Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 37A (Statistical Physics)

The microcanonical ensemble describes an isolated system with fixed energy, volume, and particle number. It assigns equal probabilities to the accessible microstates in the chosen narrow energy shell. The canonical ensemble describes a system in thermal contact with a large heat reservoir: its temperature, volume, and particle number are fixed, while its energy can fluctuate.

For the Gibbs entropyfirst maximize subject only to ${\sum }_n{p}_n=1$ over $\Omega$ accessible microstates. A Lagrange multiplier givesso every $p_n$ is equal. Normalization yields the microcanonical ensemble

For the canonical ensemble, impose both normalization and the mean-energy constraint ${\sum }_n{p}_n{E}_n=E$. Stationarity ofgives $p_n=C{e}^{-\beta E_n}$. The canonical partition function fixes $C$, soThermodynamic consistency identifies $\beta =(k_{B}T{)}^{-1}$. This is the maximum-entropy derivation of equilibrium ensembles.

Substitution of the canonical probabilities into the Gibbs entropy givesHence

Put $x=\beta ϵ$. The one-particle canonical partition function isand independence gives $Z=z^N$. Therefore the independent symmetric three-level system hasand, using part (c),As $T\to \infty$, $x\to 0$, all three levels become equally likely andAs $T\to 0^{+}$, every particle occupies the nondegenerate ground level $-ϵ$, and

Because the one-particle spectrum is bounded above, a negative temperature is possible. A population-inverted configuration with more particles in the $+ϵ$ level than in the $-ϵ$ level haswhich requires $\beta <0$. Equivalently, above the maximum-entropy energy $E=0$, adding energy reduces the number of compatible microstates, so $\partial S/\partial E=1/T<0$. The limiting configuration with every particle at $+ϵ$ has $E=Nϵ$, zero entropy, and $T\to 0^{-}$.