Application of geometric probability techniques to elementary particle and nuclear physics
Abstract
Geometric probability functions have been of interest for many years. As early as 1919 Deltheil [Deltheil, 1919] was studying the distribution of distances in a sphere. Others such as Hammersley [Hammersley, 1950] and Lord [Lord, 1954] followed his quest to find new methods for determining such probability functions. Recently Overhauser [Overhauser, unpublished] developed an interesting technique for determining the distribution of distances in a sphere. We invoke Overhauser's method to solve a number of interesting geometric probability problems. We determine the probability that two points are separated by a distance r in an ellipse with uniform density as well as in an ellipsoid with uniform density. These results are checked via a Monte Carlo simulations. In the case of the ellipsoid, we calculate the Coulomb energy of charged ellipsoid, and we then compare our results to the Coulomb energy determined by Feenberg [Feenberg, 1939]. Our interest returns then to spherical distributions as we determine the probability that two points are separated by a distance r in a sphere with layered density. This result is checked by calculating the Coulomb energy of a charged sphere with layered density and comparing it to the Coulomb energy calculated by Krause [Krause, unpublished]. Finally, we consider the distribution of three points in a sphere. We find the probability that points 1 and 2 are separated by a distance r12 and points 2 and 3 are separated by a distance r23 in a sphere with uniform density. This results along with a Monte Carlo simulation is then used to check our calculation of the probability that three form a triangle with sides r12, r23, and r13 in a sphere with uniform density. We surmise that these results will be useful to others in solving problems in the future.
Degree
Ph.D.
Advisors
Fischbach, Purdue University.
Subject Area
Particle physics|Mathematics|Nuclear physics
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