Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

TL;DR

This paper investigates the intrinsic nonlinearity in classical discrete systems caused by discretizing Gaussian measures, using Wasserstein distance and information geometry to quantify and interpret this effect.

Contribution

It introduces a novel geometric framework employing Wasserstein distance to quantify discretization-induced nonlinearity in Gaussian-based models.

Findings

Explicit expression for Wasserstein distance in the limit of small discretization scale.

Geometric interpretation of Wasserstein distance as a KL divergence with parallel translation.

Framework generalizes beyond Gaussian families to characterize discretization effects.

Abstract

When we consider canonical averages for classical discrete systems, typically referred to as substitutional alloys, the map phi 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. 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. In addition, we confirm that this W2-KL equivalence admits a natural generalization beyond Gaussian families. The correspondence reveals that the irreversible geometric distortion of the local measure induced by discretization, while extrinsic to information geometry alone, can generically be characterized by a standard KL divergence.