Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 35A (Statistical Physics)

In the microcanonical ensemble, the microcanonical energy-shell entropy is the Boltzmann entropyFor a macroscopic system, changing the small energy resolution by a fixed factor changes $\log \Omega$ only by a nonextensive additive term, while $S=O(N{k}_B)$. Its relative effect therefore vanishes in the thermodynamic limit.

The thermodynamic temperature, pressure, and chemical potential are defined in the entropy representation byThusor, equivalently, the first law of thermodynamics in fundamental form is

For the one-dimensional freely jointed chain, counting right- and left-pointing links givesThereforeChoosing which links point right gives the state degeneracyPut $x=L/(Na)$. Applying the Stirling formula to $S=k_B\log \Omega$ and discarding terms subleading in $N$ yields

Because increasing the length by $dL$ requires work $fdL$ on the chain, its first law of thermodynamics isAll configurations have the same energy, so at fixed $E$ and $N$,Differentiating the entropy found above givesand hence the entropic Hooke law for a one-dimensional chain isWhen $L\ll Na$, the Taylor expansion of $\mathrm{artanh}x$ giveswhich is Hooke's law. Conversely, at fixed force,The extension decreases as the temperature rises and tends to zero as $T\to \infty$: the rubber contracts on heating at fixed tension.