 Solution

At $k_{B}T\gg ϵ$, attractive contact energies are weak compared with thermal fluctuations, so expanded coil-like conformations are favoured and the end-to-end distribution broadens towards larger distances. Strong short-range repulsion still limits overlap over an effective, temperature-dependent diameter. The soft Lennard-Jones model is therefore not exactly identical to a temperature-independent hard-sphere chain, even when attraction is relatively unimportant.

A sufficiently stiff spring keeps the chain contour length near $(N-1)\sigma$ in all three comparisons. If temperature instead becomes comparable with $k{\sigma }^2/k_B$, increased bond stretching introduces an additional size change and the intended fixed-bond coil comparison breaks down. Every equilibrated trace should fluctuate around a stationary distribution; an expanding running mean caused by incomplete relaxation is not evidence for an equilibrium high-temperature state.