# Periodic Boundary Conditions¶

The central goal of freud is the analysis of simulations performed in periodic boxes. Periodic boundary conditions are ubiquitous in simulations because they permit the simulation of quasi-infinite systems with minimal computational effort. As long as simulation systems are sufficiently large, i.e. assuming that points in the system experience correlations over length scales substantially smaller than the system length scale, periodic boundary conditions ensure that the system appears effectively infinite to all points.

In order to consistently define the geometry of a simulation system with periodic boundaries, freud defines the freud.box.Box class. The class encapsulates the concept of a triclinic simulation box in a right-handed coordinate system. Triclinic boxes are defined as parallelepipeds: three-dimensional polyhedra where every face is a parallelogram. In general, any such box can be represented by three distinct, linearly independent box vectors. Enforcing the requirement of right-handedness guarantees that the box can be represented by a matrix of the form

\begin{eqnarray*} \mathbf{h}& =& \left(\begin{array}{ccc} L_x & xy L_y & xz L_z \\ 0 & L_y & yz L_z \\ 0 & 0 & L_z \\ \end{array}\right) \end{eqnarray*}

where each column is one of the box vectors.

As such, the box is characterized by six parameters: the box vector lengths $$L_x$$, $$L_y$$, and $$L_z$$, and the tilt factors $$xy$$, $$xz$$, and $$yz$$. The tilt factors are directly related to the angles between the box vectors. All computations in freud are built around this class, ensuring that they naturally handle data from simulations conducted in non-cubic systems. There is also native support for two-dimensional (2D) systems when setting $$L_z = 0$$.

Boxes can be constructed in a variety of ways. For simple use-cases, one of the factory functions of the freud.box.Box provides the easiest possible interface:

# Make a 10x10 square box (for 2-dimensional systems).
freud.box.Box.square(10)

# Make a 10x10x10 cubic box.
freud.box.Box.cube(10)


For more complex use-cases, the freud.box.Box.from_box() method provides an interface to create boxes from any object that can easily be interpreted as a box.

# Create a 10x10 square box from a list of two items.
freud.box.Box.from_box([10, 10])

# Create a 10x10x10 cubic box from a list of three items.
freud.box.Box.from_box([10, 10, 10])

# Create a triclinic box from a list of six items (including tilt factors).
freud.box.Box.from_box([10, 5, 2, 0.1, 0.5, 0.7])

# Create a triclinic box from a dictionary.
freud.box.Box.from_box(dict(Lx=8, Ly=7, Lz=10, xy=0.5, xz=0.7, yz=0.2))

# Directly call the constructor.
freud.box.Box(Lx=8, Ly=7, Lz=10, xy=0.5, xz=0.7, yz=0.2, dimensions=3)


More examples on how boxes can be created may be found in the API documentation of the Box class.

Note

All freud boxes are centered at the origin, so for a given box the range of possible positions is $$[-L/2, L/2)$$.