Exact and approximate methods of calculating the sum of states for noninteracting classical and quantum particles occupying a finite number of modes

TL;DR

This paper introduces exact formulas and a new fourth-order approximation for calculating the sum of states in noninteracting particles, improving accuracy and computational efficiency across classical and quantum systems.

Contribution

It provides novel exact expressions and a superior approximation method for the density of states, applicable to finite modes and different particle statistics.

Findings

The fourth-order approximation captures variance and kurtosis effectively.

The exact method reduces computational effort compared to saddle-point approximation.

Numerical tests confirm the method's broad applicability.

Abstract

We present exact expressions for the sum of states of noninteracting classical and quantum particles occupying a finite number of modes with arbitrary spacings. Exploiting a probabilistic analogy, we derive an analytic fourth-order approximation to the density of states, which captures its variance and kurtosis, and is superior to the previous, commonly used methods for all three particle statistics. Our approach employs a simple exact method of calculating the moments of the microcanonical density of states for quantum particles, which requires less computational effort than the commonly used saddle-point approximation. We test our methods numerically and discuss their applicability to various physical systems.