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We present exact expressions for the sum of states of noninteractingclassical and quantum particles occupying a finite number of modes witharbitrary spacings. Exploiting a probabilistic analogy, we derive an analyticfourth-order approximation to the density of states, which captures itsvariance and kurtosis, and is superior to the previous, commonly used methodsfor all three particle statistics. Our approach employs a simple exact methodof calculating the moments of the microcanonical density of states for quantumparticles, which requires less computational effort than the commonly usedsaddle-point approximation. We test our methods numerically and discuss theirapplicability to various physical systems.

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