The Monte Carlo Method: an Introduction Detlev Reiter Research Centre Jülich (FZJ) D Jülich Tel.: / 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

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, ) What is the probability p, that a needle (length L), which randomly falls on a sheet, crosses one of the lines (distance D)? First application of Monte Carlo Method (N trials, n „hits“)

Y t =1, if crossing, Y t =0 else, then

Today: Using a computer to generate random events: We need to be able to generate random numbers X with any given probability function f(x), or a 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

ab f(x) 1/(b-a) Random number generation

We will see next: Any continuous distribution can be generated from uniform random numbers on [0,1] Any discrete distribution can be generated from uniform random numbers on [0,1] Hence: Any given distribution can be generated from uniform random numbers on [0,1]

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

Example: Normal (Gaussian) distribution Cumulative distr. functionInverse 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 have same distrib.

Cauchy: e.g.: natural Line broadening

( stepwise constant, with steps at points T)

X y=f(x) sample x from f(x) f(x): distribution density enclosing rectangle z, uniform y uniform Reject z Accept z, take x=z Rejection

NEXT: Any Monte Carlo estimate can be regarded as a mean value, i.e. an integral (or sum) over a given probability distribution, ususally in a high dimensional 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 area known area x 1, uniform x 2 uniform miss hit Hit or Miss

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

Outlook: next lecture (tomorrow)

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