2. Experimental Method and Data Process: “Monte Carlo Method” Presentation # 1 Nafisa Tasneem CHEP,KNU. 20-03-2003 nafisa@ams1.knu.ac.kr

3. The Monte Carlo method is a method of approximately solving Mathematical and Physical a problem by generating suitable random numbers and observing that fraction of the numbers obeying some properties. It is a method which is easy to implement on a computer. It performs simulation of any process whose development is influenced by random factors, but also if the given problem involves no chance, the method enables artificial construction of a probabilistic model. The method is also useful for obtaining numerical solutions to problems which are too complicated to solve analytically. In this presentation I shall discuss briefly how to generate suitable random numbers and present an example of a MC generator which is created for calculation of decay & high energy collision processes of the elementary particles in the lowest order (tree) approximation. What is Monte Calro Method nafisa@ams1.knu.ac.kr

4. The name "Monte Carlo" comes from the city of Monte Carlo in the principality of Monaco, famous for its gambling house. The generally accepted birth date of the Monte Carlo method is 1949, when an article entitled "The Monte Carlo Method"( by N. Metropolis and S. Ulam ) appeared. The American mathematicians J. Neyman and S. Ulam are considered its originators. History of Monte Calro Method nafisa@ams1.knu.ac.kr

5. MC method can be applied to simulate any –Systems –Apparatus –Processes to gain insight in their quantitative aspects. Application of Monte Calro Method nafisa@ams1.knu.ac.kr Design of nuclear reactors: –geometry, shielding, risk-analysis Design of detectors in particle physics: –by simulating particle collisions –resolution, acceptance, disturbance Design of networks for telecom as function of load Design of operating systems of large computer systems as function of the load.

6. Why we use random numbers in MC method nafisa@ams1.knu.ac.kr A key ingredient of any MC method is a good source of randomness. Then the obvious question that will come to our mind is “What is a random number?” Is the number 2 random? Naturally, even if 2 were a random number, it would not be very useful: If we used it to simulate the rolling of a dice, the outcome would always be 2. This is why we are looking for a sequence of independent numbers with a specified distribution. In order to be truly useful, such sequences of random numbers must normally be uniformly distributed within some interval (typically [0, 1]  R). So we use random number in MC method as : Many important analytically intractable problems can be solved using Monte Carlo techniques. Many problems that can be solved by other computational methods, can be solved much more efficiently using Monte Carlo techniques. (Example - Multidimensional integration)

7. So, A random number is a number chosen from some specified distribution. A single number is not random. Only an infinite sequence can be described as being random number. As 3.141592653589793238462643………… Random number can be classified as “Truly random number” and “Pseudorandom number”. Truly random number: Truly random number sources can only be found by measuring inherently unpredictable physical processes such as background radiation, the resistively of a nonohmic electrical circuit or atmospheric noise. Whereas Pseudorandom numbers can be obtained from computer which are generated by some algebraic algorithm, which are not random in true sense. Generation of random numbers should maintain at least two steps: 1.Generated random number should be uniformly distributed between 0 and 1, rather than –1 to +1, 0 to 100, or anything else. If not, the range can be changed by a suitable linear transformation. 2. the number should indeed be (pseudo) random. Thus at least they should be approximately uniformly distributed over their range, and successive pairs of numbers should be statistically uncorrelated. Random Numbers: nafisa@ams1.knu.ac.kr

8. To generate random numbers on a computer an algorithm is needed, which can be implemented in a function. Some of those methods are Middle square method Linear congruent sequence method Middle square method: –Suppose first random number of 10 digits: 5772156649 –So, second random number: square the first number and take the 6 th – 15 th digits: 33317792380594909201 –Third random number: square the second … etc Middle Square Methods with 2 digits: Start with 99: Start with 98: Random numbers Algorithms nafisa@ams1.knu.ac.kr

9. Linear congruent sequence method: A useful and easily implemented method of generating pseudorandom sequences is the linear congruential method first suggested by Lehmer (1949). The sequence of pseudorandom numbers  X n  is given by X n+1 = (a X n + c) mod m, n  0 Where, m (modulus), a (multiplier) and c (increment) are magic integers chosen by theoretical and empirical analysis of the sequences generated. Random numbers Algorithms nafisa@ams1.knu.ac.kr

10. Every random number is determined by the previous These are not truly random numbers Series generated by a computer are not truly random With a good algorithm they can look like random numbers: pseudo-random With numbers of 10 digits not more than 10 10 different numbers: each algorithm has a maximum period If the maximum period is much larger than the number of random numbers used, there is no harm Maximum period depends on number of digits used. However,… nafisa@ams1.knu.ac.kr

11. Simple Example : Integration nafisa@ams1.knu.ac.kr

12. What are the Advantages of MC?

13. nafisa@ams1.knu.ac.kr What are the Advantages of MC?

14. Demonstration of a Monte Carlo generator: nafisa@ams1.knu.ac.kr