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Expectation value of dynamical variables in thermodynamically equilibriumstate can be typically provided through well-known canonical average. Theaverage includes tremendous number of possible states considered far beyondpractically handled, which makes it difficult to obtain analytic representationof the average to clarify how the expectation value connects with giveninteraction: i.e., the relationship is generally understood in phonomenologicalmanner except for modest, simple models. Here we show that the relationship isexplicitly clarified for discrete large systems, where the configurationaldensity of states for any single pair correlation is represented in terms oflinear combination of Dirac delta function and its derivatives. The significantadvantage of the present representation is that it can decompose contributionsto macroscopic dynamical variables into harmonicity and anharmonicity in termsof their underlying structural degree of freedom, which will lead to find a setof special microscopic state to characterize macroscopic properties inequilibrium state for classical many-body systems.

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