Nonequilibrium Bounds for Canonical Nonlinearity Under Single-Shot Work

TL;DR

The paper addresses the problem of characterizing and bounding the nonlinearity of the canonical average mapping from interatomic interactions to configurational averages in substitutional alloys, extending the concept from thermodynamic equilibrium to nonequilibrium states generated by single-shot work. It introduces a local nonlinear vector field $\vec{H}$ and a nonlocal KL-based nonlinearity $D_{\mathrm{NOL}}$, then models nonequilibrium transformations via a Gibbs-preserving map and thermo-majorization, deriving a tight upper bound on nonequilibrium nonlinearity: $D(P:R) \le \inf_{\alpha>1} [S_{\alpha}(Q:R) + \frac{\alpha}{\alpha-1}W]$, with $W$ the work and $S_{\alpha}$ the Renyi divergence. A corresponding lower bound is discussed, and the optimal $\alpha$ is shown to depend on the work, smoothly interpolating between equilibrium ($W \to 0$, $\alpha\to 1$) and highly nonequilibrium regimes. The results connect configurational geometry (CDOS) with nonequilibrium thermodynamics, offering a physically meaningful, parameter-free way to bound nonlinearity under finite work inputs.

Abstract

For classical discrete systems under constant composition (specifically substitutional alloys), canonical average acts as a map from a set of many-body interatomic interactions to a set of configuration in thermodynamic equilibrium, which is generally nonlinear. In terms of the configurational geometry (i.e., information about configurational density of states), the nonlinearity has been measured as special vector on configuration space, which is extended to Kullback-Leibler (KL) divergence on statistical manifold. Although they successfully provide new insight into how the geometry of lattice characterizes the nonlinearity, their application is essentially restricted to thermodynamic equilibrium. Based on the resource theory (especially, thermo-majorization), we here extend the applicability of the nonlinearity to nonequilibrium states obtained through single-shot work on Gibbs state. We reveal that the extended nonlinearity for nonequilibrium state is bounded from upper and lower by the information about one of the optimal Renyi divergences for equilibrium states in between practical and linear systems, and temperature and work.

Figures (2)

• Figure 1: Present setup for preparation of the nonequilibrium state from equilibrium state through the single-shot work.
• Figure 2: Schematic relationships between nonlinearity for equilibrium state $D_{\textrm{NOL}}$ and nonequilibrium state $P^{\textrm{N}}$ obtained through Gibbs preserving map $\Lambda$. Solid arrows denote taking map, and dashed arrows represents taking difference in probability distributions through (appropriate) divergence.