Configurational Entropy and Its Scaling Behavior in Lattice Systems with Number of States Defined by Coordination Numbers

Abstract: We introduce an exactly solvable lattice model that reveals a universal finite-size scaling law for configurational entropy driven purely by geometry. Using exact enumeration via Burnside's lemma, we compute the entropy for diverse 1D, 2D, and 3D lattices, finding that the deviation from the thermodynamic limit $s_{\infty} = \ln (z)$ scales as $\Delta s_{N} \sim N^{-1/d}$, with lattice-dependent higher-order corrections. This scaling, observed across structures from chains to FCC and diamond lattices, offers a minimal framework to quantify geometric influences on entropy. The model captures the order of magnitude of experimental residual entropies (e.g., $S_{\mathrm{molar}} = R \ln 12 \approx 20.7 \, \mathrm{J/mol \cdot K}$) and provides a reference for understanding entropy-driven order in colloids, clusters, and solids.