1549380323-Statistical Mechanics Theory and Molecular Simulation

Central Limit theorem 281

practice, it would take about 10^6 such dart throws to achieve a reasonable estimate of
π/4, which would try the patience of even the most avid dart player. For this reason,
it is more efficient to have the computer throw the darts. Nevertheless, this example
shows that a simple random process can be used to produce a numerical estimate of a
two-dimensional integral; no fancy sets of dynamical differential equations are needed.
In this chapter, we will discuss an important underpinning of the Monte Carlo
technique, namely the central limit theorem, and then proceed to describe a number
of commonly used Monte Carlo algorithms for evaluating high-dimensional integrals
of the type that are ubiquitous in classical equilibrium statistical mechanics.

7.2 The Central Limit theorem

The integrals that must be evaluated in equilibrium statistical mechanics are generally
of the form

I=

∫

dxφ(x)f(x), (7.2.1)

where x is ann-dimensional vector,φ(x) is an arbitrary function, andf(x) is a function
satisfying the properties of a probability distribution function, namelyf(x)≥0 and

f(x)≥ 0

∫

dxf(x) = 1. (7.2.2)

The integral in eqn. (7.2.1) represents the ensemble average of a physical observable in
equilibrium statistical mechanics. Let x 1 ,...,xMbe a set ofM n-dimensional vectors
that aresampledfromf(x). That is, the vectors x 1 ,...,xMare distributed according
tof(x), so that the probability that the vector xiis in a small region dx of the
n-dimensional space on which the vectors x 1 ,...,xM are defined isf(xi)dx. Recall
that in Section 3.8.3, we described an algorithm for sampling the Maxwell-Boltzmann
distribution, which is a particularly simple case. In general, the problem of sampling
a distributionf(x) is a nontrivial one that we will address in this chapter. For now,
however, let us assume that an algorithm exists for carrying out the sampling off(x)
and generating the vectors x 1 ,...,xM. We will establish that the simple arithmetic
average

I ̃M=^1

M

∑M

i=1

φ(xi) (7.2.3)

is anestimatorfor the integralI, meaning that

lim

M→∞

I ̃M=I. (7.2.4)

This result is guaranteed by a theorem known as thecentral limit theorem, which
we will now prove. Readers wishing to proceed immediately to the specifics of Monte
Carlo methodology can take the results in eqns. (7.2.3) and (7.2.4) asgiven and skip
to the next section.