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monte carlo simulations (nonthermal) 121This can be integrated to obtain the time dependence of the total number of particlesand of the total activity:N(t)=N(0)e −λt = N(0)e −t/τ , (5.22)dN(t)dt= −λN(0)e −λt = dN dt e−λt (0). (5.23)We see that in this limit we obtain exponential decay, which leads to theidentification of the decay rate λ with the inverse lifetime:λ = 1 τ . (5.24)So we see from its derivation that exponential decay is a good description of naturefor a large number of particles where ∆N/N ≃ 0. The basic law of nature (5.19) isalways valid, but as we will see in the simulation, exponential decay (5.23) becomesless and less accurate as the number of particles becomes smaller and smaller.5.5.3 Decay SimulationA program for simulating radioactive decay is surprisingly simple but not withoutits subtleties. We increase time in discrete steps of ∆t, and for each time intervalwe count the number of nuclei that have decayed during that ∆t. The simulationquits when there are no nuclei left to decay. Such being the case, we have an outerloop over the time steps ∆t and an inner loop over the remaining nuclei for eachtime step. The pseudocode is simple:✞☎✝input N, lambdat=0while N> 0DeltaN = 0for i = 1..Nif ( r_i < lambda) DeltaN = DeltaN + 1end fort = t +1N=N− DeltaNOutput t , DeltaN , Nend whileWhen we pick a value for the decay rate λ =1/τ to use in our simulation, weare setting the scale for times. If the actual decay rate is λ =0.3 × 10 6 s −1 and if wedecide to measure times in units of 10 −6 s, then we will choose random numbers 0 ≤r i ≤ 1, which leads to λ values lying someplace near the middle of the range (e.g.,λ ≃ 0.3). Alternatively, we can use a value of λ =0.3 × 10 6 s −1 in our simulationand then scale the random numbers to the range 0 ≤ r i ≤ 10 6 . However, unless youplan to compare your simulation to experimental data, you do not have to worry−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 121