Geometric Study on Noncommutativity in Canonical Nonlinearity

TL;DR

This paper investigates the geometric aspects of noncommutativity in canonical nonlinearity within classical discrete systems, using a coarse-grained model to relate noncommutativity to configurational geometry and thermodynamic bounds.

Contribution

It introduces a geometric approach to analyze noncommutativity in nonlinearity, employing a coarse-grained state model to connect configurational geometry with thermodynamic properties.

Findings

Noncommutativity correlates positively with asymmetric Hausdorff distance in configurational polyhedra.

Lower noncommutativity values occur when coordination number differences are small.

Including nonseparability information modifies the correlation between noncommutativity and configurational measures.

Abstract

For classical discrete systems under constant composition, canonical average provides equilibrium configuration from a set of many-body interactions, which typically acts as nonlinear map. The nonlinearity has recently been investigated in terms of configurational geometry, where two measures for the nonlinearity as vector field on configuration space and divergence on statistical manifold are introduced. Then, concepts of these measures are further unified through stochastic thermodynamic treatment. While these studies provide deeper insight into understanding the nonlinearity, thermodynamic treatment for non-separability in structural degrees of freedom (NS), still remains difficult due mainly to nonexistence of the corresponding CDOS. Our recent study partially overcome the problem, by considering additional information of non-commutativity for the nonlinearity (NC). The present study focuses on the basic behavior of the NC, in terms of the geometric information about configurational geometry, much more easily-accesible information than NC. For this purpose, we here employ the coarse-grained state model (CSM) that can qualitatively capture the nonlinearity information through the coarse-grained configuration space with tractable numbers of parameters, where the present CSM can implicitly include the difference in coordination numbers on real lattice. We find that (i) the NC exhibit positive correlation with asymmetric Hausdorff distance in configurational polyhedra between of pracitcal and separable systems in terms of SDFs, especially for partically-ordered-state-rich region, (ii) the NC take lower value when the difference in coordination number becomes small, indicating that we can obtain better thermodynamic bound for NS in such systems, and (iii) the positive correlation can be further modified when we additionally include information about nonseparability in the practical CDOS.