Distribution Functions and Kirkwood-Buff Integrals

The previous expressious involve particle number (and energy) fluctuations. It is more conunon, and totally equivalent, to use correlation/distribution functions to replace the number fluctuations. In many cases this can help to clarify the significance of the number fluctuations (correlations) as we indicate in this section. However, in doing so one has to lemanber that these distributions correspond to a systan volume that is open to aU species. 

In the grand canonical ensonble, the probability that any Ni molecules of species 1, and Aj molecules of species 2, and so forth, are within d r at r is given by p H( ) W where 

Various combinations of the above integrals are commonly encountered in statistical thermodynamic theories of solutions. The most relevant is given by, 

A set of grand canonical distribution functions can then be defined for species 

The KBIs quantify the average deviation, from a random distribution, in the distribution of j molecules surrounding a central i molecule summed over all space. In this respect they are more informative than the particle number fluctuations as they can then be decomposed and interpreted in terms of spatial contributions— using computer simulation data, for example. They clearly resemble the integrals 

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