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monte carlo simulations (nonthermal) 1195. Check the validity of the assumptions made in deriving the theoretical result(5.17) by checking how well〈∆x i ∆x j≠i 〉R 2 ≃ 〈∆x i∆y j 〉R 2 ≃ 0.Do your checking for both a single (long) run and for the average over trials.6. Plot the root-mean-square distance R rms = √ 〈R 2 (N)〉 as a function of √ N.Values of N should start with a small number, where R ≃ √ N is not expectedto be accurate, and end at a quite large value, where two or three places ofaccuracy should be expected on the average.5.5 Radioactive Decay (Problem)Your problem is to simulate how a small number N of radioactive particles decay. 3In particular, you are to determine when radioactive decay looks like exponentialdecay and when it looks stochastic (containing elements of chance). Becausethe exponential decay law is a large-number approximation to a natural processthat always ends with small numbers, our simulation should be closer to naturethan is the exponential decay law (Figure 5.5). In fact, if you go to the CD and“listen” to the output of the decay simulation code, what you will hear soundsvery much like a Geiger counter, a convincing demonstration of the realism of thesimulation.Spontaneous decay is a natural process in which a particle, with no externalstimulation, decays into other particles. Even though the probability of decay ofany one particle in any time interval is constant, just when it decays is a randomevent. Because the exact moment when any one particle decays is random, it doesnot matter how long the particle has been around or whether some other particleshave decayed. In other words, the probability P of any one particle decaying perunit time interval is a constant, and when that particle decays, it is gone forever.Of course, as the total number of particles decreases with time, so will the numberof decays, but the probability of any one particle decaying in some time interval isalways the same constant as long as that particle exists.C D5.5.1 Discrete Decay (Model)Imagine having a sample of N(t) radioactive nuclei at time t (Figure 5.5 inset).Let ∆N be the number of particles that decay in some small time interval ∆t. Weconvert the statement “the probability P of any one particle decaying per unit timeis a constant” into the equation3 Spontaneous decay is also discussed in Chapter 8, “Solving Systems of Equations withMatrices; Data Fitting,” where we fit an exponential function to a decay spectrum.−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 119