Shu-Ju Tu

Random distance distribution for spherical objects: general theory and applications to physics

A formalism is presented for analytically obtaining the probability density function, Pn(s), for the random distance s between two random points in an n-dimensional spherical object of radius R. Our formalism allows Pn(s) to be calculated for a spherical n-ball having an arbitrary volume density, and reproduces the well-known results for the case of uniform density. The results find applications in geometric probability, computational science, molecular biological systems, statistical physics, astrophysics, condensed matter physics, nuclear physics and elementary particle physics. As one application of these results, we propose a new statistical method derived from our formalism to study random number generators used in Monte Carlo simulations.

Geometric random inner products: A family of tests for random number generators

We present a computational scheme, GRIP (geometric random inner products), for testing the quality of random number generators. The GRIP formalism utilizes geometric probability techniques to calculate the average scalar products of random vectors distributed in geometric objects, such as circles and spheres. We show that these average scalar products define a family of geometric constants which can be used to evaluate the quality of random number generators. We explicitly apply the GRIP tests to several random number generators frequently used in Monte Carlo simulations, and demonstrate a statistical property for good random number generators.

A STUDY ON THE RANDOMNESS OF THE DIGITS OF π

We apply a newly-developed computational method, Geometric Random Inner Products (GRIP), to quantify the randomness of number sequences obtained from the decimal digits of π. Several members from the GRIP family of tests are used, and the results from π are compared to those calculated from other random number generators. These include a recent hardware generator based on an actual physical process, turbulent electroconvection. We find that the decimal digits of π are in fact good candidates for random number generators and can be used for practical scientific and engineering computations.

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(r12), where r12=|r2−r1|. The integrals that arise in calculating these energies can be separated into a radial piece that depends on r12 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(r12;ρ1,ρ2), which depends on r12 and the densities ρ1 and ρ2 of two interacting regions. G(r12;ρ1,ρ2) 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.