Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

TL;DR

Canonical nonlinearity in mapping from interatomic interactions to equilibrium configurations in discrete lattice systems arises from discretizing Gaussian configurational statistics. The authors propose a 2-Wasserstein transport distance with a Fisher-metric-aligned cost to quantify the unavoidable discretization contribution (UCN) and derive the universal limit $W_2^2 = d^2 \frac{1}{12} \mathrm{Tr}(\Gamma^{-1})$ as $d\to 0$, linking it to twice the KL divergence averaged over Gaussian translations. This provides a transport-information-geometric interpretation of discretization effects and separates them from intrinsic non-Gaussian features of the CDOS, with implications for configurational thermodynamics. The framework offers a principled, geometry-aware measure of discretization-induced nonlinearity that can inform both theory and practical thermodynamic modeling of configurational systems.

Abstract

When we consider canonical averages for classical discrete systems, typically referred to as substitutional alloys, the map from many-body interatomic interactions to thermodynamic equilibrium configurations generally exhibits complicated nonlinearity. This canonical nonlinearity is fundamentally rooted in deviations of the discrete configurational density of states (CDOS) from continuous Gaussian families, and has conventionally been characterized by the Kullback-Leibler (KL) divergence on discrete statistical manifold. Thus, the previous works inevitablly missed intrinsic nonlinearities induced by discretization of Gaussian families, which remains invisible within conventional information-geometric descriptions. In the present work, we identify and quantify such unavoidable canonical nonlinearity by employing the 2-Wasserstein distance with a cost function aligned with the Fisher metric for Gaussian families. We derive an explicit expression for the Wasserstein distance in the limit of vanishing discretization scale d to 0: W2 = d*sqrt(Tr(Gamma)^(-1)/12), where Gamma denotes covariance matrix of the CDOS. We further show that this limiting Wasserstein distance admits a clear geometric interpretation on the statistical manifold, equivalent to a KL divergence associated with the expected parallel translations of continuous Gaussian. Our framework thus provides a transport-information-geometric characterization of discretization-induced nonlinearity in classical discrete systems, with future potential applications to configurational thermodynamics.