Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / b / iii

Write $S_{\mathrm{conn}}(L)$ for the connected candidate andfor the $L$-independent disconnected candidate. The physical entropy is the lower envelopeNear $L=C$, the connected branch rises to meet the horizontal disconnected branch. The entropy is continuous but generically has a kink there: its derivative jumps from the positive slope of $S_{\mathrm{conn}}$ to zero. Correspondingly, $I_c=S_{\mathrm{disc}}-S(D_1\cup D_2)$ falls to zero and remains zero.