Interpretation of the Free Energy Perturbation Equation

The formulas for free energy differences, (2.8) and (2.9), are formally exact for any perturbation. This does not mean, however, that they can always be successfully applied. To appreciate the practical limits of the perturbation formalism, we return to the expressions (2.6) and (2.8). Since AA is calculated as the average over a quantity that depends only on A17, this average can be taken over the probability distribution Po(AU) instead of Pq(x. p ) [6], Then, AA in (2.6) can be expressed as a onedimensional integral over energy difference 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

Even though P,(AU) is only rarely an exact Gaussian, it is instructive to consider this case in more detail. If we substitute 

If P0(AU) is Gaussian, there is, of course, no reason to carry out a numerical integration, since the integral in (2.15) can be readily evaluated analytically. This yields 

One consequence of the positivity of a is that A A (AU)0. If we repeat the same reasoning for the backwards transformation, in (2.9), we obtain A A (AU)V These inequalities, known as the Gibbs-Bogoliubov bounds on free energy, hold not only for Gaussian distributions, but for any arbitrary probability distribution function. To derive these bounds, we consider two spatial probability distribution functions, F and G, on a space defined by N particles. First, we show that 

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