Let us consider an open region, i.e., one in which particles can come and go freely, drawn in a system of infinite extent. What will now be shown is that the fluctuation in the number of particles in this region is given by the volume integral of g(r) – 1, which is specifically the isothermal compressibility of the liquid. Another interesting example of such a relation between fluctuations and thermodynamic quantities yields the specific heat cv; this is discussed in Appendix A5.4.

One reason for the interest in the above relation between the volume integral of the radial distribution function—or, equivalently, from (2.4), the long wavelength limit of the structure factor S(k)—and the compressibility (first derived by Ornstein and Zernike) is because of the difficulty of extending diffraction experiments to very small scattering angles.

Let us consider a member of the grand canonical ensemble in which the open region, of volume V, contains exactly N particles.