# Analytic Determination of Variance for Configurational Density of States in Crystalline Solids

## TL;DR

This paper derives an analytical expression for the variance of the configurational density of states in crystalline solids, aiding the understanding of thermodynamic properties in disordered systems.

## Contribution

It provides the first analytic formula for the variance of CDOS as a function of composition in binary crystalline systems, validated by comparison with numerical sampling.

## Key findings

- Derived an explicit variance formula for CDOS in binary systems.
- Validated the formula through comparison with lattice sampling.
- Facilitates microscopic characterization of free energy in classical systems.

## Abstract

Our recent study elucidate that information of density of states in configuration space (CDOS) for non-interacting system, characterized by spatial constraint on the system, plays essential role to determine thermodynamically equilibrium properties for interacting systems. Particularly for disordered states, variance of CDOS along all possible coordinations plays significant role. Despite this fact, even for binary system of crystalline solids, analytic expression for variance of CDOS as a function of composition has not been clarified so far. Here we successfully derive variance of CDOS as a function of composition for pair correlations, whose validity is demonstrated by comparing the results with uniform sampling of CDOS on real lattices. The present result certainly advances determining special microscopic state to characterize Helmholtz free energy in classical systems, whose structure is difficult to determine by numerical simulation.