Analysing the Results of a Simulation and Estimating Errors

The chi-squared test can be used to provide a quantitative estimate of the deviation of a calculated distribution from that expected Suppose that the value of some property (A) has been calculated from the simulation at regular intervals to give a total of M values. The average value of the property A is determined together with the standard deviation. The data, comprising all of the A values from the simulation, are then divided into bins such that the number of values in each bin (M,) is approximately the same. The nmnber of values that would be expected in the ith bin is  

If is large (bigger than unity) then it is unlikely that the two distributions are the same. Any significant deviations from the expected behaviour should be investigated further to try to eliminate as much of the systematic error as possible. It is good practice to vary as 

If aU sources of systematic error can be eliminated, there will still remain statistical errors. These errors are often reported as standard deviations. What we would particularly like to estimate is the error in the average value, (A). The standard deviation of the average value is calculated as follows  

As the number of steps tj, in each block increases, so it would be expected that the block averages become uncorrelated. When this is the case, then the variance of the block averages, cr ((A)[,), will become inversely proportional to f. a (A)h) is calculated as follows  

Fig 6 27 Calculating the statistical efficiency, a A plot of ti,a (A)i,/a A) against fj shotvs a steep rise before levelling off Here the property A corresponds to the pressure calculated from the molecular dynamics simulation of argon 

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