Canonical–grand-canonical thermodynamic duality

Fix an interior attainable density $\rho >0$ and a sequence $N_L\mathrm{/}{L}^d\to \rho$. The grand sum contains the nonnegative term $e^{\alpha N_L}Z(L,N_L;\beta )$, so ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{L}^{-d}\log {\mathrm{\Xi }}_L(e^{\alpha },\beta )\ge F_{\beta }(\rho )+\alpha \rho$ by [F1]. Taking the supremum over interior densities gives the lower bound. The vacuum term $Z(L,0)=1$ gives the value $0$ at density zero. Endpoint values are limits from the interior by their definition, so they do not enlarge the supremum.

Put $q=e^{\alpha +\beta B}$. For each $N\ge 0$, stability and the box-volume formula give $e^{\alpha N}Z(L,N;\beta )\le (q{L}^d{)}^N\mathrm{/}N!$. Choose $K>eq+1$. For $N>K{L}^d$, $1\le e^{N-K{L}^d}$, so the factorial bound gives ${\sum }_{N>K{L}^d}{e}^{\alpha N}Z(L,N;\beta )\le e^{-K{L}^d}{\sum }_{N\ge 0}\frac{(eq{L}^d{)}^N}{N!}\le e^{-L^d}\mathrm{.}$ The remaining at most $K{L}^d+2$ terms are bounded by their maximum times this number; its logarithm divided by $L^d$ tends to zero.

Choose maximizing retained particle numbers and a subsequence with $N_L\mathrm{/}{L}^d\to \rho \in [0,K]$; nested-interval bisection on the compact interval $[0,K]$ supplies the subsequence. The canonical upper rate is at most $b(\rho )\le F_{\beta }(\rho )$, since the one-term mixture in the definition of $F_{\beta }$ includes $b(\rho )$. At $\rho =0$, stability gives upper rate at most $0=F_{\beta }(0)$. If $u$ is a finite upper endpoint and $b(u)>-\mathrm{\infty }$, mix a preparation approaching $b(u)$ with an interior preparation of density $r<u$, in slab fractions $1-ϵ$ and $ϵ$. The exact packing inequality in [F1] gives $F_{\beta }((1-ϵ)u+ϵr)\ge (1-ϵ)b(u)+ϵb(r)\mathrm{.}$ As $ϵ\downarrow 0$, the left side tends to the assigned interior endpoint limit; hence $b(u)\le F_{\beta }(u)$. If $b(u)=-\mathrm{\infty }$ this bound is automatic. Outside the closed effective domain, $b(\rho )=-\mathrm{\infty }$, again because a finite $b(\rho )$ would make $F_{\beta }(\rho )\ge b(\rho )$. Thus the maximum retained term has limiting rate at most ${\sup }_{\rho \ge 0}(F_{\beta }(\rho )+\alpha \rho )$. The tail estimate in step 1.2 and the lower bound in step 1.1 prove the first equality.

The extension $F_{\beta }$ is proper, upper-semicontinuous and concave. It is finite and concave on its open effective interior by [F1], hence continuous there. At zero, stability and the concavity of $g(\rho )=\rho (1+\beta B-\log \rho )$ give $F_{\beta }(\rho )\le g(\rho )\to 0$. The sparse separated-cell lower bound in [F1] gives $F_{\beta }(\rho )\ge c\rho$ for a fixed finite $c$, so $F_{\beta }(\rho )\to 0=F_{\beta }(0)$. At any finite upper endpoint use the assigned interior upper-semicontinuous limit; outside the closed effective domain set $F_{\beta }=-\mathrm{\infty }$. Hence $G=-F_{\beta }$, with $G=+\mathrm{\infty }$ on negative densities, is a proper closed convex extended-real function on $\mathbb{R}$. Its conjugate is $G^{\ast }(\alpha )={\sup }_{\rho \ge 0}\{\alpha \rho +F_{\beta }(\rho )\}=P(\alpha ,\beta )$ by steps 1.1–2.1. Applying [F3] gives $G(\rho )={\sup }_{\alpha }\{\alpha \rho -P(\alpha ,\beta )\}$, which is the second equality. The endpoint convention is needed for closedness; it asserts no finite-volume endpoint realization.