= Solution
\b[The hard-core chain does not undergo an energetic low-temperature collapse.] For allowed configurations the only finite energy is bond stretching. If $k\sigma^2\gg k_BT$, bond lengths remain near $\sigma$ and temperature mostly changes the small bond-length fluctuations, not the distribution of angular conformations. The end-to-end distance therefore fluctuates over an expanded self-avoiding-coil distribution even at low temperature, with an approximate scale $\sigma(N-1)^{\nu}$ and $\nu\simeq0.588$ for sufficiently long three-dimensional chains.
For $N=10$, the scaling is qualitative and finite-chain effects matter. An equilibrated distance-versus-sweep trace is stationary, with correlated excursions; a histogram of that trace is a distinct plot, not a curve of <probability density> against simulation step. Neither the temperature in kelvin nor the spring constant is specified, so an exact histogram cannot be calculated. The illustrative figure compares like-for-like coil traces rather than asserting a unique numerical simulation result.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2-chain-sampling.png]
{title=Original qualitative Monte Carlo traces and end-to-end-distance distributions for repulsive and attractive chains}
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