Estimating Energies and Entropies

The three quantities of interest — the free energy difference, AA, the difference in potential energies, AIJq i, and the entropy difference, AS o i, are connected through a basic thermodynamic relation 

U0 and Ui are the potential energies of the reference and the target system, respectively. Note that AUr,, is not the same quantity as the previously used (AU)0, because the latter is the average of energy differences between the target and the reference states taken over the reference ensemble. 

The difficulty in calculating a small number as a difference of two large ones, both known with limited accuracy, is not new and arises in many other fields. It often justifies using perturbation theory, which is aimed at estimating the quantity of interest directly. We will attempt to follow the same approach here. To do so, we combine (2.60) with (2.11) from Sect.2.2, in which we substitute U for This yields 

If this equation is used, A A and A IS,, can be calculated from a single simulation, or, as is usually the case, from a single series of stratified simulations. This is why it is sometimes called a single-state perturbation method. Unfortunately, the problem 

Another approach to calculating AS0 i or AU,, 1 relies on the classical thermodynamic relationships 

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