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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
122 chapter 5✞/ / Decay . java : Spontaneous decay simulationimport java . io .∗ ;import java . util .∗ ;☎public class Decay {static double lambda = 0.01; / / Decay constantstatic int max = 1000, time_max = 500 , seed = 68111; / / Paramspublic static void main ( String [] argv ) throws IOException , FileNotFoundException {int atom , time , number , nloop ;double decay ;PrintWriter w = new PrintWriter (new FileOutputStream ("decay . dat") , true );number = nloop = max ; // Initial valueRandom r = new Random( seed ) ; / / Seed number generatorfor ( time = 0; time
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COPYRIGHT 2008, PRINCET O N UNIVE R
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−101COPYRIGHT 2008, PRINCET O N U
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Copyright © 2008 by Princeton Univ
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viiicontents2.3 Experimental Error
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xcontents6.3 Experimentation 1356.4
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xiicontents9.7.1 Friction: Model an
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xivcontents12.8 Signals of Chaos: L
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xvicontents14.15.2Exercise 2: Cache
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xviiicontents18.4 Waves for Variabl
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xxcontentsC.3 DX Tools Summary 576C
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xxivprefacewhich includes understan
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2 chapter 1PhysicsCPFigure 1.1 A re
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4 chapter 1Scientific LibrariesPerf
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6 chapter 1C DWe have tried to make
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8 chapter 1possibly when installing
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10 chapter 11.4.1 Structured Progra
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12 chapter 17. Revise Area.java so
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14 chapter 1argv). Because main met
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16 chapter 1Package Class Tree Depr
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18 chapter 1Memory and storage size
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20 chapter 1TABLE 1.4The IEEE 754 S
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22 chapter 1gives 24-bit precision
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24 chapter 1TABLE 1.6Representation
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26 chapter 1The computer fetches th
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28 chapter 1Your problem is to use
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2Errors & Uncertainties in Computat
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32 chapter 2purposes, let us consid
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34 chapter 2can be avoided by combi
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36 chapter 2means that a program ru
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38 chapter 2To be more specific, le
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40 chapter 2Let us assume that an a
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42 chapter 2To see if these assumpt
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44 chapter 2computation. Accordingl
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46 chapter 3is regular practice to
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48 chapter 3to plot for x and y, we
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50 chapter 3Sample PtPlot Data file
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52 chapter 3TABLE 3.1Text Files Gra
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54 chapter 3Figure 3.4 Left: A plot
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56 chapter 3TABLE 3.2Grace Menu and
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58 chapter 33.4.1 Gnuplot Input Dat
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60 chapter 3gnuplot> set output "pl
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62 chapter 3gnuplot> set terminal e
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64 chapter 3By setting terminal to
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66 chapter 33.6 Texturing and 3-D I
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68 chapter 4R R 2RL L 2LC C 2CFigur
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172 chapter 8✞/∗ JamaFit : JAMA
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174 chapter 8( ) α β3. Consider t
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176 chapter 8discarding some inform
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178 chapter 8Lagrange interpolation
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180 chapter 8Figure 8.3 Three fits
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182 chapter 8if you have the abilit
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184 chapter 840fitNumber20dataN(t)0
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186 chapter 8theoretical curve went
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188 chapter 8This is a measure of t
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190 chapter 88.7.4 Linear Quadratic
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192 chapter 8Your problem here is t
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9Differential Equation Applications
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196 chapter 9V(x)HarmonicAnharmonic
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198 chapter 9B(t) are solutions of
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200 chapter 9This expresses the acc
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202 chapter 9derivative:dy(t)dt≃
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204 chapter 9As an example of the u
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206 chapter 9C Dhigh-precision work
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208 chapter 9Amplitude Dependence,
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210 chapter 9Here N is the normal f
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212 chapter 99.9 Unit II. Binding A
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214 chapter 9the Schrödinger equat
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216 chapter 99.11.1 Numerov Algorit
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218 chapter 9}}xl = xl0+i∗h;ul[i]
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220 chapter 9public static double d
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222 chapter 99.13 Unit III. Scatter
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224 chapter 9The equations for the
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226 chapter 9Figure 9.9 The traject
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228 chapter 9fFigure 9.10 Left: The
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230 chapter 99.17.1.1 EXPLORATION:
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232 chapter 10y(t)10Y( )10-1t-10 20
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234 chapter 10As seen in the b n co
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236 chapter 1010.4 Fourier Transfor
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238 chapter 10Regardless of the act
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240 chapter 10for the exponential a
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242 chapter 10102 4 6-1Figure 10.2
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244 chapter 10coefficients a k . If
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246 chapter 104. As always, check t
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248 chapter 10A special case of the
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250 chapter 10Figure 10.4 Input sig
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256 chapter 1010.8 Unit III. Fast F
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258 chapter 10Figure 10.9 The basic
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260 chapter 10TABLE 10.1Binary-Reve
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262 chapter 10✞/∗ FFT. java : F
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11Wavelet Analysis & Data Compressi
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266 chapter 11the second derivative
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268 chapter 1111.2.1 Wave Packet As
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270 chapter 11localized in time, su
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272 chapter 11201 Y100Signal0-10-10
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274 chapter 11you have been using f
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276 chapter 11}return Math . sin (8
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278 chapter 11FrequencyTimeFigure 1
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280 chapter 11N SamplesInputN/2(1)c
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282 chapter 11Figure 11.10 The filt
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wavelet analysis & data compression
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288 chapter 113. Reproduce the scal
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290 chapter 12discrete decay law,
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292 chapter 120.80.400 10 20Ax nn n
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294 chapter 12b. Repeat the simulat
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296 chapter 121b=1.0b=4.0b=5.0X0-1-
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298 chapter 12TABLE 12.1Several Non
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300 chapter 12✞/ / LyapLog . java
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302 chapter 1212.10 Unit II. Pendul
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306 chapter 12Figure 12.6 The data
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308 chapter 12rotating solutionsθ2
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310 chapter 12θ(t)8040.θ(0)=0.314
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322 chapter 12numbers that accounts
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324 chapter 12400200pPopulationpP00
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13Fractals & Statistical GrowthIt i
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328 chapter 1330010,000 points20010
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330 chapter 13Figure 13.2 Left: A f
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332 chapter 13copy of a frond, and
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334 chapter 13The results of this t
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336 chapter 13log N(r) = log C −
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338 chapter 13to determine the slop
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340 chapter 13✞☎import java . i
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342 chapter 13Figure 13.8 Number 8
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344 chapter 13Conway in the 1970s,
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346 chapter 13(x 0, y 1)(x 0, y 0)(
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348 chapter 13yxFigure 13.13 After
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350 chapter 13gradient // Vertical
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14High-Performance Computing Hardwa
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354 chapter 14A(1)A(2)A(3)CPUA(N)M(
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356 chapter 14ADBCFigure 14.4 Multi
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358 chapter 14as Fortran or C, tran
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360 chapter 14TABLE 14.2Computation
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362 chapter 14The processors in a p
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364 chapter 14Figure 14.7 Two views
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366 chapter 14Speedup8Amdahl's Lawp
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368 chapter 1414.11 Parallelization
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370 chapter 14The problem affects p
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372 chapter 14• A race condition
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374 chapter 14yet in order to obtai
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376 chapter 14slightly faster or sm
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378 chapter 14You see (Listing 14.1
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380 chapter 14/optimize:4/optimize:
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382 chapter 14• As indicated in
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384 chapter 1410,000Execution Time
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386 chapter 14as high-performance c
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388 chapter 14✞Dimension Vec( idi
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16Simulating Matter withMolecular D
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426 chapter 16+ +Figure 16.1 The mo
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428 chapter 16the molecules stay cl
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430 chapter 162 3 2 3 2 31 4 1 4 1
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432 chapter 16Velocity-Verlet Algor
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434 chapter 16Energy vs Timefor 36
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thermodynamic simulations & feynman
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17PDEs for Electrostatics & Heat Fl
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θpde waves: string, quantum packet
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pde waves: string, quantum packet,
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solitons & computational fluid dyna
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✞solitons & computational fluid d
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integral equations in quantum mecha
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✞integral equations in quantum me
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Appendix A: Glossaryabsolute value
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glossary 557control character — A
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glossary 559log in (on) — To sign
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glossary 561supercomputer — The c
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installing ptplot & java developer
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industrial-strength data visualizat
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Appendix D: An MPI TutorialIn this
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an mpi tutorial 595• Open a shell
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an mpi tutorial 597of all the compu
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an mpi tutorial 599TABLE D.1Some Co
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an mpi tutorial 601jobs to MPI. 2 A
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an mpi tutorial 603✞> cd $HOME Do
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an mpi tutorial 605✞/ / MPIhello
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an mpi tutorial 607Argument Namemsg
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an mpi tutorial 609D.3.4 Broadcast
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an mpi tutorial 611D.4 Parallel Tun
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an mpi tutorial 613✝}MPI_Recv ( &
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an mpi tutorial 615✞/ / Code list
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an mpi tutorial 617-1) does not sen
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an mpi tutorial 619D.6.1 Nonblockin
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an mpi tutorial 621D.9 List of MPI
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Appendix E: Calling LAPACK from CCa
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calling LAPACK from C 625E.2 Compil
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software on the CD 627JavaCodes Con
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software on the CD 629JavaCodes Con
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software on the CD 631Applets Direc
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software on the CD 633Fortran77code
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Appendix G: Compression via DWTwith
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compression via DWT with thresholdi
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compression via DWT with thresholdi
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BIBLIOGRAPHY[Abar 93] Abarbanel, H.
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ibliography 643[Erco] Ercolessi, F.
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ibliography 645[L&F 93] Landau, R.
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ibliography 647[P&W 91] Pinson, L.
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ibliography 649[VdeV 94] van de Vel
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IndexAbstract data structures,70Abs
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index 653drand, 114Driving force, 2
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index 655Libraries, see Subroutines
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index 657structured, 10for virtual
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