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. 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 ϕ : R n → R {displaystyle phi :mathbb {R} ^{n} o mathbb {R} } by for all m = 0, 1, 2, ... and for constants a i {displaystyle a_{i}} and b i {displaystyle b_{i}} . In molecular dynamics simulation, PBC are usually applied to calculate bulk gasses, liquids, crystals or mixtures. 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 electromangentics, PBC can be applied for different mesh types to analyze the electromagnetic properties of periodical structures. Three-dimensional PBCs are useful for approximating the behavior of macro-scale systems of gases, liquids, and solids. Three-dimensional PBCs can also be used to simulate planar surfaces, in which case two-dimensional PBCs are often more suitable. Two-dimensional PBCs for planar surfaces are also called slab boundary conditions; in this case, PBCs are used for two Cartesian coordinates (e.g., x and y), and the third coordinate (z) extends to infinity. PBCs can be used in conjunction with Ewald summation methods (e.g., the particle mesh Ewald method) to calculate electrostatic forces in the system. However, PBCs also introduce correlational artifacts that do not respect the translational invariance of the system, and requires constraints on the composition and size of the simulation box. In simulations of solid systems, the strain field arising from any inhomogeneity in the systemwill be artificially truncated and modified by the periodic boundary. Similarly, the wavelengthof sound or shock waves and phonons in the system is limited by the box size. In simulations containing ionic (Coulomb) interactions, the net electrostatic charge of the system must be zero to avoid summing to an infinite charge when PBCs are applied. In some applications it is appropriate to obtain neutrality by adding ions such as sodium or chloride (as counterions) in appropriate numbers if the molecules of interest are charged. Sometimes ions are even added to a system in which the molecules of interest are neutral, to approximate the ionic strength of the solution in which the molecules naturally appear. Maintenance of the minimum-image convention also generally requires that a spherical cutoff radius for nonbonded forces be at most half the length of one side of a cubic box. Even in electrostatically neutral systems, a net dipole moment of the unit cell can introduce a spurious bulk-surface energy, equivalent to pyroelectricity in polar crystals.