Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics

Michelle Parry, Ephraim Fischbach
2000 Journal of Mathematical Physics  
A number of authors have previously found the probability P n (r) that two points uniformly distributed in an n-dimensional sphere are separated by a distance r. This result greatly facilitates the calculation of self-energies of spherically symmetric matter distributions interacting by means of an arbitrary radially symmetric two-body potential. We present here the analogous results for P 2 (r;⑀) and P 3 (r;⑀) which respectively describe an ellipse and an ellipsoid whose major and minor axes
more » ... e 2a and 2b. It is shown that for ⑀ϭ(1Ϫb 2 /a 2 ) 1/2 р1, P 2 (r;⑀) and P 3 (r;⑀) can be obtained as an expansion in powers of ⑀, and our results are valid through order ⑀ 4 . As an application of these results we calculate the Coulomb energy of an ellipsoidal nucleus, and compare our result to an earlier result quoted in the literature.
doi:10.1063/1.533249 fatcat:6lxti5j42zer5ovrsvmk45vc2a