Integral Equations between Distribution Functions of Molecules

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Integral equations are derived that relate variations in the potentials of average force between molecules of a system at two different densities or activities. These permit the calculation of the change in thermodynamic properties, or of the change in the distribution of molecules in space, in a liquid or crystalline phase, if either the temperature is varied, or if the the mutual potentials between the molecules is assumed to change. The equations are in a somewhat complex, but still distinguishable, matrix form. A matrix operates on the variations in potential occurring at one activity and transforms them into those occurring at a second activity. The matrix elements are combinations of the distribution functions at the second activity, to which the transformation is made, multiplied by powers of the difference of the two activities. The matrix approaches the unit matrix in value as this activity difference approaches zero. The product of the two matrices, one which transforms from activity α to activity β, with that which transforms from activity β to α, is the unit matrix. This leads to an integral (matrix) equation between the distribution functions at any two activities. The calculation of one element of this matrix product leads to a cell type equation for the computation of the activity, or free energy, of a system in terms of the potential between a single molecule and the molecules that immediately surround it in the system. The transformation matrices have solutions, at certain values of the activity difference y, corresponding to variations in the potentials of average force, which are transformed into zero variations at the new activity. The activities at which such variations can occur are those of the phase transitions in the systems. The solutions at these special values of the activity are differences of combinations of the distribution functions in the two pure phases that can coexist at these activities.