The difference between v$_{¯}$ s and v$_{T}$ is a consequence of the finite thermal expansion. Equation (12) gives the density response for Bose and Fermi fluids described by the non-dissipative Landau two-fluid equations. The two-fluid density response function in the presence of dissipation arising from transport coefficients has been given by Hohenberg and Martin [11, 12, 15]. (See also the discussion given by Vinen [14].) For our purposes in this paper, the primary effect of dissipation is to broaden the delta function peaks in the imaginary part of the non-dissipative density response function. It does not otherwise change the structure of the two-fluid response function and in particular, does not affect in a significant way the weights of first and second sound in this function. From (12), it is straightforward to obtain the imaginary part of the density response function, which is the experimentally relevant quantity in dilute gases: Here, we have defined the “traditional” amplitudes for first and second sound, namely Using (17) in (3), the two-fluid expression for the dynamic structure factor is found to be 9 Im χ$_{nn}$ ( q, ω ) = − nπ q 2 2 m { Z$_{1}$ ω [ δ ( ω − u$_{1}$q )+ δ ( ω + u$_{1}$q )] + Z$_{2}$ ω [ δ ( ω − u$_{2}$q ) + δ ( ω + u$_{2}$q )] } . (17) Z$_{1}$ ≡ u 2 1 − v 2 u 2 1 u 2 , Z$_{2}$ ≡ v 2 − u 2 u 2 1 u 2 . (18) S ( q, ω ) = q 2 2 m [ N $^{0}$( ω ) + 1] { Z$_{1}$ ω [ δ ( ω − u$_{1}$q )+ δ ( ω + u$_{1}$q )] + Z$_{2}$ ω [ δ ( ω − u$_{2}$q ) + δ ( ω + u$_{2}$q )] } . (19) W$_{1}$ ≡ Z$_{1}$ u 2 1 , W$_{2}$ ≡ Z$_{2}$ u 2 . (20) The two-fluid dynamic structure factor in (19) turns out to satisfy both the f -sum and compressibility sum rules, given by (7) and (11) [23]. This makes use of the identity N $^{0}$( ω )+ 1 + N $^{0}$( − ω ) + 1 = 1, satisfied by the Bose distribution. Substituting (19) into the left-hand side of (7), one sees that the amplitudes Z$_{1}$ and Z$_{2}$ describe the weights of first and second sound, respectively, in the f -sum rule. That the sum of these two contributions saturate the f -sum rule is a consequence of the fact that Z$_{1}$ + Z$_{2}$ = 1. Similarly, substituting (19) into the left-hand side of (11), one sees the relative contributions of first and second sound to the isothermal compressibility are given by Z$_{1}$/u 2 1 and Z$_{2}$/u 2 $_{2}$, respectively. We will denote these weights by W$_{i}$ :