Analysis of errors

It is often stated that a computer simulation generates ``exact'' data for a given model. However, this is true only if we can perform an infinitely long simulation, which is impossible in practice. Therefore, the results of simulations are always subject to statistical errors which have to be estimated. Besides that, finite-size effects, unreliable generator of random numbers, inaccurate potential and numerical techniques have to be taken as sources of systematic errors into account.

The main source of systematic errors could be the interatomic potential, which is in our case the Finnis-Sinclair potential. The validity of the potential is tested in our simulations, and it was found that the potential is sufficiently reliable and gives adequate description of vanadium. The predictor-corrector methods is accurate enough, provided the time step is chosen appropriately [55,59]. A random number generator proposed by Ziff et al. [60] which is used in our simulations was checked earlier and considered to be very dependable [60].

Special attention was payed to statistical errors. In order to gather better statistics, we repeated our simulations with different initial conditions several times, e.g. the distribution of the initial velocities of all atoms, and positions of point defects were changed using different seeds for the random number generator. Various correlation functions were monitored during our simulations, and corresponding characteristic decay time of these functions were estimated to make sure that equilibrium is indeed achieved. An average statistical error is calculated according to recipes of statistical analysis:

$$\Delta \overline x =\frac{s}{\sqrt{k}}$$ (4.22)

where $s$ is a standard deviation:

$$s=\sqrt{\frac{\sum\limits_{i=1}^k (x_i -\overline x)^2}{k-1}}$$ (4.23)

and $\overline x$ is a simple arithmetic mean value:

$$\overline x = \frac{1}{k} \sum_{i=1}^k x_{i}.$$ (4.24)