Cumulant Expansion of the Free Energy

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting 

Cumulants have several interesting properties. All k for n 1 are shift-independent, i.e., they do not depend on the value of AU)0. Homogeneity, expressed by the relationship 

The first four terms called, respectively, the average (or expectation value), variance, skewness, and kurtosis, are equal to 

As may be seen, the formulas for higher-order cumulants become more complicated. More importantly, they are increasingly difficult to estimate accurately from simulations. 

If the expansion is terminated after the second order, the free energy takes the form  

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G. Archontis and M. Karplus, J. Chem. Phys., 105,11246 (1996). Cumulant Expansion of the Free Energy Application to Free Energy Derivatives and Component Analysis. 

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