Exact Microcanonical Formulation and Thermodynamics of Equispaced Finite-Level Systems

We present an exact microcanonical formulation, in the thermodynamic limit, for a system of $N$ noninteracting particles with $p$ equally spaced energy levels $\{0,\epsilon,2\epsilon,\ldots,(p-1)\epsilon\}$. Writing the microcanonical multiplicity $\Omega_p(E,N)$ as the coefficient of a generating function and evaluating the resulting representation by saddle-point analysis, we derive analytical expressions for the entropy per particle $s(u,p)$ and inverse temperature $\beta(u,p)$, with $u=E/(N\epsilon)$ in the interval $[0,p-1]$. The formulation applies to arbitrary $p$ and recovers the known cases $p=2$, $p=3$, and $p\to\infty$. For finite $p$, the bounded spectrum implies an entropy maximum at $u_c=(p-1)/2$, where $\beta$ vanishes and changes sign. In the limit $p\to\infty$, the upper spectral bound is lost, the finite-energy entropy maximum disappears, and no negative-temperature branch remains. To our knowledge, this is the first general thermodynamic-limit microcanonical solution for arbitrary $p$. It therefore provides a unified framework for the thermodynamics of equispaced finite-level systems and their bounded-spectrum crossover with increasing $p$.