Periodic boundary conditions: Difference between revisions
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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.
List of periodic boundary conditions[edit]
Cubic[edit]
Orthorhombic[edit]
Parallelepiped[edit]
Truncated octahedral[edit]
Rhombic dodecahedral[edit]
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).