The Monte Carlo Method: an Introduction

Detlev Reiter

Research Centre Jülich (FZJ)D -52425 Jülichhttp://www.fz-juelich.dee-mail: d.reiter@fz-juelich.deTel.: 02461 / 61-5841

Vorlesung HHU Düsseldorf, WS 07/08 March 2008

There are two dominant methods of simulation for complex many particle systems

1) Molecular Dynamics• Solve the classical equations of motion from mechanics.• Particles interact via a given interaction potential.• Deterministic behaviour (within numerical precision).• Find temporal evolution.

2) Monte Carlo Simulation• Find mean values (expectation values) of some system components.• Random behaviour from given probability distribution laws.

The Monte Carlo technique is a very far spread technique, because it is not limited to systems of particles.

This lecture

•Brief introduction: simulation

•What is the Monte Carlo Method

•Random number generation

•Integration by Monte Carlo

Tomorrow: one (of many) particular application:

•particle transport by Monte Carlo

4

ASDEX-UPDRADE (IPP Garching)

Monte Carlo particle trajectories, ions and neutral particles

Trilateral Eureg io Cluster

TEC

Inst itu t f ü r PlasmaphysikA ssoziat ion EU RA TO M -Fo rschungszentrum Jü l ich

Basic principle of the Monte Carlo method

• The task: calculate (estimate) a number I (one number only. Not an entire functional dependence).

Historic example: A dull way to calculate – Numerically: look for an appropriate convergent series and

evaluate this approximately– by Monte Carlo: look for a stochastic model (i.e.: (p, X): probability space with random variable X)

Example: throw a needle an a sheet with equidistant parallel stripes. Distance between stripes: D, length of needle: L, L<D.

The needle experiment of Comte de Buffon, 1733(french biologist, 1707-1788)

What is the probability p, that a needle (length L), which randomly fallson a sheet, crosses one of the lines (distance D)?

First application of Monte Carlo Method

(N trials, n „hits“)

Yt =1, if crossing, Yt=0 else, then

Today:

Using a computer to generate random events:

We need to be able to generate random numbers Xwith any given probability function f(x), ora given cumulative distribution F(x) .

1) Uniformly distributed random numbers 2) General random numbers: can be obtained from a sequence of independent uniform random numbers

a b

f(x)

1/(b-a)

Random number generation

We will see next:

Any continuous distribution can be generated fromuniform random numbers on [0,1]

Any discrete distribution can be generated fromuniform random numbers on [0,1]

Hence:

Any given distribution can be generated fromuniform random numbers on [0,1]

Strategy: try to transform F to another distribution, such thatinverse of new F is explicitly known.

Example: Normal (Gaussian) distribution

Cumulative distr. function Inverse cumul. distr. fct.

best format of storing distributions for Monte Carlo applications:„Inverse cumulative distribution function F-1(x)“, x uniform [0,1]

Exercise (and most important example:)

Generate random numbers from a Gaussian.

Let X, Y two independent Gaussian random numbers.

Transform to polar coordiantes (Jacobian!) R, Φ

Sample Φ (trivial, it is uniform on 2π)Apply inversion method for R

Transform sampled Φ, R back to X, Y.This is a pair of Gaussians. (Box-Muller Method)

Exponential distribution by „inversion“

(see tomorrow)

Note:Z and 1-Z havesame distrib.

Cauchy:e.g.: naturalLine broadening

(stepwise constant, with steps at points T)

X

y=f(x)

sample x from f(x)

f(x): distribution densityenclosing rectangle

z, uniform

yuniform

Reject zAccept z, take x=z

Rejection

NEXT:

Any Monte Carlo estimate can be regarded asa mean value, i.e. an integral (or sum) over a given probability distribution, ususally in a highdimensional space (e.g. of random walks….)

Generic Monte Carlo: Integration

Hence: How does Monte Carlo integration work?

X

f(x)

I = ∫ f(x) dx

I: unknown areaknown area

x1, uniform

x2

uniform

misshit

Hit or Miss

Suggestion: try again with previous example from dull and crude Monte Carlo

Outlook: next lecture (tomorrow)

END