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 evaluate 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 energies can be separated into a radial piece that depends on r(12) and a nonradial piece that describes the geometry of the system, including the density 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 then used to evaluate interaction energies for several cases of interest. Our results find application in elementary particle, nuclear, and atomic physics. (C) 1999 American Institute of Physics. [S0022-2488(99)00102-4].