The location of the zeros of the partition function for Ising models having four-, six-, or eight-body interactions are studied. For quasi-one-dimensional models the zeros are found to lie on intersecting circles in the y plane where y=exp (2 beta h). These circles are part of a family of circles which includes the unit circle of Lee and Yang. Two, three, and four circles are needed for the four-body, six-body, and eight-body interaction systems, respectively. A counter example is found to show this behaviour does not hold for all such systems.
It is shown by a rigorous development from first principles that a relatively simple form for the partition function of an imperfect Bose-Einstein gas can be obtained. This form is that of a
3N-dimensional integral (
N being the total number of particles) the integrand of which can be completely determined for any given interparticle interaction provided certain, in general small, many-body quantum mechanical effects are ignored. The final evaluation of the partition function in closed form requires that this integrand be approximated by functions sufficiently tractable that the
3N-dimensional integral can be performed.