Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(iii) / 17

Interpret the given noise correlation as the three-dimensional dot-product correlation. Each Cartesian component then has $⟨{\xi }_i(t){\xi }_j(t^{\prime })⟩=2\gamma k_{B}T{\delta }_{ij}\delta (t-t^{\prime })$. For the usual Gaussian thermal-noise idealization, each coordinate is an Ornstein-Uhlenbeck process. With $\tau =\gamma /k$,For a particle initially at ${\mathbf{x}}_0=0$,Hence its long-time displacement about the trap/elastic equilibrium hasThis also follows from equipartition theorem: each of three quadratic coordinates contributes $k_{B}T/2$.

A different common experimental quantity is the stationary time-lag displacement. It obeyswith long-lag RMS $\sqrt{6k_{B}T/k}$. The factor of two distinguishes fluctuations about a mean position from differences between two decorrelated positions. The observation convention must be stated rather than silently interchanged.

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.