Time and size limitations

Typical MD simulations can be performed on systems containing thousands--or, perhaps, millions--of atoms, and for simulation times ranging from a few picoseconds to hundreds of nanoseconds. While these numbers are certainly respectable, it may happen to run into conditions where time and/or size limitations become important.

A simulation is ``safe'' from the point of view of its duration when the simulation time is much longer than the relaxation time of the quantities we are interested in. However, different properties have different relaxation times. In particular, systems tend to become slow and sluggish in the proximity of phase transitions, and it is not uncommon to find cases where the relaxation time of a physical property is orders of magnitude larger than times achievable by simulation.

A limited system size can also constitute a problem. In this case one has to compare the size of the MD cell with the correlation lengths of the spatial correlation functions of interest. Again, correlation lengths may increase or even diverge in proximity of phase transitions, and the results are no longer reliable when they become comparable with the box length.

This problem can be partially alleviated by a method known as finite size scaling. This consist of computing a physical property A using several box with different sizes L, and then fitting the results on a relation

$$A(L) = A_\circ + \frac{c}{L^n}$$

using $A_\circ, c, n$ as fitting parameters. $A_\circ$ then corresponds to $\lim_{L\rightarrow\infty} A(L)$, and should therefore be taken as the most reliable estimate for the ``true'' physical quantity.