Exact expression for the number of states in lattice models

TL;DR

This paper derives a precise combinatorial formula for counting states in lattice models, linking partition functions to internal configurations, with validation on classical distributions and the 1D Ising model.

Contribution

It introduces an exact low-temperature power series expansion for the partition function, providing new insights into the combinatorial structure of statistical mechanics.

Findings

Derived a closed-form expression for the number of states.

Connected the logarithm of the partition function to generating functions.

Validated the approach with classical distributions and the 1D Ising model.

Abstract

We derive a closed-form combinatorial expression for the number of states in canonical systems with discrete energy levels. The expression results from the exact low-temperature power series expansion of the partition function. The approach provides interesting insights into basis of statistical mechanics. In particular, it is shown that in some cases the logarithm of the partition function may be considered the generating function for the number of internal states of energy clusters, which characterize system's microscopic configurations. Apart from elementary examples including the Poisson, geometric and negative binomial probability distributions for the energy, the framework is also validated against the one-dimensional Ising model.