A Monte Carlo Primer: A Practical Approach to Radiation by Stephen A. Dupree, Stanley K. Fraley

By Stephen A. Dupree, Stanley K. Fraley

The mathematical means of Monte Carlo, as utilized to the shipping of sub-atomic debris, has been defined in several stories and books in view that its formal improvement within the Forties. almost all these tutorial efforts were directed both on the mathematical foundation of the procedure or at its useful program as embodied within the numerous huge, formal computing device codes on hand for acting Monte Carlo shipping calculations. This ebook makes an attempt to fill what seems to be a spot during this Monte Carlo literature among the math and the software program. therefore, whereas the mathematical foundation for Monte Carlo shipping is roofed in a few aspect, emphasis is put on the applying of the strategy to the answer of sensible radiation shipping difficulties. this is often performed through the use of the computer because the simple instructing instrument. This booklet assumes the reader has a data of essential calculus, neutron delivery thought, and Fortran programming. It additionally assumes the reader has on hand a laptop with a Fortran compiler. Any notebook of average dimension may be enough to breed the examples or clear up the routines contained herein. The authors think it's important for the reader to execute those examples and workouts, and by means of doing so that you could turn into finished at getting ready acceptable software program for fixing radiation shipping difficulties utilizing Monte Carlo. The step from the software program defined during this ebook to using creation Monte Carlo codes might be straightforward.

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Unbiased results stdev = DSQRT(DABS(sumgsq/j - (sumg/j)**2)) ~(6,14)j,sumg/j,stdev,stdev/SQRT(~T(j)) ! 8) 13 IX) STOP END To use the mathematics associated with random variables, it has been convenient to consider a new random variable that is defined for a modified pdf. 92, V'(x) = V(x)f(x)/g(x). The ratio off(x)/g(x) is then the weighting factor that is applied to the random variable V to compensate for selecting from the modified pdf. This concept has direct application to the Monte Carlo solution of particle transport problems.

Biasing is an important part of Monte Carlo and will be discussed in Chapter 6. There are, from the above definitions, two funptions that may be available for selecting an event from a sample space: the cumulative distribution function and the pdf. The sampling of events uses the following information: • The sample space from which a particular event or sample is to be selected, • The value of the random variables associated with every event in the sample space (this is required so that the Monte Carlo model can deal with real numbers), • The cumulative distribution and/or the pdf for the random variables involved in the problem, and • A method of obtaining a sequence of random numbers.

A Monte Carlo Primer © Springer Science+Business Media New York 2002 22 Chapter 2 possible events accessible to the random variable is called the sample space for the random variable. The mathematical definition of a random variable is: Definition: A random variable V on a sample space U is a function from U into the set of real numbers R such that the inverse image of every interval in R is an event in U. b The argument on the random variable V, defined on the sample space U, refers to the fact that the random variable V is taken on the sample space variable u in U.

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