Thermodynamic Approach for Nonlinearity within Canonical Ensemble

TL;DR

This work addresses how nonlinearity in canonical ensembles for classical discrete systems (e.g., substitutional alloys) can be formulated beyond single-configuration analysis by introducing a stochastic thermodynamic framework. It defines a nonlinear map $\phi_{\textrm{th}}$ from many-body interactions $\vec{U}$ to equilibrium configurations $\vec{Q}_{Z}$ and distinguishes local nonlinearity via a vector field $\vec{H}(\vec{q})$ from nonlocal nonlinearity via the KL-divergence $D_{\textrm{NOL}}$ on a statistical manifold, both grounded in the CDOS $g(\vec{q})$. The core idea is to transform the configuration-space transition driven by nonlinearity into a stochastic transition in thermal contact, with transition probabilities $R(\vec{q}_{B}|\vec{q}_{A})$ and entropy production $\sigma$, linking changes in nonlinearity to heat inflow and to the covariance structure (matrix $\Gamma$) of CDOS. The authors prove that the average change in nonlinearity across configurations is bounded by entropy production in a corresponding linear system, implying lattice geometry constrained by the CDOS covariance governs nonlinearity across configurations. This thermodynamic framing enables systematic analysis of multi-configuration nonlinearity and offers a practical pathway to study complex configurational nonlinear behavior in classical discrete systems.

Abstract

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.

Figures (4)

• Figure 1: Relationship between nonlinearity on configuration space as a vector $\vec{H}$ at configuration $\vec{q}_{A}$ and nonlinearity on statistical manifold as KL divergence $D_{\textrm{NOL}}$. Thin arrows denote taking map (e.g., $\vec{V}_{A} = \left( -\beta\Gamma \right)^{-1}\cdot \vec{q}_{A}$ and $\vec{q}_{B} = \phi_{\textrm{th}}\cdot \vec{V}_{A}$), dashed arrows denote the operation of canonical average with $\vec{V}_{J}$ and CDOS ($g$ or $g_{\textrm{G}}$) (i.e., corresponding to Eq. \ref{['eq:cdoss']}), and bold arrows represent differences on configuration space (as vector) or on statistical manifold (as Kullback-Leibler divergence), i.e., $\vec{H}\left( \vec{q}_{A} \right) = \vec{q}_{B} - \vec{q}_{A}$ and $\Delta D_{\textrm{NOL}}=D_{\textrm{NOL}}^{B}-D_{\textrm{NOL}}^{A}$.
• Figure 2: Schematic illustration of (a): System transition from $\vec{q}_{A}$ to $\vec{q}_{B}$ on configuration space driven by nonlinearity, transformed into (b): System transition contacting with a thermal bath. To achieve the transformation, extension of the deterministic system transition to stochastic one is required, as described in Eqs. \ref{['eq:gp']}-\ref{['eq:dt-r']}.
• Figure 3: Schematic illustration of coarse-grained configuration space with the resultant CDOS $g\left(q\right)$ (vertical bold lines) and vector field $\vec{H}$ (dashed curves) for equiatomic binary alloys on representative lattice (including fcc and bcc).
• Figure 4: Summary of the representative results for nonlinearity character obtained through the present transformation. (a) Brief concept of the present setup, measuring the transformed thermodynamic quantities on practical system from those on linear system. The practical and linear system exhibit transition from initial $\vec{q}_{A}$ to final configuration $\vec{q}_{B}$, respectively, contacting with a thermal bath at inverse temperature $\beta$. (b1) Relationship between $\Delta D_{\textrm{NOL}} = D_{\textrm{NOL}}^{B}-D_{\textrm{NOL}}^{A}$ and heat inflow to the system, bridging the concept of nonlinearity on statistical manifold and nonlinearity on configuration space through the transformed thermodynamic quantities (corresponding to Eq. \ref{['eq:nol-Q']}). (b2) From (b1), upper bound for average nonlinearity along forward transition is clarified, which is characterized by entropy production for the linear system (corresponding to Eq. \ref{['eq:nol-change']}).