The mean molecular field

It is the essential feature of Rayleigh s treatment (and of that of van der Waals which we examine in more detail in the next diapter) that U(z) can also be expressed in terms of the equation of state of the hypothetical homogeneous fluid represented by the continuous van der Waals loops in Fig. 1.8. Thus (1.37) can be written 

It remains to calculate the surface tension for this continuous profile. Rayleigh tackled this by calculating the pressure inside a liquid drop, bounded by a surface in which p is a continuous function of the distance from the centre. It is easier, however, to obtain his result by using the fact that, on this molecular model, the surface tension is twice the excess energy per unit area of the surface (see 3.1). That is, it is given by 

The results above summarize the position reached by the molecular theory of capillarity by 1892. The attractive forces between molecules are responsible for a high, but not directly measurable, internal pressure in a 

There is, however, one further approxhnation that has been used above, and that is the one we now call a mean-field appmxinuitwn, or the assumption of the existence of a mean molecular field. It is an approximation which is sometimes difficult to avoid even in modern statistical mechanics, and is worth full discussion. 

Once the problem was so dearly recognized then a solution, at least in principle, was to hand. Instead of aamniing a random or uniform distribution of molecules we introduce distribution or correlation functions into our expressions for the mean interactions of molecules at two positions in the fluid. These functions are measures of the conditional probability of the occurrence of pairs (or larger groups) of molecules at specified points. Their calculation is one of the principal aims of modem theories liquids. However there are still many problems, particularly those connected with phase transitions, which we cannot solve expliddy in terms of dosed expressions for these distribution functions, and we often have recourse to mean-field approximations even today. 

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In 1.6 we saw how important was the role played in the historical development of ideas about the constitution of fluids generally, and that of the liquid-gas interface particularly, by the mean-molecular-field (or mean-field) approximation. We asserted in 3.5 (anticipating the development in the present chapter) that this approximation contains the assumption that within the range of the attractive forces of every molecule there is always found the same number of neighbouring molecules. This means that the potential energy of attraction felt by the molecules is assumed to be a constant, not varying from molecule to molecule at any one time or with time for any one molecule. 

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