Application of the Percus-Yevick Integral Equation

The Percus-Yevick equation, adapted to the two-dimensional system of nonspherical particles, is 

The details of the derivation of the Percus-Yevick equation and the numerical procedure for its solution are highly technical and will not be presented here. [Details may be found in Appendices 9-D and 9-E and in Ben-Naim (1971c, 1972c,d).] We note, however, that each pairwise function in (6.122) depends only on three coordinates, which we can choose as follows R is the distance between the centers of the two particles, = I — RJ, and is the angle between the vector ii and the direction of Ry = Rj — Rj, measured counterclockwise. The full pair correlation function is thus a function of three variables, g(R, a2) Because of the special symmetry of the pair potential, it is clear that all of the pairwise functions, such as U, y, or g, will be invariant to a rotation of the particle 

The total density is = 0.6. The angle-average pair correlation function is defined by 

Note that because of the symmetry of the function g for rotations by 2tiI3, it is sufficient to integrate for each angle in the range 0 a 2jr/3. 

As in the previous illustrations, we use here a dimensionless unit of length for and (Twl hence, the density q is the number of particles in the corresponding unit of area. 

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