Tropical Diagram for Linear/Nonlinear Boundary in Canonical Ensemble

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 microscopic configuration thermodynamic equilibrium, is characterized by underlying lattice, from tropical geometry. We here show applying suitable limit multiple coordinate transform to time evolution dynamical system, reflecting geometric aspect the nonlinearity, successfully construct diagram caputuring universal character linear/nonlinear region on space for f = 2 structural degree freedoms (SDFs). The indicates near disordered state mainly dominated constraints individual SDF, while quasi apart local non-separability SDFs.