The Variational Approach to Approximations

The calculation of the quantum correction to the excess chemical potential is of the form 

We will consider expression (11.23) as a schematic for the inner ak averaging in (11.22). Let us now explore an approximation for this expression. First, expand the exponential and take the averages of the terms in increasing powers of . We will consider terms here up to second order in / . Then, expand the logarithm and we get 

These are the first two terms in a cumulant expansion [50]. We note here that the convergence of cumulant expansions is a subtle issue. Generally, if the statistics are nearly Gaussian, the cumulant expansion yields a good approximation. If the statistical distribution is not Gaussian, however, the cumulant expansion diverges with the inclusion of higher-order terms. See [29] and references therein for more discussion of this point. 

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