Periodic Boundary Condition       Molecular simulations are used to ultimately shed light on macroscopic behavior and to obtain information that is not easily obtained from experiments. However, computers still cannot model more than a few million atoms at one time, despite the rapid advancement of computer power. These numbers are still far below the real size of most systems, which contain on the order of Avogadro's number (6.023×1023) of particles. The behavior at the system boundaries is therefore also an issue. The states of particles near the boundaries are different from those of particles more fully surrounded by other particles, because the particles surrounded by boundaries have deficient bonds. In order to model a macroscopic system in terms of a finite simulation system of N particles, the concept of periodic boundary conditions is often employed. Figure 1. Schematic idea of periodic boundary conditions.       According to periodic boundary conditions, the same boxes are replicated in one, two or three dimensions. In Fig. 1, four two-dimensional boxes are shown, where just one box corresponds to the original system. When the interactions of the vertically elliptic particle in the left-upper box are calculated, those with neighboring particles inside the dotted circle representing truncated (or cutoff) distance beyond which interactions are small enough to be neglected.       If a particle i is located at position vector r in a box, the identical particles are assumed to be located at r + (ix + jy + kz), where i, j, and k are integer numbers varied from -∞ to +∞, and x, y, and z are the vectors corresponding to the edges of the box. Some particles are considered to be far from other particles in the same box, but they may be very close to particles in adjacent boxes. Furthermore, the interactions between same particles may be calculated several times in the same or opposite directions if the periodic boundary conditions are too short. Even though the original volume and the number of particles are very small compared to macroscopic systems, using the periodic boundary conditions can help to simulate much larger systems. References Frenkel, D. and B. Smit, Understanding Molecular Simulation. 1996, San Diego: Academic Press. 28.