Periodic boundary conditions

Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a unit cell. PBCs are often used in computer simulations and mathematical models. The topology of two-dimensional PBC is equal to that of a world map of some video games; the geometry of the unit cell satisfies perfect two-dimensional tiling, and when an object passes through one side of the unit cell, it re-appears on the opposite side with the same velocity. In topological terms, the space made by two-dimensional PBCs can be thought of as being mapped onto a torus (compactification). The large systems approximated by PBCs consist of an infinite number of unit cells. In computer simulations, one of these is the original simulation box, and others are copies called images. During the simulation, only the properties of the original simulation box need to be recorded and propagated. The minimum-image convention is a common form of PBC particle bookkeeping in which each individual particle in the simulation interacts with the closest image of the remaining particles in the system.

One example of periodic boundary conditions can be defined according to smooth real functions ${\displaystyle \phi :\mathbb {R} ^{n}\to \mathbb {R} }$ by

${\displaystyle {\frac {\partial ^{m}}{\partial x_{1}^{m}}}\phi (a_{1},x_{2},...,x_{n})={\frac {\partial ^{m}}{\partial x_{1}^{m}}}\phi (b_{1},x_{2},...,x_{n}),}$

${\displaystyle {\frac {\partial ^{m}}{\partial x_{2}^{m}}}\phi (x_{1},a_{2},...,x_{n})={\frac {\partial ^{m}}{\partial x_{2}^{m}}}\phi (x_{1},b_{2},...,x_{n}),}$

${\displaystyle ...,}$

${\displaystyle {\frac {\partial ^{m}}{\partial x_{n}^{m}}}\phi (x_{1},x_{2},...,a_{n})={\frac {\partial ^{m}}{\partial x_{n}^{m}}}\phi (x_{1},x_{2},...,b_{n})}$

for all m = 0, 1, 2, ... and for constants ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$.

In molecular dynamics simulations and Monte Carlo molecular modeling, PBCs are usually applied to calculate properties of bulk gasses, liquids, crystals or mixtures.^[1] A common application uses PBC to simulate solvated macromolecules in a bath of explicit solvent. Born–von Karman boundary conditions are periodic boundary conditions for a special system.

In electromagnetics, PBC can be applied for different mesh types to analyze the electromagnetic properties of periodical structures.^[2]