General Matrix Formulation of Fluctuation Theory

The number of simultaneous equations provided by Equation 1.43 increases as the number of components increases. It then becomes more efficient to express the results in terms of matrices. Again, there are many equivalent matrix formulations that have been presented (Kirkwood and Buff 1951 O Connell 1971b Ben-Naim 2006 Nichols, Moore, and Wheeler 2009). Here, we present one of the simplest. A general formulation is easiest starting from the first expression in Equation 1.44. Writing the number fluctuations in matrix form for an component system where we also include the number densities (GD expression at constant T) in the first row provides. 

By taking derivatives of the GD expression with respect to pressure with all molalities (and T) constant, together with a series of derivatives of Equation 1.44 with respect to each species molality at constant p (and T), the results can be expressed in the general form. 

A general formulation of KB/FST theory in terms of direct correlation functions is also possible. Because the DCFIs are related to TCFIs through a matrix inversion, their general multicomponent formulas do not involve matrix cofactors (O Connell 1971b). The equivalents of Equations 1.54 and 1.55, which apply over the entire composition range are then 

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