A calculation given in [52] yields the desired relationship:

Here S′(q, ω − ω₀) is the so-called dynamic structure factor. It is interpreted as the spatial and temporal Fourier transform of the autocorrelation function of refractive-index fluctuations δn. This quantity plays a central role in connecting theory and experiment. The final relation describing this connection is:

Here S(q) is the static structure factor and I(q) the static scattering intensity. The physical role of the dynamic structure factor is discussed in the following section.

3.2.2 Scattering Spectrum of a Liquid

Access to the information contained in Eq. (3.59) is provided by the mode-coupling theory already mentioned in Chapter 1. This requires solving the conservation and continuity equations of hydrodynamics. Only a brief summary of the procedure is given here.

The conservation and continuity equations must be linearized. Furthermore, to account for the thermodynamic parameters of the system, the relationship between the time-dependent energy density e(r, t) arising from the continuity equation and the heat density q(r, t) must be established:

Here P denotes pressure. After linearization and several mathematical transformations, described in full in [52], the desired connection is obtained between the dynamic structure factor and thermodynamic quantities such as the heat capacities C_V and C_p.

Original mathematical typography retained. .