Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A3 / e / Solution

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.