Application of geometric probability techniques to the evaluation of interaction energies arising from a general radial potential

A formalism is developed for using geometric probability techniques to eval uate interaction energies arising from a general radial potential V(r(12)), where r(12) =\r(2)-r(1)\. The integrals that arise in calculating these en ergies can be separated into a radial piece that depends on r(12) and a non radial piece that describes the geometry of the system, including the densi ty distribution. We show that all geometric information can be encoded into a "radial density function'' G( r(12); rho(1), rho(2)), which depends on r (12) and the densities rho(1) and rho(2) of two interacting regions. G(r(12 ); rho(1), rho(2)) is calculated explicitly for several geometries and is t hen used to evaluate interaction energies for several cases of interest. Ou r results find application in elementary particle, nuclear, and atomic phys ics. (C) 1999 American Institute of Physics. [S0022-2488(99)00102-4].