Inversion of Fluctuation Theory

The KB/FST inversion procedure is the process of obtaining expressions for the particle number fluctuations or KBIs in terms of experimentally available (isothermal-isobaric) data. Again, there are multiple approaches to the inversion procedure (Ben-Naim 1977 O Connell 1994 Smith 2008). Arguably, the simplest approach involves the pseudo chemical potential and partial molar volumes (Ben-Naim 2006). First, we note that combining Equations 1.46 and 1.47 provides 

FIGURE 1.2 The Kirkwood—Buff inversion approach for obtaining KBIs from the available experimental data. See Prolegomenon for symbol definitions. (Ploetz, E. A., and Smith, P. E., 2011, Local Fluctuations in Solution Mixtures, Journal of Chemical Physics, 4, 135. With permission of the American Institute of Physics.) 

The above equations can then be expressed in matrix form. 

Alternatively, one can simply use the matrix relation between the A and B matrices, and a thermodynamic relation between the isochoric and isobaric chemical potential derivatives to provide the elements of the A matrix. The relevant expressions are 

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