Monte Carlo Methods in Statistical Mechanics: Foundations and ...

elevant details of apparatus (program) design, comparisons with previous
investigations, discussion of systematic errors, etc. Only if these are provided
will the results be trustworthy guides to improved theoretical understanding.
2 Dynamic Monte Carlo Methods:
General Theory
All Monte Carlo workhas the samegeneral structure: given some probabilitymeasure
on some con guration space S, wewishto generate many random samples from .
How isthis to be done?
Monte Carlo methods can be classi ed as static or dynamic. Static methods are
those that generate a sequence of statistically independent samples from the desired
probability distribution .These techniques are widely used in Monte Carlo numerical
integration in spaces of not-too-high dimension [2]. But they are unfeasible for most
applications in statistical physics andquantum eld theory,where is the Gibbs measure
of some rather complicated system (extremely many coupled degrees of freedom).
The idea of dynamic Monte Carlo methods is to invent astochastic process with
state space S having as its unique equilibrium distribution. We then simulate this
stochasticprocessonthecomputer, starting from an arbitrary initial con guration once
the system has reached equilibrium, wemeasure time averages, which converge (as the
runtimetends to in nity) to -averages. In physical terms, weareinventingastochastic
time evolution for the given system. Let us emphasize, however, that this timeevolution
need not correspond to any real \physical" dynamics: rather, the dynamics is simply a
numerical algorithm, and itistobechosen, like allnumerical algorithms, on the basis
of its computational e ciency.
In practice, the stochastic process is always taken to beaMarkov process. So our
rst order of business is to review some ofthe general theory of Markov chains. 2 For
simplicity let us assumethatthestate space S is discrete (i.e. niteorcountably in nite).
Much ofthe theory for general state space can be guessed from the discretetheory by
making the obvious replacements (sums by integrals, matrices by kernels), although the
proofs tend tobeconsiderably harder.
Loosely speaking, a Markovchain (= discrete-timeMarkov process) withstate space
S is a sequence of S-valued random variables X0X1X2::: such that successive transitions
Xt ! Xt+1 are statistically independent (\the future depends on the past only
through the present"). More precisely, aMarkov chain is speci ed by two ingredients:
2 The books of KemenyandSnell [4] and Iosifescu [5] are excellent references on the theory of Markov
chains with nite state space. At a somewhat higher mathematical level, the books of Chung [6]and
Nummelin [7] deal with the cases of countable and general state space, respectively.
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