Order-Parameter Fluctuations in a Weakly Interacting Bose Gas near the Superfluid Transition

The contribution of order-parameter fluctuations to the static and dynamic properties of a weakly interacting Bose gas near the Bose-Einstein condensation temperature $T_c$ is investigated via a novel approach. Self-consistent approximations for the propagator of order-parameter fluctuations $G$ are generated in successive temperature ranges closer and closer to $T_c$. Mean-field theory corresponds to assuming that the correlation of fluctuations $G_2$ is given by ${G_2}^{\mathrm{}\left(1\right)\mathrm{}}=GG$. This first approximation is valid in the range ${\tau }_1<\tau =\frac{(T-T_c)}{T_c}<1\mathrm{}$, where at $\tau ={\tau }_1$ the neglected terms are comparable to those included. In this paper we consider the second approximation ${G_2}^{\mathrm{}\left(2\right)\mathrm{}}=GGVGG$, which involves the first-order interaction between the fluctuations. Two independent criteria for the range of validity ${\tau }_2<\tau <{\tau }_1$ of the second approximation are established. It is conjectured that the qualitative results of the present model calculation are applicable to liquid helium in the presently accessible temperature range. The predicted temperature dependences of the specific heat, superfluid density, and fourth sound are in agreement with the scaling hypothesis and available experiments in helium. The condensate density is predicted to vanish linearly with $\tau$. It is shown that the contribution of order-parameter fluctuations to the thermal conductivity diverges as ${\tau }^{\frac{-1\mathrm{}}{3}}$ above $T_c$, and it is suggested that the absence of logarithmic factors is due to the inconsistent treatment of entropy fluctuations.