Tropical Diagram for Linear-Nonlinear Boundary in Canonical Ensemble

TL;DR

This paper introduces a tropical geometric approach to characterize the linear-nonlinear boundary in the canonical ensemble of discrete systems, revealing universal structural features related to constraints and non-separability.

Contribution

It develops a novel tropical diagram method to analyze the geometric structure of the linear-nonlinear boundary in discrete systems with two degrees of freedom.

Findings

Boundary near disordered state dominated by individual SDF constraints

Quasi linear-nonlinear boundary influenced by local non-separability

Tropical diagram captures universal geometric features of the boundary

Abstract

For classical discrete systems under constant composition, we re-examine how linear-nonlinear boundary in canonical ensemble, connecting a set of potential energy surface and that of microscopic configuration in thermodynamic equilibrium, is characterized by underlying lattice, from tropical geometry. We here show that by applying suitable tropical limit and multiple coordinate transform to time evolution of discrete dynamical system, reflecting geometric aspect of the nonlinearity, we successfully construct tropical diagram caputuring the universal character of linear/nonlinear region on configuration space for f = 2 structural degree of freedoms (SDFs). The diagram indicates that the boundary near disordered state is mainly dominated by constraints to individual SDF, while quasi linear-nonlinear boundary apart from disordered state is dominated by local non-separability in SDFs.