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Periodic boundary conditions

The boundary conditions are periodic. This means that the area is in fact a hyper-torus: a particle going past the boundary on one side of the box results in the particle being warped back to the corresponding opposite side.

periodic4Pairing with periodic boundary conditions

The periodicity of the box applies to the pairing as well. In order to deal with pairings that may pass through the periodic boundary, the periodic box is replicated once in all six directions, producing 27 boxes. When the distance matrix is computed, all 27 combinations of a quark in the middle box, and an anti-quark in one of each of the 27 boxes is calculated. Only the one with the smallest distance will be important.

The nature of the permutation must be recorded for later use by both the force direction calculation, and by the viewer (for drawing parings). Three small integers valued take care of recording which box the partner is found in. Once the pairing is complete, the anti-quark must also be considered to be in the middle box, and its quark must be permuted in the opposite direction. Figure shows a two-dimensional slice of the boxes, and shows the direction of the force vectors.