content_image/374970.jpg content_image/374969.jpg content_image/374968.jpg content_image/374967.jpg content_image/374966.jpg
where we have introduced the isothermal sound velocity v$_{T}$ . 

The sum rules in (7) and (11) are general and may be viewed as two constraints which S ( q , ω ) must always satisfy. The f − sum rule is used frequently in discussions of inelastic neutron scattering as a check on the experimental data. As we discuss in section III, the compressibility sum rule has special significance when we discuss the two-fluid hydrodynamic region, as first emphasized by Nozi` eres and Pines [23]. 

## III. DYNAMIC STRUCTURE FACTOR IN THE TWO-FLUID REGIME 

The Landau two-fluid equations for an isotropic (such that χ$_{nn}$ ( q , ω ) = χ$_{nn}$ ( q, ω )) super- fluid (Bose or Fermi) describe the dynamics when collisions are strong enough to produce local equilibrium. Using the non-dissipative two-fluid equations, one finds the density re- sponse function (see Refs. [11, 12] and page 138 of [3]), 

where we have defined a new velocity 

<img src='content_image/374970.jpg'>

<img src='content_image/374969.jpg'>

Here, ¯ s$_{0}$ ≡ S$_{0}$/Nm is the equilibrium entropy S$_{0}$ per unit mass and ¯ c$_{v}$ ≡ T ( ∂ ¯ s/∂T )$_{ρ}$ is the specific heat per unit mass at constant volume. The equilibrium superfluid and normal fluid densities are denoted by ρ$_{s}$$_{0}$ and ρ$_{n}$$_{0}$ , respectively. In (12), u$_{1}$ and u$_{2}$ are the well- known exact first and second sound velocities given by Landau’s hydrodynamics. For later purposes, we note that these velocities satisfy the exact relations 

<img src='content_image/374968.jpg'>

where v$_{T}$ is defined in (11) and the adiabatic sound velocity is defined as 

<img src='content_image/374967.jpg'>

We note that thermodynamic relations [24] show that, quite generally, the ratio of the specific heats can be related to the isothermal and adiabatic sound velocities, 

<img src='content_image/374966.jpg'>

