COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

5Monte Carlo Simulations (Nonthermal)Unit I of this chapter addresses the problem of how computers generate numbers thatappear random and how we can determine how random they are. Unit II shows howto use these random numbers to simulate physical processes. In Chapter 6, “Integration,”we see show how to use these random numbers to evaluate integrals, and inChapter 15, “Thermodynamic Simulations & Feynman Quantum Path Integration,”we investigate the use of random numbers to simulate thermal processesand the fluctuations in quantum systems.5.1 Unit I. Deterministic RandomnessSome people are attracted to computing because of its deterministic nature; it’s niceto have a place in one’s life where nothing is left to chance. Barring machine errorsor undefined variables, you get the same output every time you feed your programthe same input. Nevertheless, many computer cycles are used for Monte Carlo calculationsthat at their very core include elements of chance. These are calculationsin which random numbers generated by the computer are used to simulate naturallyrandom processes, such as thermal motion or radioactive decay, or to solveequations on the average. Indeed, much of computational physics’ recognition hascome about from the ability of computers to solve previously intractable problemsusing Monte Carlo techniques.5.2 Random Sequences (Theory)We define a sequence of numbers r 1 ,r 2 ,... as random if there are no correlationsamong the numbers. Yet being random does not mean that all the numbers in thesequence are equally likely to occur. If all the numbers in a sequence are equallylikely to occur, then the sequence is said to be uniform, and the numbers can berandom as well. To illustrate, 1, 2, 3, 4, ... is uniform but probably not random.Further, it is possible to have a sequence of numbers that, in some sense, are randombut have very short-range correlations among themselves, for example,r 1 , (1 − r 1 ),r 2 , (1 − r 2 ),r 3 , (1 − r 3 ),...have short-range but not long-range correlations.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 109