Integral Equations between Distribution Functions of Molecules

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We may define distribution functions for given number, n, of molecules as being proportional to the probability density of finding n molecules in a certain configurational position, averaged over all positions of the other molecules of the system. These functions are the exponent of minus the potential of average force between molecules.. devided by kT. Mayerll derived a set of integralequations which relate variations in the potentials of average force at two different fugacities. On the other hand, Kirkwood!!} evolved an integral equation for the radial distribution function, and Born and Green3) obtained a similar equation. 'But the relations between these integral equations and Mayer's general ,variational method are apparently not known. The purpose of this paper is to attempt to clarify these relations, at the same time, to derive new integral equation which is a generalization of the Born-Green equation. We will deal with a multicomponent system, because the final equations are in no way more complicated if a system of many components is considered than if a single component system is handled.