# Periodic boundary conditions

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).