Thermodynamic Approach for Nonlinearity within Canonical Ensemble

In the field of classical discrete systems, specifically substitutional alloys, this study introduces a stochastic thermodynamic approach to address nonlinearity within a canonical ensemble. This approach establishes a nonlinear relationship between a spectrum of many-body interactions and the corresponding equilibrium configuration, as determined through the canonical average. The proposed method facilitates the analysis of nonlinearity across multiple configurations via newly introduced thermodynamic functions. These functions enable the formulation of nonlinearity in the configuration space, previously conceptualized as local, and extend it to nonlocal nonlinearity within statistical manifolds. The present findings indicate that the average nonlinearity disparity between partially ordered and other configurations is constrained by the entropy production in an ideal linear system. This system is comprehensively described by a covariance matrix of the density of states in the configuration space. Practically, this approach could significantly advance the analysis of nonlinearity for various classical discrete systems.

💡 Research Summary

The paper presents a novel stochastic thermodynamic framework for analyzing non‑linearity within the canonical ensemble of classical discrete systems, with a focus on substitutional alloys. Traditional alloy modeling techniques—such as Metropolis Monte‑Carlo, multihistogram reweighting, multicanonical sampling, and entropy‑based methods—have succeeded in predicting equilibrium properties by fitting many‑body interaction parameters to first‑principles data. However, these approaches treat the canonical average ⟨q⟩ as a black‑box map from interaction space to configuration space and do not reveal the intrinsic structure of the non‑linear mapping ϕ_th that underlies equilibrium.

The authors begin by representing a configuration on a fixed lattice as an f‑dimensional vector q = (q₁,…,q_f), where each component is a multisite correlation function (e.g., pair, triplet). The many‑body interaction vector U is similarly defined. The canonical average can be written as a map ϕ_th : U → Q_Z, which is generally non‑linear because the density of states (CDOS) in configuration space deviates from a multivariate Gaussian. To quantify this non‑linearity the paper introduces two complementary measures:

1. Local non‑linearity – a vector field H(q) that captures the deviation of ϕ_th from its linear approximation. Formally, H(q) = n ϕ_th(β) ∘ (−β Γ)⁻¹ · q − q, where Γ is the f × f covariance matrix of the CDOS and n denotes the composite map. When H(q) = 0 the map is locally linear; importantly, H depends only on the CDOS and is independent of temperature β and interaction strengths.

2. Non‑local (statistical) non‑linearity – the Kullback‑Leibler (KL) divergence D_NOL between the actual canonical distribution c_A(q) and the distribution c_GA(q) that would arise if the CDOS were exactly Gaussian with the same covariance Γ. The KL divergence is expressed as D_NOL = Σ_q c_A(q) ln