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Most path integral calculations work with a Metropolis rejection
algorithm (Metropolis et al., 1953), in which a Markov process is constructed in
order to generate a random walk through state space,
. A transition rule
depending on the initial state and
the final state is exploited to step from to
, which is chosen in such a way that the distribution of
converges to a given distribution . If the
transition rule is ergodic and fulfills the detailed balance

(59) 
then the probability distribution converges to an equilibrium
state satisfying,

(60) 
The transition probability
can be split
into two parts, the sampling distribution that
determines how the next trial state is selected in state
and the acceptance probability for the
particular step,

(61) 
The detailed balance can be satisfied by choosing
to be,

(62) 
One starts the MC process at any arbitrary state . Most likely
this state has only a very small probability because in thermodynamic system, is
a sharply peaked function that usually spans many orders of
magnitude. Therefore, it would be overrepresented in averages
calculated from,

(63) 
In the averages, one notices a transient behavior that eventually
reaches a regime, where it fluctuates around a steady mean. From
approximately that point on, one starts to collect statistics. For
uncorrelated measurements, the estimator for the standard deviation and the
error bars can be determined from
However, in most MC simulations the events are correlated because one
only moves a small fraction of the particles at a time. The correlation time can be shown to be estimated by

(66) 
The true statistical error considering correlations is given by

(67) 
Alternatively, it can be obtained from a blocking
analysis. There, one averages over events
before
calculating the error bar from Eq. 2.56. This error will grow
as a function of and eventually converge when the interval
is long enough that the averages can be considered to be statistically
independent. It should be noted that there can be different reasons
for correlations in MC simulations that can occur on different time
scales. In certain cases, it becomes difficult to estimate the
correlation time from Eq. 2.55 because of long correlations
that can only be determine accurately from very long series of
simulations data.
The aim of an efficient MC procedure is to decrease the error bars as
rapidly as possible for given computer time. The efficiency is defined
by,

(68) 
where is the computer time per step.
For certain applications, the sampling distribution leads to
error bars for a subset of observables that are too large. A typical example
in classical MC is the pair correlation function at small distances.
In those cases, importance sampling can be applied. One employs an
importance function to generate a Markov chain according to modified
distribution

(69) 
rather than to . In the end, one divides it out and calculates averages from

(70) 
This method will be applied to the sampling with open paths in chapter
5. It works well as long
as the modifications to the sampling distribution are not too
disruptive. Otherwise, the variance
grows or even becomes
infinite. A sufficient condition for the applicability is that
stays finite.
Next: Single Slice Moves
Up: Monte Carlo Sampling
Previous: Monte Carlo Sampling
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Burkhard Militzer
20030115