Periodic boundary conditions

From SklogWiki

A liquid, in the thermodynamic limit, would occupy an infinite volume. It is common experience that one can perfectly well obtain the thermodynamic properties of a material from a more modest sample. However, even a droplet has more atoms or molecules than one can possibly hope to introduce into ones computer simulation. Thus to simulate a bulk sample of liquid it is common practice to use a 'trick' known as periodic boundary conditions. If one has a cube of atoms/molecules, the molecule leaving one side enters on the diametrically opposite side. This is analogous to the arcade video game Asteriods ^[1], where one can imagine the action takes place on the surface of a torus. In general, a simulation box whose dimensions are several times the range of the interaction potential works well for equilibrium properties, although in the region of a phase transition, where long-range fluctuations play an important role, problems may arise. In confined systems periodicity is only required in some spacial dimensions.

Contents 1 List of periodic boundary conditions 1.1 Cubic 1.2 Orthorhombic 1.3 Parallelepiped 1.4 Truncated octahedral 1.5 Rhombic dodecahedral 1.6 Slab 1.7 Hexagonal prism 2 See also 3 References 4 External resources

[edit] List of periodic boundary conditions

[edit] Cubic

[edit] Orthorhombic

[edit] Parallelepiped

[edit] Truncated octahedral

^[2]

[edit] Rhombic dodecahedral

^[2]

[edit] Slab

[edit] Hexagonal prism

[edit] See also

• Binder cumulant
• Finite size scaling
• Lees-Edwards boundary conditions
• System-size dependence

[edit] References

1. ↑ play the official on-line version from Atari
2. ↑ ^{2.0 2.1} W. Smith; D. Fincham "The Ewald Sum in Truncated Octahedral and Rhombic Dodecahedral Boundary Conditions", Molecular Simulation 10 pp. 67-71 (1993)

Related reading

• M. J. Mandell "On the properties of a periodic fluid", Journal of Statistical Physics 15 pp. 299-305 (1976)
• Lawrence R. Pratt and Steven W. Haan "Effects of periodic boundary conditions on equilibrium properties of computer simulated fluids. I. Theory", Journal of Chemical Physics 74 pp. 1864- (1981)
• M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989) Section 1.5.2
• Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition pp. 32-35 (2002) ISBN 0-12-267351-4
• Phil Attard "Non-periodic boundary conditions for molecular simulations of condensed matter", Molecular Physics 104 pp. 1951-1960 (2006)

[edit] External resources

• Periodic boundary conditions in various geometries sample FORTRAN computer code from the book M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989).