7 Monte Carlo
7.1 Introduction to the Monte Carlo method
In our treatment of the equilibrium ensembles, we have, thus far, exclusively developed
and employed dynamical techniques for sampling the phase space distributions. This
choice was motivated by the natural connection between the statistical ensembles
and classical (Hamiltonian or non-Hamiltonian) mechanics. The dynamical aspect of
these approaches is, however, irrelevant for equilibrium statistical mechanics, as we
are interested only in sampling the accessible microscopic states of the ensemble.
In this chapter, we will introduce another class of sampling techniques known as
Monte Carlomethods. As the name implies, Monte Carlo techniques are based on
games of chance (driven by sequences of random numbers) which,when played many
times, yield outcomes that are the solutions to particular problems.The first use of
random methods to solve a physical problem dates back to 1930 when Enrico Fermi
(1901-1954) employed such an approach to study the propertiesof neutrons. Monte
Carlo simulations also played a central role in the Manhattan Project. It was not
until computers could be leveraged that the power of Monte Carlo methods would
be realized. In the 1950s, for example, Monte Carlo methods were carried out on
the MANIAC at Los Alamos National Laboratory in New Mexico for research on the
hydrogen bomb. Eventually, it was determined that Monte Carlo techniques constitute
a powerful suite of tools for solving statistical mechanical problems involving integrals
of very high dimension.
As a simple illustrative example, consider the evaluation of the definiteintegral
I=
∫ 1
0dx∫√ 1 −x 20dy=π
4. (7.1.1)
The resultπ/4 can be obtained straightforwardly, since this is an elementary integral.
Note that the answerπ/4 is also the ratio of the area of a circle of arbitrary radius
to the area of its circumscribed square. This fact suggests that the following game
could be used to solve the integral: Draw a square and an inscribed circle on a piece
of paper, tape the paper to a dart board, and throw darts randomly at the board.
The ratio of the number of darts that land in the circle to the numberof darts that
land anywhere in the square will, in the limit of a very large number of dart throws,
yield a good estimate of the area ratio and hence of the integral in eqn. (7.1.1).^1 In
(^1) Kalos and Whitlock (1986) suggested putting a round cake panin a square one, placing the
combination in a rain storm, and measuring the ratio of raindrops that fall in the round cake pan to
those that fall in the square one.