Bounds and Inequalities The Bogoliubov Inequality

The basic idea is as follows we assume that the free energy of real interest is that associated with a Hamiltonian H, for which the exact calculation of the free energy is intractable. We then imagine an allied Hamiltonian, denoted Hq, for which the partition function and free energy may be evaluated. The inequality that will serve us in the present context asserts that 

Presently, we consider a few of the details associated with the derivation of the Bogoliubov inequality. Our treatment mirrors that of Callen (1985) who, once again, we find gets to the point in the cleanest and most immediate fashion. As said above, we imagine that the problem of real interest is characterized by a Hamiltonian H which may be written in terms of some reference Hamiltonian Ho 

The significance of the latter equation is that it demonstrates that for all X, the curvature of the function FexactiF) is negative. Why should this observation fill us with enthusiasm The relevance of this insight is revealed graphically in fig. 3.21, where it is seen that we may construct an approximate free energy (a first-order Taylor expansion in powers of A) of the form 

What has been done, in essence, is to exploit our certainty about the sign of the curvature of the free energy with respect to X to carry out a Taylor expansion of that free energy with respect to X on the hope that the leading order expansion will be sufficiently accurate. Note that when the slope of Fexact X) is evaluated explicitly, we find 

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