Page 14 -

Monte Carlo methods 1

Starting with this chapter, weembark ona journey into the fascinating
realms of statistical mechanics and computational physics.Weset outto 1.1 Popular games in Monaco 3
study ahost ofclassical and quantumproblems, all of valueas models 1.2 Basic sampling 27
and with numerousapplications and generalizations.Many computa- 1.3 Statistical data analysis 44
tional methods will be visited, by choice orby necessity. Notall ofthese 1.4 Computing 62
methods are, however, properly speaking, computer algorithms.Never- Exercises 77
theless, theyoften help ustackle, and understand, properties ofphysical References 79
systems.Sometimes wecan even say that computational methods give
numerically exact solutions, because fewquestions remain unanswered.
Among all the computational techniques in this book, one stands out:
the Monte Carlo method.Itstems fromthe same roots as statistical
physics itself, it is increasingly becoming part of the discipline it is meant
to study, and it is widely applied in the natural sciences, mathematics,
engineering, and even the social sciences.The Monte Carlo methodis
the first essential stop on ourjourney.
Inthe most general terms, the Monte Carlo methodis a statistical—
almost experimental—approach to computing integrals using random 1
1
positions, called samples, whose distributioniscarefully chosen.Inthis
chapter, weconcentrate onhow toobtain these samples, how to process
them in order to approximately evaluate the integral in question, and
how to get goodresults with as few samples as possible.Starting with
very simpleexample, weshall introduce to the basic sampling techniques
forcontinuous and discrete variables, and discuss the specific problems
ofhigh-dimensional integrals.Weshall also discuss the basic principles
of statistical data analysis: how to extract results from well-behaved
simulations.Weshall also spend much time discussing the simulations
where something goes wrong.
The Monte Carlo methodis extremely general,and the basic recipes
allow us—in principle—to solveany problem in statistical physics.In
practice, however, much effort has to be spent in designing algorithms
specifically geared to the problem at hand.The design principles are
introduced in the present chapter; theywill come up time and again in
the real-world settings of later parts ofthis book.

1 “Random”comes from the old French word randon (to run around); “sample” is
derived from the Latin exemplum (example).