 Passive microrheology

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.