  The fluctuations of a number of particles in the Bose-Einstein condensate are
studied in the grand canonical ensemble with an effective single-mode
Hamiltonian, which is derived from an assumption that the mode corresponding to
the Bose-Einstein condensate does not asymptotically correlate with other
modes. The fluctuations are evaluated in the dilute limit with a proposed
simple method, which is beyond the mean-field approximation. The accuracy of
the latter is estimated; it is shown that the mean-field scheme does not work
for the single-mode Hamiltonian, while for the Hartree Hamiltonian it allows us
to estimate the condensate fluctuations up to a numerical factor. As a
hypothesis, a formula is proposed that relates the fluctuations in the
canonical ensemble with that of the grand canonical one.
