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, with a focus on the temperature-dependent speed of second sound. Starting from the one-loop Bogoliubov action, we compute the condensate gradient stiffness B from the thermal self-energy, and we evaluate the condensate-normal fluid coupling lambda from the standard matching integral. Our main contribution is the numerical evaluation of the hydrodynamic coefficients for rubidium-87 across a broad temperature range, using stable dimensionless variables to compute the temperature-dependent specific heat, entropy, normal density, and second-sound speed. We provide the first temperature scan of the coherence-factor suppression ratio R(T) and quantify the effect of the full epsilon_k/E_k factor on lambda(T) from 10 to 350 nK. This yields a quantitative benchmark for future precision measurements on rubidium gases, with the c2(T) trend—a non-monotonic shape with a maximum followed by a drop to zero near T_c—in qualitative agreement with the homogeneous Bose gas experiment of Hilker et al. For 87Rb at T=100 nK and n0=10^20 m^-3, we find lambda = (2.59 ± 0.52) x 10^-22 kg s^-1 and c2 = 1.09 mm/s. These results provide a directly usable reference for future experiments on rubidium gases.