressible linear medium at equilibrium, the bead's displacement per unit step force is $J_b(t)=J(t)/(6\pi a)$, where $J$ is the bulk shear creep compliance. The fluctuation-dissipation theorem gives one-coordinate displacement variance $2k_{B}T{J}_b(t)$. Adding the three equal Cartesian contributions proves the displayed relation. In a Newtonian fluid, $J=t/\eta$, so it becomes $⟨\Delta r^2⟩=6Dt$ with the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

The particle must sample a continuum response on its scale and remain in thermal equilibrium. A two-dimensional image records only two of the three displacement components; isotropy then gives $⟨\Delta r^2{⟩}_{3d}=(3/2)⟨\Delta r^2{⟩}_{2d}$. Localization noise, confinement and unresolved fast relaxation can otherwise be mistaken for elastic response.

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.