Deviations from the Boltzmann distribution in small microcanonical quantum systems: Two approximate one-particle energy distributions

Abstract

The Boltzmann distribution, which accurately describes the exponential energy dependence of the canonical ensemble, only describes the distribution of one-particle energies for a microcanonical system in the large system limit. We present two distribution functions which closely approximate the distribution of allowed one-particle energies in weakly coupled microcanonical quantum systems. One function is exact for a set of identical harmonic oscillators. The second function is a generalization of work by Andersen et al. [J. Chem. Phys. 114, 6518 (2001)] and is exact for a system with constant microcanonical heat capacity. We compare these two functions with enumerated probabilities for three model systems. The model system distributions and both approximate functions become exponential for large systems but differ from the Boltzmann distribution most dramatically at high energy, for which states can be considerably less populated than predicted by the Boltzmann distribution. Corrections to the Boltzmann distribution may be important in unimolecular reactions, fragmentation dynamics, and in the spectroscopy of nanoclusters.