Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 35C (Statistical Physics)

The latent heat for converting liquid to gas at coexistence iswith all quantities taken per mole or per particle consistently. Along the phase coexistence curve, the two Gibbs free energy values agree:Sincedifferentiating the equality along the curve givesTherefore the Clausius-Clapeyron relation isAt the critical point the liquid and gas become the same phase, so their entropy discontinuity disappears and

PutThe one-spin partition function isThe spins are independent, so the spin-1 paramagnet hasand free energy

The canonical heat capacity can be obtained from the energy variance:Becausewe obtain

At high temperature, $x\to 0$, and hence

Immediately after the reversal the state populations have not changed, but the new Hamiltonian satisfiesThusso the same state is canonical for the reversed field withThis is the magnetic-field reversal and negative temperature construction. Sincethe partition function is unchanged. Therefore

A gas has kinetic energy unbounded above. Its partition function diverges for $\beta <0$, so it cannot have a normalizable negative temperature canonical state and cannot satisfy the zeroth-law equilibrium condition ${\beta }_{\mathrm{gas}}={\beta }^{\prime }<0$.

If an amount $\delta Q>0$ flows from the spins into a positive-temperature gas, the total entropy change isThe second law therefore selects heat flowThis is the general ordering expressed by heat flow from negative to positive temperature.