  This paper presents a conserving gapless mean-field theory for weakly
interacting Bose gases. We first construct a mean-field Luttinger-Ward
thermodynamic functional in terms of the condensate wave function $\Psi$ and
the Nambu Green's function $\hat{G}$ for the quasiparticle field. Imposing its
stationarity respect to $\Psi$ and $\hat{G}$ yields a set of equations to
determine the equilibrium for general non-uniform systems. They have a
plausible property of satisfying the Hugenholtz-Pines theorem to provide a
gapless excitation spectrum. Also, the corresponding dynamical equations of
motion obey various conservation laws. Thus, the present mean-field theory
shares two important properties with the exact theory: ``conserving'' and
``gapless.'' The theory is then applied to a homogeneous weakly interacting
Bose gas with s-wave scattering length $a$ and particle mass $m$ to clarify its
basic thermodynamic properties under two complementary conditions of constant
density $n$ and constant pressure $p$. The superfluid transition is predicted
to be first-order because of the non-analytic nature of the order-parameter
expansion near $T_{c}$ inherent in Bose systems, i.e., the Landau-Ginzburg
expansion is not possible here. The transition temperature $T_{c}$ shows quite
a different interaction dependence between the $n$-fixed and $p$-fixed cases.
In the former case $T_{c}$ increases from the ideal gas value $T_{0}$ as
$T_{c}/T_{0}= 1+ 2.33 an^{1/3}$, whereas it decreases in the latter as
$T_{c}/T_{0}= 1- 3.84a(mp/2\pi\hbar^{2})^{1/5}$. Temperature dependences of
basic thermodynamic quantities are clarified explicitly.

Statistical Mechanics