 Solution

For a genuinely equilibrated low-temperature attractive chain, many nonbonded contacts lower the energy. Compact conformations normally dominate, giving a smaller characteristic end-to-end distance than the high-temperature coil. The trace can remain near particular compact basins for long periods but must, under the stated premise, sample their correct equilibrium weights. The end-to-end distribution may have structure associated with distinct compact shapes; the end beads need not be nearest neighbours, so a compact globule does not force $d_{\mathrm{ETE}}=0$.

For $N=10$ there is a rounded finite-system crossover rather than a sharply defined bulk coil–globule phase transition. The exact histogram requires the dimensionless temperature, stiffness and treatment of bonded-pair Lennard-Jones interactions, none of which is numerically specified.