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

摘要

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.