  We discuss the mean-field theories obtained from the leading order in a
large-$N$ approximation for one- and two- component dilute Bose gases. For a
one-component Bose gas this approximation has the following properties: the
Bose-Einstein condensation (BEC) phase transition is second order but the
critical temperature $T_c$ is not shifted from the non-interacting gas value
$T_0$. The spectrum of excitations in the BEC phase resembles the Bogoliubov
dispersion with the usual coupling constant replaced by the running coupling
constant which depends on both temperature and momentum. We then study
two-component Bose gases with both inter- and intra- species interactions and
focus on the stability of the mixture state above $T_c$. Our mean-field
approximation predicts an instability from the mixture state to a
phase-separated state when the ratio of the inter-species interaction strength
to the intra-species interaction strength (assuming equal strength for both
species) exceeds a critical value. At high temperature this is a structural
transition and the global translational symmetry is broken. Our work
complements previous studies on the instability of the mixture phase in the
presence of BEC.
