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Short Time Observables


To measure the numerical accuracy of the force approximation we follow the suggestion in ref. [19]. We consider the root mean square (rms) error of forces,
where tex2html_wrap_inline1409 and tex2html_wrap_inline1411 denote the approximated and the exact Coulomb force, respectively, acting on atom i at time t, and where the forces are calculated using the trajectory of the reference simulation. As an accuracy measure we take the mean tex2html_wrap_inline1417 of the rms error tex2html_wrap_inline1419 for the simulated trajectory. The rms error of FAMUSAMM tex2html_wrap_inline1419 will fluctuate in the course of a simulation, because, as is apparent from Figure 4, the accuracy of the force approximation should vary from step to step. The size of these fluctuations is measured by the standard deviation tex2html_wrap_inline1423.

Taking the point of view of numerical mathematics one might expect that tex2html_wrap_inline1417 and tex2html_wrap_inline1427 represent measures for the algorithmic noise associated with a given approximation method. However, as discussed in refs. [16] and [20] there are physical considerations contradicting that expectation. For instance, the DC-1d extrapolation scheme has been designed to guarantee optimal energy conservation and other useful properties. Correspondingly, that method has been demonstrated to entail smaller algorithmic noise in the framework of pure force extrapolation than the linear extrapolation scheme, although it exhibits larger values of tex2html_wrap_inline1417 and tex2html_wrap_inline1427 than the latter [16, 20]. It remains to be checked whether this is also the case in our framework of local Taylor expansion extrapolations. Therefore we also have to consider the size of uncontrolled algorithmic noise. In the microcanonical simulations at hand the estimation of that size is trivial. As algorithmic noise is the sole cause for energy transfer into such systems, one simply has to monitor the drift of the total energy tex2html_wrap_inline1433.

Helmut Grubmueller
Wed Apr 30 15:40:09 MET DST 1997