Basic MonteCarlo concepts

Transcription

1 Departamento de Física Atómica, Molecular y Nuclear Universidad de Sevilla Introductory tutorial to Geant4 ITA/IEav, São José dos Campos, SP, Brazil July August 1 st, 2014

2 Index 1 Probability distributions Binomial Poisson Normal

3 Index 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method

4 Index 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

5 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept Normal distribution

6 Normal distribution It is a discrete distribution which was introduced by the swiss mathematician Jacob Bernoulli ( ). Let s consider a discrete variable, which has two possible results, with probabilities p and q = 1 p. The probability to obtain x results of the first type after N measurements of the variable is P(x) = N! x!(n x)! px q (N x) with the following mean value and standard deviation x = Np σ = Npq = Np(1 p) Therefore it s a discrete distribution defined by two parameters: N and p.

7 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept Normal distribution

8 Normal distribution It was introduced by Siméon Denis Poisson ( ) during its studies on criminal matter. It describes a discrete random variable x which tallies the number of occurrences of a given type which take place during a time interval. It s the limiting case of a binomial distribution when N, p 0, being x = Np its only parameter. The Poisson distribution is discrete and is fully characterized by it s mean value x. p x (x) = x x x! e x

9 Normal distribution Let s consider the disintegration of a radiactive atomic nucleus with probability per unit time λ. Ultimately, we will resort to differential calculus, therefore being dt the differential time interval and t the total elapsed time. t = N dt N dt 0 each sampling of the random variable (disintegration: does it happen? or doesn t it happen?) corresponds to a time interval dt with probability of success (i.e. it happens) which evidently fulfills p = λ dt p 0

10 Normal distribution In these conditions ( N and p 0 ) the binomial distribution tends to the Poisson distribution (also known as Law of rare events). The condition to satisfy this statistics is that the probability of success in each sampling be small, which is obviously accomplished since each samplig corresponts to a differential time interval dt. Let p x (t) be the probability of x successes (disintegrations) in a given interval [0, t]. In this case from which it follows dp x dt which solution is p x (t + dt) = p x (t)(1 λdt) + p x 1 (t)λdt = p x(t + dt) p x (t) dt p x (t) = (λt)x e λt x! = λ [p x 1 (t) p x (t)]

11 Evidently the average number of successes is Normal distribution x = λt which leads to the the most usual representation p x (x) = x x It s straightforward to demonstrate x! e x x 2 = x 2 + x i.e. σ = x. The Poisson distribution is discrete and fully characterized by its mean value x. The rare events condition has been implicitly applied when the probability of simulataneous occurrence of two disintegrations in the same time interval dt has been neglected.

12 Normal distribution In our differential treatment this condition is met, since p = λdt 0 For this reason the Poisson distribution plays a key rôle in the simulations by the MonteCarlo method of processes in which the probability of occurrence per unit time or unit lenght is known (for instance, the radiactive disintegration and processes taking place during radiation transport through matter).

13 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept Normal distribution

14 Normal distribution The normal distribution (also called Gauss distribution) p x,σ (x) = 1 e (x x)2 2σ 2 2πσ is characterized by two parameters: the mean value x and the standard deviation σ. The Poisson distribution tends to the normal for large values of the mean value x, therefore becoming : p x (x) = x x x! e x 1 e (x x)2 2x 2πx The standard normal distribution corresponds to the situation where x = 0 and σ = 1 p estndar (x) = 1 2π e x

15 Calculations with Mathematica Normal distribution Figura: x = 5 Figura: x = 100

16 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

17 The goal is to generate values of ξ with probability density p(x), defined in the interval [x 1, x 2 ]. It can be shown that ξ x 1 p(x)dx = γ (1) where γ is uniformly in [0, 1]. Therefore, provided a random number generator is available, the problem reduces to solve the above integral equation.

18 PROOF let s define x y(x) = p(x)dx x 1 which is monotonically increasing (y (x) = p(x) > 0 ) with y(x 1 ) = 0 and y(x 2 ) = 1. It follows that y(x) = valor has a unique solution If x (a, b) y(x) [y(a), y(b)] P [a < ξ < b] = P [y(a) < γ < y(b)] Since γ is uniformly distributed: y(b) y(a) P [y(a) < γ < y(b)] = y(x 2 ) y(x 1 ) = b y(b) y(a) = p(x)dx a Therefore P [a < ξ < b] = b a p(x)dx

19 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

20 The inverse transform method may be unpractical if: The integral of p(x) cannot be solved analitically. The probability density p(x) is known grafically or is tabulated. Method: let s consider ξ defined in [x 1, x 2 ] and its probability density p(x) M 0, then the values of ξ can be sampled as follows: 1 A point Γ of coordinates (η, η ) is built from two values γ and γ of the random variable uniformly distributed in [0, 1] : η = x 1 + γ (x 2 x 1 ) η = γ M 0 2 If the point Γ is below the curve y = p(x), then ξ = η. Otherwise a new pair of values (γ, γ ) is sampled and the procedure is repeated until the above condition is fulfilled.

21 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

22 Generation of a gaussian distribution (inverse transform method) Figure: Inverse transform method. CPU time Mathematica: 103 seconds

23 Generation of a gaussian distribution (acceptance-reject method) Figure: Acceptance-rejection method. CPU time Mathematica: 10 seconds

24 2D uniform distribution 1 = R 2π P(r, φ)drdφ = R 2π Uniform distribution: p(r, φ) r drdφ p(r, φ) = C C = 1 πr 2 1 = C R 2π r drdφ 0 0 P(r, φ)drdφ = r drdφ π R 2 Figura: Polar coordinates The distribution in r: [ 2π ] P r (r)dr = P(r, φ)dφ dr = 2rdr 0 R 2 The distribution in φ: [ R ] P φ(φ)dφ = P(r, φ)dr dφ = dφ 0 2π

25 2D uniform distribution (cont.) Therefore for sampling the r and φ values, being γ a continuous uniformly distributed random variable in [0, 1] (random number generator in mathematical libraries available in all major scientific languajes, as FORTRAN and C++): r 0 φ 0 P r (r )dr = γ = 2 r R 2 P φ (φ )dφ = γ = 1 2π 0 φ 0 r dr = r 2 R 2 r = R γ dφ = φ 2π φ = 2πγ

26 Mathematica calculations Figura: Uniform 2D distribution Figura: Pseudouniform 2D distribution

27 3D uniform distribution Figura: Spherical coordinates The distribution in r: [ π ] 2π P r (r)dr = P(r, θ, φ)dθdφ dr 0 0 = 3r 2 dr R 3 The distribution in θ: [ R ] 2π P θ(θ)dr = P(r, θ, φ)drdφ dθ 0 0 = sen(θ)dθ 2 1 = R π 2π 0 0 R π 2π P(r, θ, φ)drdθdφ = p(r, θ, φ) r 2 sen(θ) drdθdφ Uniform distribution: p(r, θ, φ) = C 1 = C R π 2π r 2 sen(θ) drdθdφ C = 3 4πR 3 P(r, θ, φ)drdθdφ = 3r 2 sen(θ) drdθdφ 4π R 3 The distribution in φ: [ R ] π P φ(φ)dφ = P(r, θ, φ)drdθ dφ 0 0 = dφ 2π

28 3D uniform distribution (cont.) Therefore, for sampling the r, θ and φ values, being γ a continuous uniformly distributed random variable in [0, 1] (random number generator generador in mathematical libraries available in all major scientific languajes, as FORTRAN and C++): r 0 P r (r )dr = γ = 3 r R 3 r 2 dr = r 3 0 R 3 r = Rγ1/3 θ 0 P θ (θ )dθ = γ = 1 2 φ 0 θ 0 sen(θ )dθ = P φ (φ )dφ = γ = 1 2π 1 2(1 cos(θ)) θ = acos(2γ 1) φ 0 dφ = φ 2π φ = 2πγ

29 Mathematica calculations Figura: Uniform distribution Figura: Pseudouniform distribution

30 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

31 Theorem The MonteCarlo method Given N independent, equal, random variables: ξ 1, ξ 2,...ξ N ξ 1 = ξ 2 =... = ξ N = m ξ 1 = ξ 2 =... = ξ N = σ and being ρ their sum N N ρ N = ξ i ρ N = ξ i = Nm i=1 i=1 ρ N = Nσ for large N, ρ is a random normal variable (i.e. distributed acoording to Gauss law) with the above parameters P[a ρ N b] b a 1 p ρn (x) dx con p ρn (x) = e (x ρ N ) 2 ρ 2 N 2π ρn 2

32 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

33 It is a direct application of the Central Limit Theorem. The goal is to calculate a magnitude m (for instance, the number of neutrons which get through a given material thickness) The method consists in: 1 Take N equal and independent random variables (which ultimately are the N Monte Carlo samplings or histories). ξ 1, ξ 2,...ξ N ξ 1 = ξ 2 =... = ξ N = m 2 Calculated their mean value: ξ 1 = ξ 2 =... = ξ N = σ ξ = ρ N N N = i=1 ξi N According to the Central Limit Theorem, it follows: P[Nm 3σ N ρ N Nm + 3σ N] P[m 3 σ N ρ N N m + 3 σ N ] P[ 1 i=n ξ i m < 3 σ ] N i=1 N

34 Basically, a MonteCarlo calculation is a numerical experiment which uses random number generators to simulate the physical quantities, which are sampled according to probability distributions (Physics models)

35 Evidently, the larger the number N of samplings, the closer will be the experimental average (by MonteCarlo) to the theoretical mean value, which strictly would ask for N. In a typical application a huge number of events takes place, for instance a beam of particles with intensity p/s impinges on a a macroscopic target, which contains of the order of the Avogadro number ( ) of scattering centers. As a direct consequence of the Central Limit Theorem the quality of the Montecarlo simulation is proportional to N, since the relative error goes as σ/ N, where σ is the standad deviation; from this follows the limitation coming from the available computational power.

36 Index The MonteCarlo method 1 Probability distributions Binomial Poisson Normal 2 Generation of a continuous random variable Inverse transform method Acceptance-rejection method 3 The MonteCarlo method Central Limit Theorem Concept

37 Calculation of π The MonteCarlo method The numerical integration by geometrical (acceptance-rejection) method was at the origin of the MonteCarlo calculations. We will start with a simple geometrical example for calculating the value of π. From the figure, it s obvious that the area of the cicle within one of the quadrants is r=1 es π/4.

38 Let s imagine that we throw darts against the figure. After a high enough number of throws, the number of hits (impacts within the circle) will be approx. proportional to the area of the circle. In other words, the ratio of hits over total number of throws will be π/4. If we actually carry ouit this experiment, we woud verify that a very large number of throws (well above 1000) is needed in order to reach an acceptable estimation of π. For this very reason it s much better to carry the experiment virtually with a computer by using use a random number generator withing this very simple algorithm. The quality of the estimation will depend on the quality of the random number generator and the number of iterations (virtual throws).

39 Distribution law from Poisson statistics If Σ is the probability per unit lenght for a certain process to occur, that leads to the Poisson distribution of the number of occurrences which take place within the length interval [0, l]: p x (l) = (Σl)x e Σl x! Nevertheless, in a numerical simulation by the MonteCarlo method what we do really need is the occurrence probability of the first success within the interval [l, l + dl]: Distribution law of the lenght where I 1 (l)dl = p 0 (l) Σdl = e Σl Σdl p 0 (l) is the non-occurrence probability within the inverval [0, l]. Σdl is the occurrence probability of a single success within the interval [l, l + dl].

40 Distribution law from Poisson statistics (cont.) By applying the inverse transform method: Sampling algorithm l = 1 Σ lnγ where γ is a random variable uniformly distributed in [0, 1]. The same applies for distribution of occurrence times (for instance, in radiactive disintegration) provided that the disintegration constant (probability per unit time) is known.

41 Neutron transport The MonteCarlo method In this simple example only two processes are considered: elastic scattering absorption Therefore there are 3 possibilities after transportation through the slab: reflection (probability: p ) absorption (probability: p 0 ) transmission (probability: p + ) The macroscopic cross section is defined as the probability per unit path length of a neutron experiencing a collision. Σ = σρ, where σ is the cross section of the process and ρ is the density of targets. Figura: Neutrons through a slab Σ = 1 λ = Σ abs + Σ disp (2) being λ the mean free path of a neutron in the slab material.

42 sampling of λ λ: random variable with distribution function sampling of θ θ: random variable with distribution function: p(λ) = 1 λ e 1 λ λ = Σe Σλ By applying the inverse transform method: λ = 1 Σ lnγ P θ (θ)dr = sen(θ)dθ 2 By applying the inverse transform method: θ = acos(1 2γ) Method (after the k-th collision the neutron is in position z k ) 1 Sampling of λ: λ k = 1 Σ lnγ 2 Sampling of θ: θ k = acos(2γ 1) 3 The new position is calculated: z k+1 = z k + λ k cos(θ k ) 4 If z k+1 > h transmitted neutron N + = N If z k+1 < 0 reflected neutron N = N If 0 < z k+1 < h the neutron is still inside: If γ < Σ abs Σ N0 = N (captured neutron) Otherwise return to point 1 and repeat the process Finally: p + = N + /N ; p 0 = N 0 /N ; p = N /N.

43 Potential problem A problem may arise if after N repetitions few particles (if any) survive (bad statistics). Solution: weighting method 1 Instead of considering individual neutrons, an initial bunch made of ω 0 neutrons is created In a given interaction, (k), in average 2 ω k 1 Σ abs Σ Σabs ω k 1 Σ Σdisp ω k 1 Σ are absorbed. is are scattered. is added to N0 (absorbed particles) and the process is Σ repeated with the remaining bunch, with weight ω disp k 1 Σ, moving in the same direction. Σ All former formulae apply, susbtituting ω k+1 = ω disp k Σ as the part of the bunch which survives after each collision (ω k is the weight of the neutron. the initial weight is ω 0 = 1. In this way a trajectory never ends in an absorption: there is always a remnant, which is a fractionary quantity, obviously proportional to the transmission probability. In these circumstances, this method noticeably improves the efficiency of the simulation.