Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section G / 31 / d

With an attractive Lennard-Jones potential and $k_{B}T\ll ϵ$, a short run can remain trapped in an initial conformation or a metastable compact state. A distance trace may show a long apparently stationary segment, an initial collapse, or occasional jumps between local structures; its histogram depends on initialization and is not necessarily an equilibrium distribution. An extended start need not collapse on the available run length, and a compact start need not already be the global minimum.

Accepting only downhill moves would make this trapping worse and would not sample the specified finite-temperature canonical ensemble. The sketch should therefore label this trace as unequilibrated and avoid assigning its mean a thermodynamic interpretation.

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.

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.