Exact Microcanonical Formulation and Thermodynamics of Equispaced Finite-level Systems

Abstract

We develop a unified microcanonical formulation for \(N\) noninteracting particles with an arbitrary number \(p\) of equally spaced local energy levels, \(0,\varepsilon ,2\varepsilon ,\dots ,(p-1)\varepsilon \). The exact microcanonical multiplicity is expressed as a coefficient of a local generating function, providing a finite-\(N\) representation that can also be systematically analyzed in the thermodynamic limit. By converting this coefficient extraction into a contour integral and applying saddle-point analysis, we obtain analytical expressions for the entropy per particle \(s(u,p)\) and the inverse temperature \(\beta (u,p)\), where \(u=E/(N\varepsilon )\in \left [ 0,p-1\right ] \). The main result is a single arbitrary-\(p\) solution that contains the two-level and three-level systems as special cases and connects them continuously to the unbounded limit \(p\rightarrow \infty \). For every finite \(p\), the upper spectral bound produces an entropy maximum at \(u_\mathrm {c}=(p-1)/2\), where the inverse temperature vanishes and the positive-temperature branch changes into a negative-temperature branch. As \(p \) increases, this characteristic energy is displaced toward higher values, and in the limit \(p\rightarrow \infty \) the upper bound is removed, the finite-energy entropy maximum disappears, and only positive temperatures remain. The formulation therefore provides an analytically controlled description of the crossover between bounded- and unbounded-spectrum thermodynamics. Beyond the present model, the method supplies an exact microcanonical template for discrete additive systems whose local spectra can be encoded by generating functions, including finite-level quantum and spin systems. Abstract Published by the Jagiellonian University 2026 authors