Computational Physics

296 The Monte Carlo method

Direct Monte Carlo is a powerful method which can be applied to a wide variety
of problems inside and outside physics. There is, however, not much to be said
about this method as its implementation is as direct as the name suggests. The diffi-
culty usually resides in the modelling aspect: how to represent certain phenomena
using random numbers. The implementation of the method is then usually rather
straightforward. In the next section we shall briefly discuss MC integration. The
Metropolis sampling method will be discussed in the remainder of this chapter. The
MC techniques which will be discussed in this chapter are essential for much of
the material covered in Chapters 12 and 15 on quantum Monte Carlo methods and
lattice field theory simulations.
A general reference on MC techniques is the book by Hammersley and Hand-
scomb [1]. More detailed material concerning Metropolis Monte Carlo methods
can be found in the book by Allen and Tildesley [2], two review volumes by Binder
[ 3 , 4 ] and the books by Kalos and Whitlock [5], Binder and Heermann [6] and
Barkema and Newman [7].^2

10.2 Monte Carlo integration

Suppose we want to calculate the integral of a smooth functionfon the interval
[a,b]on the real axis:

I=

∫b

a

dxf(x). (10.1)

Standard numerical methods for this problem are discussed inAppendix A6, and
they usually boil down to calculating the function on a set of equally spaced values
xi(except for Gaussian integration, where the points are not equidistant) and then
evaluating the sum

I=

(b−a)

N

∑N

i= 1

wif(xi), (10.2)

where the weightswido not depend onf – they determine the accuracy of the
method. Usually such methods are based on polynomial approximations of the
integrand and their accuracyσis expressed in terms of a power of the separation
hof the integration points:σ∝hk∝N−k, wherekis a positive integer. In Monte
Carlo integration we also use Eq. (10.2), with the weightswiall equal to 1 but the
xinow chosenrandomly.
It will be clear that if the random coordinatesxiare homogeneously distributed
on[a,b], and ifNis sufficiently large, the sum (10.2) yields a result close to the

(^2) Lecture notes by Frenkel[8]have been helpful in writing part of this chapter.