chapter 15 in [3]) Here, g$_{n}$ ( z ) = ∑ ∞ l $_{=1}$z $^{l}$/ln is the usual Bose-Einstein function with fugacity z . In the limit of small g , the normal fluid density ρ$_{n}$$_{0}$ reduces to the density m ˜ n$_{0}$ ≡ ρ − mn$_{c}$$_{0}$ of atoms thermally excited out of the condensate. The superfluid density ρ$_{s}$$_{0}$ is likewise given by the density mn$_{c}$$_{0}$ of condensate atoms. To lowest order in g , n$_{c}$$_{0}$ is the condensate density of the ideal Bose gas. In figure 6, we show the temperature dependence of u$_{1}$ and u$_{2}$ , satisfying (14), in the case of weak interactions ( n$_{B}$a 3 B = 10 − $^{5}$). The thermodynamic functions used to determine these sound velocities have been calculated using the Hartree–Fock–Popov (HFP) microscopic model for the thermal excitations [3]. The velocities are normalized in terms of a Fermi velocity defined by v$_{F}$ = (6 π $^{2}$n$_{B}$ ) 1 / $^{3}$/m (equivalent to the Fermi velocity of a two component Fermi gas through the BCS-BEC crossover). We also show for comparison the values of u$_{1}$ and u$_{2}$ given by the lowest order solutions described by (32). One sees that these uncoupled modes of oscillations are a good first approximation to the full solutions of the two-fluid equations in a weakly interacting BEC at all temperatures outside a small temperature region at low temperatures where first and second sound hybridize, exhibiting an avoided crossing. At this avoided crossing, the natures of first and second sound are interchanged. As T → 0, the first sound velocity coincides with the phonon velocity, as is the case with strongly interacting Fermi gases and superfluid $^{4}$He. For a dilute gas, this phonon velocity is the Bogoliubov–Popov phonon velocity √ gn$_{c}$$_{0}$/m$_{B}$ , and corresponds to an oscillation of the condensate component. Above the hybridization point, to a good approximation, second sound propagates with this velocity. Consequently, second sound in a dilute Bose gas (above the hybridization temperature) is essentially an oscillation of the condensate at all temperatures, with a static thermal cloud. In contrast, first sound describes an oscillation of the thermal cloud in the presence of a stationary condensate. We note that HFP approximation we have used to calculate thermodynamic quantities has the well-known problem near T$_{c}$ of predicting a spurious first-order transition (see, for instance, [34]). As touched on in section IV, this is actually a very general problem, and arises in any theory that treats bosonic phase fluctuations of the order parameter as a small parameter, including the HFP and NSR theories. Of course, this includes all perturbation 18 u 2 1 = 5 T 3 m$_{B}$ g$_{5}$$_{/}$$_{2}$ (1) g$_{3}$$_{/}$$_{2}$ (1) + 2 g ˜ n$_{0}$ m$_{B}$ , u 2 = gn$_{c}$$_{0}$ m$_{B}$ . (32)