Ensemble average statistical error

The statistical error can thus be reduced by averaging over a larger ensemble. How well the calculated average (from eq. (16.9)) resembles the true value, however, depends on whether the ensemble is representative. If a large number of points is collected from a small part of the phase space, the property may be calculated with a small statistical error, but a large systematic error (i.e. the value may be precise, but inaccurate). As it is difficult to establish that the phase space is adequately sampled, this can be a very misleading situation, i.e. the property appears to have been calculated accurately but may in fact be significantly in error. 

Because of the high correlation between successive configurations generated by molecular simulations, an ensemble cannot be treated as a statistically random set of configurations. As a result, the calculation of statistical errors of properties determined as ensemble averages must be evaluated with a method that accounts for this correlation. This requirement applies to any property evaluated as an ensemble average, and several techniques to determine the realistic statistical error have been suggested. - ... 

After convergence has been obtained, the statistical error should be calculated in a manner that takes into account the possibility that the data obtained from successive molecular simulation steps will be highly correlated. A correlation analysis should be carried out for each ensemble average calculated. Several methods have been suggested for performing such a correlation analysis to obtain statistical errors from molecular simulations. " - i29 pj-om the statistical errors calculated for the contribution at each Xj, assuming that the ensemble averages themselves are independent, the total statistical error % is found from... 

One last note about statistically based error estimates because they are based on determining the error in an ensemble average, these estimates are not applicable at all to simulations run using the slow growth approximation, where only a single point is used to approximate the ensemble at each window. This is yet another reason why slow growth simulations are less appealing than standard TI and FEP for many applications. 

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