Correlation Functions The Kinematics of Order

Conventional kinematic notions are built around the idea that the system of interest can be characterized by a series of simple states without any reference to microscopic fluctuations or disorder. On the other hand, in the analysis of systems at finite temperature or in the presence of the effects of disorder, it is essential to build up measures that characterize the disposition of the system while at the same time acknowledging the possibility of an incomplete state of order within that system. Just as strain measures served as the primary kinematic engine used in our analysis of continua, the notion of a correlation function will serve to characterize the disposition of those systems that are better thought of statistically. Indeed, we will find that in problems ranging from the spins introduced earlier in conjunction 

From a structural perspective, these ideas can be couched in slightly different terms by considering the two-particle distribution function defined by 

The two-particle distribution tells us the probability that particles will be found at ri and r2 and will thereby serve as the basis for determining the energy of interaction between different pairs of particles when the best statements we can make about the system are statistical. For example, when describing the energetics of liquids our statements about the structure of that liquid will be couched in the language of distribution functions. In preparation for our later work on the free energy of liquids (see chap. 6), we find it convenient to introduce an auxilliary 

This definition simplifes considerably in the case of an isotropic liquid for which the single particle density is a constant (i.e. p(r) = N/V), resulting in the radial distribution function g(r). As said above, these definitions will form a part of our later thinking. 

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