Analytic Representation of Canonical Average From Fine Structure of Density of States

TL;DR

This paper presents an analytic method to represent the canonical average in large discrete systems, clarifying the relationship between microscopic interactions and macroscopic properties by decomposing contributions into harmonic and anharmonic parts.

Contribution

It introduces a novel analytic representation of the configurational density of states using delta functions and derivatives, enabling explicit connection between microscopic structure and macroscopic averages.

Findings

Explicit representation of density of states for large systems.

Decomposition of contributions into harmonic and anharmonic components.

Potential to identify microscopic states characterizing macroscopic properties.

Abstract

Expectation value of dynamical variables in thermodynamically equilibrium state can be typically provided through well-known canonical average. The average includes tremendous number of possible states considered far beyond practically handled, which makes it difficult to obtain analytic representation of the average to clarify how the expectation value connects with given interaction: i.e., the relationship is generally understood in phonomenological manner except for modest, simple models. Here we show that the relationship is explicitly clarified for discrete large systems, where the configurational density of states for any single pair correlation is represented in terms of linear combination of Dirac delta function and its derivatives. The significant advantage of the present representation is that it can decompose contributions to macroscopic dynamical variables into harmonicity and anharmonicity in terms of their underlying structural degree of freedom, which will lead to find a set of special microscopic state to characterize macroscopic properties in equilibrium state for classical many-body systems.