Statistical mechanics of the $N$-queens problem

We investigate the $N$-queens problem as a lattice gas – a model in which $N$ queens are placed on an $N\times N$ chessboard with pairwise repulsive interactions along shared rows, columns, and diagonals – from the perspective of statistical mechanics. The ground states are exactly the $Q(N)$ solutions of the classical $N$-queens problem, with entropy per queen $s_0\approx \ln N-\gamma$ ($\gamma \approx 1.944$). This entropy reflects a characteristic constraint hierarchy: each successive geometric constraint – columns, then diagonals – reduces the entropy from the free-placement value $\ln N$ by a definite constant. We derive the exact high-temperature energy $E\mathrm{/}N\to 5\mathrm{/}3$ as $N\to \mathrm{\infty }$. Extensive Monte Carlo simulations with ${10}^8$ sweeps per temperature point for $N=8$–$1024$ reveal that the specific heat per queen $C_v\mathrm{/}N$ converges to a universal function of $T$ as $N\to \mathrm{\infty }$. The converged curve features a non-divergent peak $C_v^{\max }\mathrm{/}N\approx 1.63$ at $T^{\ast }\approx 0.235J$, establishing the absence of a thermodynamic phase transition. Combined with the trivially exact high-temperature entropy $S(\mathrm{\infty })\mathrm{/}N=(1\mathrm{/}N)\ln \left(\genfrac{}{}{0px}{}{N^2}{N}\right)$, the convergence of $C_v\mathrm{/}N$ enables a thermodynamic integration of $C_v\mathrm{/}T$ from $T=\mathrm{\infty }$ to $T=0$ that recovers the ground-state entropy – and hence the Simkin constant $\gamma$ – purely from Monte Carlo data. This provides an independent thermodynamic route to a fundamental combinatorial constant. Thermodynamic integration yields ${\gamma }_{\mathrm{M}\mathrm{C}}=1.946\pm 0.003$ at $N=1024$, within $0.1\mathrm{\%}$ of the precise combinatorial value $\gamma =1.94400(1)$. We further present a transfer-matrix-based tensor network formulation that encodes the non-attacking constraints into a rank-9 site tensor with 17 nonzero elements, providing a complementary exact-enumeration route.