Computational Physics - Department of Physics

10.5 Exercises 329 if( (i-1)*(h/2.0) < X_0 ){ lambda[i] = G*H_0; // lambda dependsonlyon x }else{ lambda[i] = G*(H_0 - ((i-1) ...

330 10 Partial Differential Equations Fig. 10.6Drawing of a synapse. The axon terminal is the knoblike structure and the spine o ...

10.5 Exercises 331 u(x,t= 0 ) =Nδ(x), (10.24) whereNis the number of particle released into the synaptic cleft per area of membr ...

332 10 Partial Differential Equations u( 0 ,t) = 1 t> 0 , and u(d,t) = 0 t> 0. In this project we want to study the numeri ...

10.5 Exercises 333 c) Find the truncation errors of these three schemes and investigate their stability properties. d) Implement ...

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Part IV Monte Carlo Methods ...

We start with an introduction to the basic Monte Carlo philosophy in the first chapter. Thereafter we discuss Markov chains and ...

Chapter 11 Outline of the Monte Carlo Strategy ’Iacta Alea est’, the die is cast, is what Julius Caesar is reported by Suetonius ...

338 11 Outline of the Monte Carlo Strategy The outcomes of these random samplings, or trials, must be accumulated or tallied in ...

11.1 Introduction 339 Eleven new attempts may results in a totally different sequence of numbers and so forth. Repeating this ex ...

340 11 Outline of the Monte Carlo Strategy Also of interest to us is thecumulative probability distribution function (CDF),P(x), ...

11.1 Introduction 341 Qualitatively it represents the centroid or the average value of the PDF and is therefore often simply cal ...

342 11 Outline of the Monte Carlo Strategy 〈H〉X 1 ...Xn= ∫ ··· ∫ H(x 1 ,...,xn)P(x 1 ,...,xn)dx 1 ...dxn If we want to find the ...

11.1 Introduction 343 wherexi− 1 / 2 are the midpoint values ofx. Introducing the concept of the average of the functionffor a g ...

344 11 Outline of the Monte Carlo Strategy However, Monte Carlo integration is more efficient in higherdimensions. To see this, ...

11.1 Introduction 345 Choose the number of Monte Carlo samplesN. Perform a loop overNand for each step generate a a random numb ...

346 11 Outline of the Monte Carlo Strategy Note that we transfer the variableidumin order to initialize the random number genera ...

11.1 Introduction 347 Choose the number of particlesN. Make a loop over time, where the maximum time (or maximum number of step ...

348 11 Outline of the Monte Carlo Strategy return0; }// end main function Figure 11.1 shows the development of this system as fu ...