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

TL;DR

This paper presents an exactly solvable lattice model demonstrating a universal finite-size scaling law for configurational entropy, driven solely by geometric factors across various lattice types, with implications for understanding residual entropy in materials.

Contribution

The paper introduces a minimal, exactly solvable lattice model that reveals a universal geometric scaling law for configurational entropy in diverse lattice systems.

Findings

Scaling deviation from thermodynamic limit: $ o N^{-1/d}$

Model captures magnitude of experimental residual entropies

Universal finite-size scaling law observed across lattice types

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.