The Bogoliubov variational theorem

As wc arc ultimately interested in the phase behavior of the confined lattice fluid at arbitrary temperatures we are seeking solutions of Eq. (1.76a), where, of course, a = on account of the discreteness of the lattice. Because 

Let us assume that we can split the Ilamiltouian function into two contributions according to 

Assuming that the perturbation is not too large we may expand fl (T, /x A) in a MacLaurin series in terms of the coupling parameter A, that is, as a Taylor expansion in A around A = 0 (i.e., the reference system). Retaining in this expansion terms up to first order, we obtain 

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