Using Random numbers in modelling and simulation

Transcription

1 Using Random numbers in modelling and simulation Determination of pi by monte-carlo method Consider a unit square and the unit circle, as shown below. The radius of the circle is 1, so the area of the entire circle is π Hence the area of the circle contained within the unit square is π/4 If you scatter points at random within the unit square, approximately π/4 will fall within the circle N := 1 Number of points to consider i :=.. N 1 X i := rnd( 1) Fill two vectors with evenly distributed random numbers. Y i := rnd( 1) These represents points in the unit square Create a third vector with 1s and s depending on whether each point is inside the unit circle. Inside i := if X i ( Y i ) + < 1, 1, Inside i PiApprox := 4 i PiApprox = 3. N PiApprox π FractionalError := FractionalError =.496 % π Or we can write this as a single expression, without the need to use intermediate vectors N ( rnd( 1) + rnd( 1) ) < 1 i = 1 PiApprox := 4 PiApprox = 3.9 N Experiment with chhanging the value of N and see how the value converges to π for large values of N (you may need to wait a while...) The values of PiApprox and PiApprox are not exactly the same, why might this be? How can you test your ideas? 1/7

2 PH41 Use of random numbers in simulations This worksheet shows how we can use random numbers to simulate a system. In this case we will investigate the scattering of light passing through a medium. Our model of scattering takes place in -dimensions, to make for ease of programming and visualisation. It is possible to extend the example presented here into the third dimension - this is left as an exercise for the advanced student. The first thing we will consider is how far a photon will travel before it undergoes a collision. It turns out that the distribution of distances travelled by photons before undergoing a collision forms a decaying exponential curve. σ := Decay constant for curve CollisionDist ( x) := σ exp σ x Form of exponential decay curve ln σ Rate of Collision Collision Rate Distance You should fix the axes limits on the graph so that the x-axis goes from to some sensible value and the y axis goes from to 1 Experiment with changing the value of σ to change the mean path length. The mean path length before collision (ie where 5% of particle will have collided) is given by λmean := ln σ CollisionDist( λmean) = 1 (C) DPL 7 /7

3 PH41 You can verify this, by getting mathcad to calculate the area under the curve from x= to x=λmean λmean CollisionDist ( x) dx =.5 The deifininte integral operator is from the calculus toolbar MathCAD has a built in random number generator which will produce a vector full of random numbers with the statistics of the exponential decay curve. NPts := 1 RanVec := rexp( NPts, σ) mean( RanVec) =.499 In order to verify the mean path length is the same for our random number generator, we need a function that will count how many elements in a vector are less than a certain target number. ProportionLessThan( Vec, x) := Vec < x length( Vec) ProportionLessThan RanVec, ln =.51 σ Experiment with changing the value of σ and verify that the proportion does indeed stay constant. Exercise Using the histogram techniques we investigated previously, draw a histogram of the distribution of values from the rexp() random number generator and determine that they do indeed follow the expontial distribution. Can you plot the two curves ion the same graph? (C) DPL 7 3/7

4 PH41 Applying random numbers to problem solving We will now use the random number generator to start modelling the scattering of photons in a medium. We will start by modelling a simple -dimensional system, where our photons are confined to a plane. We will create a function which models the path of a single photon, scattering around until it leaves our region of interest ("The box"). Each time the photon scatters we will record the x- and y- coordinates where the scattering occurs, when the photon leaves the box, the function will exit and return a matrix of co-ordinates showing each point in its path that it scattered. We will keep a record of the last X- and Y- co-ordinates of the photon and also the direction it is moving in. In order to make the coding easier to follow, we will first create some functions that tell us if the photon is in the box or not BoxSize := 1 BoxSize BoxSize BoxSize MinX := MaxX := MinY := MaxY := BoxSize InBox( x, y) := ( MinX < x < MaxX) ( MinY < y < MaxY) Experiment with different values for x and y and convince yourself that InBox() works as intended. It should return a 1 for values inside the box and for values outside InBox(, ) = 1 InBox(, ) = We will also create a function that gives us the angle that the photon will scatter at. Initially we will assume isotropic scattering, that is any photon will scatter at a random angle of between -π and π regardless of the angle it started at. Later we will experiement with different forms of an-isotropic scattering, by changing this function. ScatterAngle θ := runif ( 1, π, π) We use the random number generator to return a vector of 1 number, with a distribution of between -π and π and then use the indexing operator to extract that single number. The distance our photon travels before the collision is given by using the rexp() random number generator in the same manner. λpath := 1 mean free path ln σpath := λpath PathLength( x) := rexp 1, σpath (C) DPL 7 4/7

5 PH41 Now that we have the auxialliary functions, we can put together our program. We will supply the program with the starting co-ordinates and direction of our photon and it will return a vector showing the start location and all subsequent collision points until it leaves the box. i OnePath InitX, InitY, Initθ := Index to resulting points x InitX y InitY θ Initθ while InBox( x, y) Result i, x Result i, 1 y P PathLength( ) θ ScatterAngle θ x x + P cos θ y y + P sin θ i i + 1 Result i, x Result i, 1 y Result X&Y co-ordinates of starting position Start angle Loop round while still in the box Record position now Generate random length of path and angle of scatter Do simple trig to calculate finishing position Increment result index Now out of box, so record last point and return Result vector Having created the data in MyFirstWalk, we can now plot the path taken by our photon on an X-Y Graph MyFirstWalk := OnePath(,, ) rows( MyFirstWalk) = 1 MyFirstWalk MyFirstWalk Put the cursor in the expression where MyFirstWalk is calculated and press F9, you should see a new walk calculated each time. Scroll up the worksheet and change the value to lpath to something smaller (e.g. 1) and note how the path becomes more convoluted. (C) DPL 7 5/7

6 PH41 Having created the sturcture of our program, we can now experiment with changing the Scattering distribution to something more realistic. As a first approximation, we can assume that scattering will form a normal distribution centred around the incoming direction. Given a 'scattering constant' θ we can write our ScatterAngle function as follows: θ :=. ScatterAngle θ := rnorm( 1, θ, θ) Before you use this function in your simulation, create a histogram of the scattering, plotting the range of anlges from -π to +π When you are happy with the distribution, we can incorporate the new distribution into the scattering simulation. To do this you will need to put this in your worksheet somewhere above the OnePath() program and disable evaluation of the original ScatterAngle function. Scroll down to see the change to the graph. You can press F9 repeatedly to see the different paths. It is instructive to plot the paths taken by a large number of photons on the same graph so we can see the general trend. Create a function ManyWalks(InitX,InitY,Initθ,NWalk) which will produce a matrix holding the collision points for a large number of walks. := Result OnePath( InitX, InitY, Initθ) ManyWalks InitX, InitY, Initθ, NWalk for i 1.. NWalk 1 ( ) Result stack Result, OnePath InitX, InitY, Initθ Result The stack function creates a matrix by putting one matrix on top of the other. For instance: 1 stack , = This is just to show how the stack() function works, you can always do little experiments like this in your worksheets if you are not sure exactly how a particular function will work. So in the example above, the matrix gets bigger every time around the loop as each set of points are added to it. (C) DPL 7 6/7

7 PH41 θ := deg Starting Angle ManyPaths := ManyWalks MinX + 1,, θ, 1 Format the graph as points to avoid the line from the end of one trace back to the start of the next ManyPaths ManyPaths Experiment with changing the value of σ to show how the "jet" widens out. Also change the starting angle and veify that the jet points in the appropriate direction. If you have time,.and you are feeling adventurous, you can try taking this example out into 3-dimensions. You will need to do the following steps: InBox() will now need to take x,y & z parameters. Your result matrix will have 3 columns, x, y & z You will need to keep track of angles, θ & φ in the ScatterAngle() function You will need to use spherical co-ordinates when calculating the new x,y & z You will need to use a 3D scatter plot to plot your result. Here is a result showing the track of 1 particles launched viertically into our simulation space. (C) DPL 7 7/7