An Introduction to Monte Carlo Methods

摘要

Monte Carlo computer programming is becoming increasingly popular to those who use it, due to the ease with which complex problems may be formulated and solved. However, the growth of MC programming for small projects is inhibited by a frequent misconception of difficulty, inferred from the high level of complexity of problems solved in High Energy and Nuclear Physics using MC methods. In addition, few students of science and engineering are receiving exposure to the basic issues involved in the Monte Carlo process despite the ease with which MC can be used to solve classical physics problems, especially those problems with little symmetry or unusual geometry. Few upper-division or graduate students have begun to exploit this approach, even in research projects. Thus, an introduction to Monte Carlo methods would be valuable, even for the beginning science or engineering student. The present work introduces integration of area and volume, then expands this effort to include surface and volume integrals of scalar and vector functions. Next, integration over unusual geometries introduces programs which convert the geometries defined by CAD (Computer Aided Design) to geometries convenient to the Monte Carlo process. Finally, Gauss's Law uses MC to calculate the size of an asymmetrically positioned charge and a classic example from Sir Isaac Newton uses MC to calculate the effect of a spherically symmetric shell of mass on an exterior field point where the average force components (F_x, F_y, F_z) are calculated. These final examples introduce singularities and convergence problems arising in the Monte Carlo averaging process.