Correlation inequalities for noninteracting Bose gases

TL;DR

This paper establishes new correlation inequalities for noninteracting Bose gases, revealing monotonicity and convexity properties of occupation numbers and free energy, which deepen understanding of Bose-Einstein statistics.

Contribution

It derives novel inequalities for occupation numbers and free energy in noninteracting Bose gases, based on convexity arguments, extending theoretical understanding of quantum statistical mechanics.

Findings

<N_i>_N < <N_i>_{N+1}

<N_i N_j>_N < <N_i>_N <N_j>_N for i≠j

∂<N_0>_N/∂β > 0

Abstract

For a noninteracting Bose gas with a fixed one-body Hamiltonian H^0 independent of the number of particles we derive the inequalities <N_i>_N < <N_i>_{N+1}, <N_i N_j>_N < <N_i>_N <N_j>_N for i\neq j, \partial <N_0>_N/\partial \beta >0 and <N_i>^+_N < <N_i>_N. Here N_i is the occupation number of the ith eigenstate of H^0, \beta is the inverse temperature and the superscript + refers to adding an extra level to those of H^0. The results follow from the convexity of the N-particle free energy as a function of N.