Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 35C / c / Solution

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.