Periodic Boundaries for Off-Lattice Simulations. Potential cut-off.

As the attractive part is relatively short-ranged, it is useful to cut-off the potential at certain distance, neglecting the contribution from other distances. (See the Homework Problem 5) The correction for the error introduced by the cut-off can be made after the simulations, as we shall see later.

We model the whole bulk system as a lattice, build from the periodic images of the ``primary'' box. This ``primary'' box, that contains simulated (filled at fig. 2) particles is surrounded by it's periodic images, obtained by its translation by the vectors $n_x \mathbf{b}_x + n_y \mathbf{b}_y + \mathbf{b}_z$, where $\mathbf{b}_x$ and $\mathbf{b}_y$ are the vectors along the box edges.

  

Figure 2: Periodic Boundary Conditions

If the potential cut-off is used, and the simulation box size is at least twice the cut-off radius, interactions between any two particles are calculated as the interactions between their closest periodic images (see fig. 2). If the coordinates of the particles within the ``primary'' box are stored in memory, every time the particle leaves it during the simulation, it reenters from the opposite side. This can be also viewed as if it's corresponding periodic image from the adjacent box enters the ``primary'' box.

When it is impossible for some reason e.g. if there are strong electrostatic interactions in the system, to use the cut-off less than the half of the simulations box size, this approach does not work and more complex algorithms has to be used. See for instance ``Computer simulations of liquids'' by M. P. Allen and D. J. Tildesley.