Interdisciplinary Applied Mathematics

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Error Estimation

Sources of systematic error in MD include system-size dependence, possible effects of random number generators, and poor equilibration (Allen and Tildesley, 1994). These should be estimated and eliminated wherever possible. It is also essential to obtain an estimate of the statistical significance of the results. Simulation averages are taken over runs of finite length, and this is the main cause of statistical imprecision in the mean values so obtained.

1. Errors in Equilibrium Averages: Suppose that we are analyzing some simulation results that contain a total of rrun time steps, or configurations. The average of some property A is (Allen and Tildesley, 1994)

If it is assumed that each quantity A(r) is statistically independent of the others, then the variance in the mean would simply be given by

ff2((A)run) = ^2(A)/Trun,    (16.24)


1 Trun

а2(Л) = (M2)run = — E (Л(т) — (Л)гип)2.

1 run

T =1

The estimated error in the mean is given by a( (A)run). However, the data points are not usually independent, because configurations are often stored sufficiently frequently, so that those are highly correlated with each other. In those cases, the number of steps for which these correlations persist must be built into equation (16.24) (Allen and Tildesley, 1994). For example, suppose that the Trun configurations actually consist of blocks, each containing 2тд identical configurations. For large тд, this corresponds to a correlation “time” тд. Then

a2((A) run) = 2TA^2(A)/Trun-

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