Theoretical study on density of microscopic states in configuration space via Random Matrix

TL;DR

This paper provides a theoretical analysis of the density of microscopic states in configuration space using Random Matrix theory, revealing how spatial constraints influence macroscopic properties in disordered classical systems.

Contribution

It introduces a novel theoretical approach employing Random Matrix theory to understand the statistical independence of microscopic states in large systems.

Findings

Density of microscopic states approaches statistical independence at thermodynamic limit.

Lower-order moments of eigenstates' density show asymptotic behavior consistent with Random Matrix predictions.

Supports decomposition of macroscopic properties into contributions from a few microscopic states.

Abstract

In classical systems, our recent theoretical study provides new insight into how spatial constraint on the system connects with macroscopic properties, which lead to universal representation of equilibrium macroscopic physical property and structure in disordered states. These important characteristics rely on the fact that statistical interdependence for density of microscopic states (DOMS) in configuration space appears numerically vanished at thermodynamic limit for a wide class of spatial constraints, while such behavior of the DOMS is not quantitatively well-understood so far. The present study theoretically address this problem based on the Random Matrix with Gaussian Orthogonal Ensemble, where corresponding statistical independence is mathematically guaranteed. Using the generalized Ising model, we confirm that lower-order moment of density of eigenstates (DOE) of covariance matrix of DOMS shows asymptotic behavior to those for Random Matrix with increase of system size. This result supports our developed theoretical approach, where equilibrium macroscopic property in disordered states can be decomposed into individual contribtion from each generalized coordinate with the sufficiently high number of constituents in the given system, leading to representing equilibrium macroscopic properties by a few special microscopic states.