Computational modeling

Transcription

1 Computational modeling Lecture 8 : Monte Carlo Instructor : Cedric Weber Course : 4CCP1000

2 Integrals, how to calculate areas? 2

3 Integrals, calculate areas There are different ways to calculate areas

4 Integration F = b a f(x) dx. f(x) area a b x

5 Method 1 : Discretization * Discretize the horizontal axis, approximate the surface enclosed by the function f(x) by a set of rectangles * Example: f(x) = cos(x) 1.0 f(x) * width of a rectangle: * Discretization x: x = b n a * Area=sum of rectangles: x i = x 0 + i x nx 1 F n = f(x i ) x 0 π/4 π/2 x i=0

6 Method 2 : cutting and weighting v Plot the function f(x) on a piece of paper, from a to b, and from 0 to F max v Weight the full sheet of paper, we define this as Ws v Plot the function on a sheet of paper v Cut the sheet in paper along the function v Weight the sheet of paper contained between the horizontal axis and the function f(x), we define this as Wf area F max f(x) r : paper weight density Ws = (b- a) x F max x r Wf = integral x r Integral = Wf/Ws x (b- a) x F max b

7 Monte Carlo 7

8 The area of a circle, method 1 Area of circle = sum of squares contained within the circle (light gray)

9 Monte Carlo : The Casino

10 2D Integration : The area of a circle Game : children throw stones in a square, try to hit the circle. Reference: W. Krauth Introduction to Monte Carlo algorithms, Springer 1998

11 Question: * Group question: How can you calculate the area of the circle by using the game that children are playing?

12 Monte Carlo algorithm * Randomly throw stones within the squares * We count the number of hits within the circle * Hits within the circle: * Total number of trials: Ncircle Ntot * We define probabilities associated with the number of hits (as defined by Bernoulli in the last lecture, e.g. with coin flips). * Probability to hit the square : * Probability to hit the circle : Psquare = 1 Pcircle? * Pcircle =. [ Fill in ] * If the hits are uniformly distributed within the square, we also have that : * Pcircle / Psquare = Area of circle / Area of square = [fill the blank] * Question: can you use this algorithm to extract p?

13 Monte Carlo : pseudo- code 1. Program montecarlo 2. Nhits initialized to zero 3. Do i=1,ntot X a random number in [- 1,1] Y a random number in [- 1,1] If( x 2 +y 2 <1) Nhits=Nhits+1 4. End do 5. Output: Nhits 6. Probability to hit the circle : Nhits/Ntot 7. Area of circle : Nhits/Ntot x Area of square = Nhits/Ntot x 4 8. Compare with exact result, area of the circle = p 9. Output p Reference: W. Krauth Introduction to Monte Carlo algorithms, Springer 1998, cond- mat/961218

14 Periodic tiling / Random tiling N squares in total = 64 (8x8 grid) N squares contained in circles=32 Area circle / Area total = 0.5 Area total = 4 è Estimated Area circle=2 Exact Area circle= N hits = 47 N total = 64 Area circle / Area total = Nhits/Ntotal= Area total = 4 è Estimated Area circle=2.93 Exact Area circle= squares within the circle

15 Simulation - Movie Conclusion: Monte Carlo is better when Ntotal is small, periodic mesh is better when Ntotal is large

16 Why Monte Carlo? Monte Carlo is an extremely bad method; it should be used only when all alternative methods are worse. Alan Sokal The error is only shrinking as Monte Carlo methods in statistical mechanics, / p N Other simple methods for integrations in two dimensions: 1/N 3/2 In more than two dimensions (2D), e.g. in three dimensions (3D) (volume of a sphere, ): 1/N 3/D Which method is better for D=10? Answer = [ FILL IN].

17 Quizz : to test your knowledge * If the error in Monte Carlo is: and the error for a mesh method: 1/N 3/D 1/N 3/D * Question 1: What is N in this formula? * Question 2: If we do an integral in three dimension (D=3), which method is better, mesh or Monte Carlo? * Question 3: for which number of integration variables (dimension D) is Monte Carlo better?

18 How does Monte Carlo work From a large ensemble we picked up randomly a small selection out of it We did this in such a way that the chosen sample is representative of the full population For the volume of a circle : we chose uniformly points belonging to the square, uniformly chosen points are representative of all the points within the square

19 How to do an integral of a function? * We want now to integrate a function f(x)=sin 4 (x), from x=0 to x=p * How can I do this? * Question for the groups: think how you can use the same method (area of circle) but for integrating a function

20 * Step 1: Define a suitable box Example: sin 4 (x) * Step2: choose random points (a,b) within this box * Step3: check if the point (a,b) falls below the function y(x)=sin 4 (x), so if : b<sin 4 (a) * Step4: integral of the function is, number of points below the curve, divided by total number of points, times the area of the box

21 Further example:

22 Modelling Materials 22

23 Today s experiment 23

24 4.8 Some physics in there? x P (x) = e * Counting the number of balls in slices, starting from the ground * x=height, bounce distance of the ball 24

25 Boltzmann: * 1. In a system at equilbrium, the distribution of the energies of the particle is according to the Boltzman distribution * 2. If the system does not have particles, but configurations, the distribution of the possible configurations is according to the Boltzman distribution

26

27 Example: gas

28 Averages * For a gas of particles, the average velocity is: * If you do a simulation, you would simply do the sum: v average =<v>= X (snapshot of gas at time t) X (particle i) v i e mv2 i /kt

29 Averages in general * P(x) is a distribution * x are configurations of the system (snapshots of the gas, positions/velocity of all gas particle) * Problem: many (MANY!) configurations, too many to count! * Idea: Could we approximate this sum by only looking at a few 29 configuration (snapshots of the gas? )

30 Averages in general * What we want : Something we want to measure a probability, distribution Configuration 30 of gas, snapshots

31 Averages in general * What we want : X X P (x)f (x) F (x i ) x i {x} * Solution: choose a small sample of configurations: 31

32 Metropolis algorithm Does something strike you here?. Hint: title.. * N. Metropolis et al. * Equation of state calculations by fast computing machines. * The Journal of Chemical Physics, 32 21(6): , 1953.

33 Monte Carlo simulation * Do a simulation where you choose only some configurations of the system * We choose these configurations such that they represent very well what happens in reality * So we choose them randomly according to the Boltzmann distribution