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

An entropic network has approximately one thermal energy of elastic cost per mesh volume, giving $G_0\sim k_{B}T/{\xi }^3$. At $T=300\mathrm{K}$, the supplied mesh size and viscosity giveThe intermediate-time plateau in the three-dimensional mean squared displacement isAfter the omitted fast initial transient, the log-log sketch is approximately flat for $t\ll \tau$, crosses over near $\tau$, and grows linearly for $t\gg \tau$. The figure shows the Maxwell compliance prediction over this range.

An ordinary 25–30 frame-per-second recording samples intervals of about $0.03$–$0.04\mathrm{s}$, comparable to $\tau$. It therefore has little access to a clear plateau at lags well below the crossover, even if its positional precision resolves a displacement of a few tenths of a micrometre. Higher frame rates and calibrated localization noise would improve the modulus measurement. The bead radius is only twice the mesh size, so treating the network as a homogeneous continuum is also an approximation.