# Difference between revisions of "Periodic boundary conditions"

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− | + | 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 techniques | 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 <ref>[http://www.atari.com/arcade/asteroids#!/arcade/asteroids/play play the official on-line version from Atari]</ref>, 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 transitions |phase transition]], where long-range fluctuations play an important role, problems may arise. In [[confined systems]] periodicity is only required in some spacial dimensions. | |

− | *[[ | + | ==List of periodic boundary conditions== |

− | *[[ | + | ====Cubic==== |

− | *[ | + | ====Orthorhombic==== |

− | *[ | + | ====Parallelepiped==== |

− | *[ | + | ====Truncated octahedral==== |

+ | <ref name="multiple1">[http://dx.doi.org/10.1080/08927029308022499 W. Smith; D. Fincham "The Ewald Sum in Truncated Octahedral and Rhombic Dodecahedral Boundary Conditions", Molecular Simulation '''10''' pp. 67-71 (1993)]</ref> | ||

+ | ====Rhombic dodecahedral==== | ||

+ | <ref name="multiple1"></ref> | ||

+ | ====Slab==== | ||

+ | ====Hexagonal prism==== | ||

+ | ==See also== | ||

+ | *[[Binder cumulant]] | ||

+ | *[[Finite size scaling]] | ||

+ | *[[Lees-Edwards boundary conditions]] | ||

+ | *[[System-size dependence]] | ||

+ | ==References== | ||

+ | <references/> | ||

+ | '''Related reading''' | ||

+ | *[http://dx.doi.org/10.1007/BF01023055 M. J. Mandell "On the properties of a periodic fluid", Journal of Statistical Physics '''15''' pp. 299-305 (1976)] | ||

+ | *[http://dx.doi.org/10.1063/1.441276 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)] | ||

+ | *[http://www.oup.com/uk/catalogue/?ci=9780198556459 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 | ||

+ | *[http://dx.doi.org/10.1080/00268970600744768 Phil Attard "Non-periodic boundary conditions for molecular simulations of condensed matter", Molecular Physics '''104''' pp. 1951-1960 (2006)] | ||

+ | *[http://dx.doi.org/10.1063/1.4916294 Dhairyashil Ghatage, Gaurav Tomar and Ratnesh K. Shukla "Soft-spring wall based non-periodic boundary conditions for non-equilibrium molecular dynamics of dense fluids", Journal of Chemical Physics '''142''' 124108 (2015)] | ||

+ | |||

+ | ==External resources== | ||

+ | *[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.01 Periodic boundary conditions in various geometries] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)]. | ||

+ | [[category: Computer simulation techniques]] |

## Latest revision as of 13:27, 18 December 2017

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

## List of periodic boundary conditions[edit]

#### Cubic[edit]

#### Orthorhombic[edit]

#### Parallelepiped[edit]

#### Truncated octahedral[edit]

^{[2]}

#### Rhombic dodecahedral[edit]

^{[2]}

#### Slab[edit]

#### Hexagonal prism[edit]

## See also[edit]

## References[edit]

**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) - Dhairyashil Ghatage, Gaurav Tomar and Ratnesh K. Shukla "Soft-spring wall based non-periodic boundary conditions for non-equilibrium molecular dynamics of dense fluids", Journal of Chemical Physics
**142**124108 (2015)

## External resources[edit]

- 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).