Nonequilibrium molecular dynamics 523As shown in Appendix B, the application of periodic boundary conditions in a
cubic box of sideLproceeds as follows: Given a pair of particles with indicesiandj,
we first calculate the vector differencerij=ri−rjand then apply the minimum-image
convention
xij←−xij−L×NINT (xij/L)
yij←−yij−L×NINT (yij/L)
zij←−zij−L×NINT (zij/L). (13.5.14)Here, NINT(x) is the nearest integer function. As Fig. 13.6 (right) suggests, the appli-
cation of periodic boundary conditions in the Lees–Edwards schemerequires a modi-
fication in the minimum-image procedure along thex-direction. For a shear rateγ, a
row of boxes will be displaced with respect to a neighboring row by an amountγtLin
timet. Hence, we must replace eqn. (13.5.14) by
yij′ = L×NINT (yij/L)
xij←−xij−L∗NINT[(
xij−γty′ij)
/L
]
yij←−yij−yij′
zij←−zij−L×NINT (zij/L). (13.5.15)Note that whenγt= 1, neighboring rows come back into register with each other,
and the periodic array will produce the same set of minimum images as att= 0.
Consequently, we could resetγt= 0 at this point and continue the simulation.
As noted previously, Lees–Edwards boundary conditions are not useful for systems
with long range interactions using Ewald summation techniques, as there is no well-
defined box matrix, hence no simple definition of the reciprocal-space vectors. In such
systems, the use of a time-dependent box matrix is preferable. InAppendix B, it is
shown that for a given box matrixh, the application of periodic boundary conditions
involves the four steps:
si = h−^1 ri
sij = si−sj
sij←−sij−NINT(sij) (Minimum-image convention)
rij = hsij, (13.5.16)where the third line is a reassignment ofsijbased on the minimum-image convention.
In the time-dependent box-matrix method, the above prescription is performed using
h(n∆t) at each time step. However, as eqn. (13.5.1) suggests, eventually the box will
become long and skinny in thex-direction, causing the system to distort severely. In
order to avoid this scenario, we may use a property of the minimum-image convention
known asmodular invariance(Mundyet al., 2000). Note that the inverse of the matrix
in eqn. (13.5.1) is
h−^1 (t) =
1 /L −γt/L 0
0 1 /L 0
0 0 1 /L