Fluctuations in the Bardeen-Cooper-Schrieffer Hamiltonian induced by finite particle number

The fluctuations properties of the BCS Hamiltonian which are induced in finite systems are explored. It is shown that for very large particle numbers, the static-path approximation is an excellent approximation, while for small particle numbers, the random-phase-approximation corrections should be taken into account, especially for particle numbers small enough to cause the normal- to superconducting-phase transition. A correlated mean field containing quantal fluctuations is introduced and compared with the Hartree-Bogoliubov mean field