Notation and the theory of molecular liquids

Our principal interest in this book will be the molecular basis of the theory of molecular solutions. Because the molecular components may themselves be complicated, the notation can be complicated. This can be a huge, non-physical 

One perspective is that graph theory methods of the equilibrium theory of classical liquids are merely simple, common, intuitive notations. It is then ironic that these notations are typically a denouement of an extended theoretical development (Uhlenbeck and Ford, 1963 Stell, 1964). 

We expect that the typical initial response to the fundamental statistical mechanical formulae presented as in Eq. (2.15), p. 26, will be that this notation is schematic. We reply that for molecular liquids any communicative notation will be schematic. For example, cartesian positions of molecular centers is schematic, and for our purposes misses the important point of molecular liquids. We hope that the schematic notation that is employed will communicate satisfactorily. 

Finally, we adopt a notation involving conditional averages to express several of the important results. This notation is standard in other fields (Resnick, 2001), not without precedent in statistical mechanics (Febowitz et al, 1967), and particularly useful here. The joint probability P A, B) of events A and B may be expressed as P A, B) = P A B)P B) where P B) is the marginal distribution, and P A B) is the distribution of A conditional on B, provided that P B) 0. The expectation of A conditional on B is A B, the expectation of A evaluated with the distribution P(A B) for specified B. In many texts (Resnick, 2001), that object is denoted as E(A B) but the bracket notation for average is firmly established in the present subject so we follow that precedent despite the widespread recognition of a notation (A B) for a different object in quantum mechanics texts. 

The initial introduction of conditional probabilities is typically associated with the description of independent events, P A, B) = P A)P B) when A and B are independent. Our description of the potential distribution theorem will hinge on consideration of independent systems first, a specific distinguished molecule of the type of interest and, second, the solution of interest. We will use the notation ((... ))o to indicate the evaluation of a mean, average, or expectation of ... for this case of these two independent systems. The doubling of the brackets is a reminder that two systems are considered, and the subscript zero is a reminder that these two systems are independent. Then a simple example of a conditional expectation can be given that uses the notation explained above and in Fig. 1.8  

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