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Out One Side and In the Other -- Periodic Boundary Conditions

© Adam Hughes

This week, instead of taking a look at computational efficiency and accuracy in the use of neighbor lists, we're going to take a quick aside into periodic boundary conditions. Periodic boundary conditions probably won't be a major factor in the simulation of the planets, but they are very important in more condensed systems, and they also are closely tied to the use of neighbor lists in such systems.

By necessity, computer simulations are spatially limited to some finite region. In general, "boundary conditions" refers to how interactions are handled at the outermost limits of this limited problem space. Boundary conditions can have quite a profound effect on the calculated properties seen in a bulk system.

In the case of a molecular dynamics-type simulation that employs neighbor lists, some of the pratfalls of the finite simulation space can be seen quite easily. One of them is the fact that particles close to the "edge" of the simulation space will have fewer neighbors than those in the middle of the space. Because the simulation space is usually quite small compared to reality, this is not a very good approximation. For instance, if you had a drop of water, you would expect that within a volume the size corresponding to the simulation box every water molecule would be surrounded by roughly the same number of other water molecules. Without some special boundary conditions, though, this will not occur. Enter periodic boundary conditions.

Periodic boundary conditions allow the simulation to proceed as if every simulation box is surrounded on each side by an identical box of whatever is being simulated. So, in the case of water, the simulation cell is surrounded by 6 other boxes of water, each with the same configuration. In practice, there aren't actually representations for all of these water molecules, as the required computational time would be astronomical. In effect, though, the simulated water molecules act as if they are surrounded by an infinite bath of other water molecules. This presents a much more realistic simulation environment and allows the scientists to employ other innovative techniques.

More on periodic boundary conditions and neighbor lists next week.

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