Recursive calculation of the microcanonical density of states

TL;DR

This paper develops recursive integral equations to calculate the microcanonical density of states for noninteracting classical particles, simplifying solutions especially when single-particle states follow a power law, and demonstrates methods for general cases.

Contribution

It introduces recursive integral equations for the microcanonical density of states and provides a straightforward solution approach using Laplace transforms for noninteracting systems.

Findings

Recursive equations for density of states derived

Simplified solutions for power-law single-particle states

Laplace transform method applicable for general cases

Abstract

For a classical system of noninteracting particles we establish recursive integral equations for the density of states on the microcanonical ensemble. The recursion can be either on the number of particles or on the dimension of the system. The solution of the integral equations is particularly simple when the single-particle density of states in one dimension follows a power law. Otherwise it can be obtained using a Laplace transform method. Since the Laplace transform of the microcanonical density of states is the canonical partition function, it factorizes for a system of noninteracting particles and the solution of the problem is straightforward. The results are illustrated on several classical examples.