Osmotic pressure and generating function of the vertex functions

2 Osmotic pressure and generating function of the vertex functions 

As was previously shown (Section 5.2.2), the Green s functions can be expressed in terms of simpler functions called vertex functions. The generating function r M of these vertex functions is a functional of the magnetization A/,(r), and is related to G B by a Legendre transformation (11.5.14). 

We saw above that the osmotic pressure can be expressed in terms of the function G(B) associated with G B. In the same way for 

The functions G(B) and T(Af) are connected to each other by a Legendre transformation (11.5.25) 

Consequently, by combining (11.5.40) and (11.5.42), we can express the osmotic pressure and the concentrations, in terms of the parameter M. Thus, we obtain (des Cloizeaux 1975).11 

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