Computational Physics - Department of Physics

342 11 Outline of the Monte Carlo Strategy

〈H〉X 1 ...Xn=

∫

···

∫

H(x 1 ,...,xn)P(x 1 ,...,xn)dx 1 ...dxn

If we want to find the expectation value of an arbitrary functionh(xi)on the domain of just
one stochastic variableXi, we must still use the joint PDFPand remember to integrate over
the total domain of allXi

〈h〉X 1 ...Xn=

∫

···

∫

h(xi)P(x 1 ,...,xn)dx 1 ...dxn (11.4)

We will now define the property of correlation, of great importance for our study of random
numbers. Let us continue with the same set ofnstochastic variables{Xi}as above. The
variables areuncorrelated(or independent) ifPmay be factorized in the following form

P(x 1 ,x 2 ,...,xn) =

n

∏

i= 1

pi(xi)

wherepi(xi)is the univariate PDF ofXi. Notice, that if allXiare uncorrelated, then the above
equation for the expectation value of the univariate functionh, eq. (11.4) reduces, nicely to
the familiar simple univariate form of eq. (11.2).
To understand the definition of independence qualitatively, consider a process ofnse-
quential events determined by the stochastic variablesXi∀i∈{ 1 , 2 ,...,n}. The PDFpi(xi)
determines the probability density that thei-th event (governed byXi) will have the outcome
xi. If the individual events are to be independent, then the joint probability density should
intuitively be just the product of the individual densities. The events receive no information
about each other. The probability to get some particular outcome of an event is independent
of whether other events are happening at all or not.

11.1.2First Illustration of the Use of Monte-Carlo Methods.

With this definition of a random variable and its associated PDF, we attempt now a clarifica-
tion of the Monte-Carlo strategy by using the evaluation of an integral as our example.
In chapter 5 we discussed standard methods for evaluating anintegral like

I=

∫ 1

0

f(x)dx≈

N

∑

i= 1

ωif(xi),

whereωiare the weights determined by the specific integration method (like Simpson’s or
Taylor’s methods) withxithe given mesh points. To give you a feeling of how we are to eval-
uate the above integral using Monte-Carlo, we employ here the crudest possible approach.
Later on we will present slightly more refined approaches. This crude approach consists in
setting all weights equal 1,ωi= 1. That corresponds to the rectangle method presented in
Eq. (5.5), displayed again here

I=

∫b

a

f(x)dx≈h

N

∑

i= 1

f(xi− 1 / 2 ),

wheref(xi− 1 / 2 )is the midpoint value offfor a given valuexi− 1 / 2. Settingh= (b−a)/Nwhere
b= 1 ,a= 0 , we can then rewrite the above integral as

I=

∫ 1

0

f(x)dx≈

1

N

N

∑

i= 1

f(xi− 1 / 2 ),