Boundary Conditions in Space and Time


Spatial Boundary Conditions

What spatial boundary conditions should we use for the simulations? Boundary conditions strongly influence structure. Some possibilities:

These are different ways of representing the system.
If a particle goes out one side of the box,
it should eventually be put back in the other side
(although sometimes one doesn't want to do this.)

How do we calculate distances and energies in periodic boundary conditions? In the minimum image convention we use the nearest distance among all the images. This implies that we must smoothly truncate the potential at L/2. Otherwise energy will not be conserved as particles switch from one image to another. See the pseudocode for force calculation in periodic boundary conditions. We will discuss next week what happens to long range potentials in PBC.

It is possible to use any regular space filling lattice (Bravais) for boundary conditions such as the truncated octahedron since it is more spherical. This is also necessary when simulating non-cubic crystal structures.

It has been found that one hundred atoms in PBC can behave like an infinite system in many respects. The corrections are order (1/N) and the coefficient can be small. One cannot calculate long distance behavior (rL/2) or at large times (the traversal time for a sound wave.) Angular momentum is no longer conserved. Also PBC can influence the stability of phases. For example, PBC favor cubic lattice structures that fit well into the box.

 

Boundary Conditions in Time

How do we start up the system? What positions and velocities do we use? In Molecular dynamics one normally works at constant energy. Remember that in classical statistical mechanics that the average kinetic energy is (3/2)kBT.

The best boundary conditions in both space and time critically depend on the physics of the system. If you make the wrong choice, the simulation may no longer be feasible!


Calendar

Sept. 11, 1998 by David Ceperley

Sept. 4, 1999 D.D. Johnson