Nonequilibrium molecular dynamics 519Fig. 13.7 Periodic boundary conditions under the box evolution of eqn. (13.5.1).matrixhin time so as to “shear” the fluid in a particular direction (see Fig. 13.7).
For example, if we begin with a cubic box for whichh= diag(L,L,L), then a shear
flow profile can be established by allowinghto evolve in time according to
h(t) =
L γtL 0
0 L 0
0 0 L
. (13.5.1)
This form of the box matrix allows the simulation cell to shear along thex-direction
without changing the volume (since deth(t) = deth(0)). As a practical matter, the
box must not be allowed to “roll over” indefinitely but needs to be reset periodically
using a property of the equations of motion known asmodular invariance, which
we will discuss in the next subsection. Note that when using either Lees–Edwards
boundary conditions or eqn. (13.5.1), eqn. (13.3.8) is no longer conserved because the
inter-particle distances become time dependent, which in turn causes the potential to
become time dependent.
Another subtlety associated with the nonequilibrium molecular dynamics approach
concerns the time scales that can be accessed in a typical simulation. The driving forces
used in actual experiments are orders of magnitude smaller than those that can be
routinely employed in calculations. For planar Couette flow, this means that a typi-
cal experimental shear rate might beγ= 10^2 s−^1 while the simulation might require
γ= 10^9 s−^1. Under such enormously high shear rates, it is conceivable that thebe-
havior of the system could differ significantly from the experiment. Consequently, a
careful extrapolation to the experimental limit from simulations performed at several
different shear rates is generally needed in nonequilibrium molecular dynamics calcu-
lations. Clearly, this problem does not arise in the Green–Kubo approach because of
the absence of explicit driving forces.
Finally, when using nonequilibrium molecular dynamics, we must ensure that con-
ditions of linear response theory are valid in the molecular dynamics calculation. This