General derivation of the Kirkwood-Buff theory

The derivation of the relationship between the thermodynamic quantities and KBIs consists of two parts. First, we use the normalization conditions for the singlet and the pair distribution functions in the T, V, p ensemble. This provides relationships between the KBIs and the fluctuations in the number of the particles in the open system. Next, by differentiation of the grand partition function, we obtain relationships between thermodynamic quantities and fluctuations in the number of particles. Finally, by eliminating the fluctuations in the number of particles, we obtain the required relations between thermodynamic quantities and the KBIs. 

We start by considering the grand canonical ensemble characterized by the variables T, V, and fi where p = (/q, p2. pc) is the vector comprising the chemical potentials of all the c components of the system. The normalization conditions for the singlet and the pair distribution functions follow directly from their definitions. Here, we use the indices a and [l to denote the species a, jS = 1, 2. c. The two normalization conditions are (for particles not necessarily spherical) 

pa is the average number density of molecules of species a, i.e., pa = Na)/V, with V the volume of the system. We also recall the definition of the spatial pair correlation function 

This concludes the first part of the derivation of the KB theory. 

Equation (4.12) is a connection between the cross fluctuations in the number of particles of various species, and integrals involving only the spatial pair correlation functions for the corresponding pairs of species a and p. 

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