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Williams and Holland's Law:
If enough data is collected, anything may be proven by
statistical methods.

Suppose we are analyzing the set of configurations, produced in either
MC or MD simulations. The total number of configurations is
The time average of a property *A* is then

| |
(1) |

If each were statistically independent of the others,
the variance of the mean would simply be:
| |
(2) |

and gives the estimated error in average.
But usually the data produce in MD and MC simulations are
*correlated*. As a limiting case we can assume, that our set of
configurations actually consists of blocks of identical values (Jacucci and Rahman, 1984, corresponds to
the correlation time in the MD simulations). Then
| |
(3) |

To estimate the real error bars we split our data in blocks, varying
the block length , calculate the block averages
and then estimate
the variance, using these block averages:
| |
(4) |

As the block size increases, block averages become less and less
correlated, and becomes proportional to the
number of blocks, or inversely proportional to . This means,
that the value becomes
independent of . Each *S*-th configuration of all produced
can be taken as a set of uncorrelated configurations. Thus the error
estimate should be based on the configurations, and
not itself. That is why efficient MC algorithms are
developed to produced less correlated configurations in the same total
number of MC steps.

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© 1997
* Boris Veytsman
and ** Michael Kotelyanskii
*

Tue Dec 2 20:15:24 EST 1997
*
*