Numerical Evaluation of the Two-Fluid Hydrodynamic Coefficients and Temperature-Dependent Second Sound in a Finite-Temperature Bose Gas

We present a numerically stable evaluation of the two-fluid hydrodynamic coefficients for a dilute Bose gas of rubidium-87 at finite temperature. Our contribution is threefold: (i) we compute the condensate gradient stiffness $B$, the entropy density $S$, and the specific heat $c_v$ directly from the one-loop Bogoliubov action; (ii) we outline the key steps of the density-response matching that yields the condensate--normal fluid coupling $\lambda$, making the calculation transparent; and (iii) we quantify the systematic error incurred by the common approximation $\epsilon_k/E_k \to 1$ in $\lambda(T)$. We provide a numerical temperature scan of the hydrodynamic coefficients for rubidium-87 across a broad range, using stable dimensionless variables that ensure ultraviolet convergence. For $^{87}\mathrm{Rb}$ at $T=100\,\mathrm{nK}$ and $n_0=10^{20}\,\mathrm{m^{-3}}$, we find $\lambda = (2.59 \pm 0.52) \times 10^{-22} \, \mathrm{kg\,s^{-1}}$ and $c_2 = 1.09\,\mathrm{mm/s}$ (the phonon-limit value $c_s/\sqrt{3}$). The temperature dependence of $c_2$ exhibits a non-monotonic shape with a maximum followed by a drop to zero near $T_c$, qualitatively consistent with the universal two-fluid behavior observed in the homogeneous $^{39}$K Bose gas experiment of Hilker et al. \cite{Hilker2022}. The quantitative results are reliable for $T\lesssim 0.5T_c$; higher-temperature entries indicate the expected qualitative trend. These results provide a quantitative benchmark for comparison with future experiments on rubidium gases.