Unavoidable Canonical Nonlinearity Induced by Gaussian Measures Discretization

When we consider canonical averages for classical discrete systems, typically referred to as substitutional alloys, the map ϕ 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 inevitably 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 considering a transport cost induced by the specific local-map (LM) discretization scheme, based on the 2-Wasserstein framework with a cost function aligned with the Fisher metric for Gaussian families. In the limit of vanishing discretization scale d → 0, we derive an explicit expression for this specific transport cost: [Formula: see text], where Γ denotes covariance matrix of the Gaussian. We show that this limiting transport cost admits a clear geometric interpretation on the statistical manifold, corresponding to a KL divergence associated with the expected parallel translations of continuous Gaussian. In addition, we confirm that this [Formula: see text]-KL correspondence admits a natural generalization beyond Gaussian families, provided that the cost function is aligned with the Fisher metric of an underlying statistical submanifold and the discretization scale links to infinitesimal parameter variations. The correspondence demonstrates that the geometric distortion of the local measure induced by discretization — while extrinsic to information geometry alone — can be naturally characterized by a standard KL divergence.