Occupation Probabilities and Free Energy of Cavity Formation

Occupation Probabilities and Free Energy of Cavity Formation 

The implicit assumptions made in these calculations are that each of the possible outcomes is an elementary event and that all elementary events have equal probabilities i.e., each possible outcome in tossing a die has probability likewise each specific configuration of the system has probability of These implicit assumptions are valid for systems for which all the elementary events have equal probabilities. In our system, each of the (n) configurations has the same probability, and therefore the method used in (2.8.31) is applicable. 

This method becomes invalid when there are interactions among the ligands, in which case different configurations do not have the same probability. Nevertheless, the result (2.8.31) is still valid even when there are interactions. To prove that, we need a different argument. This is presented below. A similar argument applies in the theory of homogeneous fluids, to which we return in Chapter 5. 

Suppose that we have labeled all the N ligands. Let Pr (/th) be the probability of finding a specific ligand say, the kth, on a specific site, the iih site. Since the kih ligand must be found somewhere on one of the M sites, we must have the normalization condition 

Since the M sites are equivalent, the probabilities Pr (tth) are independent of the index /. Therefore the sum over i in (2.8.33) produces M equal terms. 

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