tailieunhanh - Computational Physics - M. Jensen Episode 1 Part 8

Tham khảo tài liệu 'computational physics - m. jensen episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . INTRODUCTION 129 Eleven new attempts may results in a totally different sequence of numbers and so forth. Repeating this exercise the next evening will most likely never give you the same sequences. Thus we say that the outcome of this hobby of ours is truly random. Random variables are hence characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding PDF. To give you another example of possible random number spare time activities consider the radioactive decay of an I-particle from a certain nucleus. Assume that you have at your disposal a Geiger-counter which registers every say 10ms whether an I-particle reaches the counter or not. If we record a hit as 1 and no observation as zero and repeat this experiment for a long time the outcome of the experiment is also truly random. We cannot form a specific pattern from the above observations. The only possibility to say something about the outcome is given by the PDF which in this case the well-known exponential function Aexp Atr with A being proportional with the half-life. First illustration of the use of Monte-Carlo methods crude Integra. on. . With this definition of a random variable and its associated PDF we attempt now a clarification of the Monte-Carlo strategy by using the evaluation of an integral as our example. In the previous chapter we discussed standard methods for evaluating an integral like 1 f z dz . j 1 . . . where Wj are the weights determined by the specific integration method like Simpson s or Taylor s method s with .r the given mesh points. To give you a feeling of how we are to evaluate the above integral using Monte-Carlo we employ here the crudest possible approach. Later on we will present slightly more refined approaches. This crude approach consists in setting all weights equal 1 1. Recall also that dx h b a N where b 1 a 0 in our case and is the step size. We can then rewrite the above integral as 1 f x dx