524 Classical time-dependent statistical mechanics
Applying the prescription in eqn. (13.5.16) using eqns. (13.5.1) and (13.5.17) gives the
three components ofrijfor a cubic box as
y′ij = L×NINT (yij/L)
xij←−xij−L∗NINT [(xij−γtyij)/L] +γtyij′
yij←−yij−y′ij
zij←−zij−L×NINT (zij/L). (13.5.18)In much the same way as occurs for Lees–Edwards boundary conditions, the same set
of images is obtained from eqns. (13.5.16) whenγt= 1 as whenγt= 0. Thus, as in the
Lees–Edwards scheme, whenγt= 1 we can reset the time-dependent elementh 21 back
to 0, thereby preventing the simulation cell from becoming overly distorted. Note that
a similar modular invariance condition exists ifh 21 is allowed to vary between−L/ 2
andL/2 rather than between 0 andL. Thus, by resettingh 21 to−L/2 whenγt= 1/2,
the box can be kept closer to cubic, thereby allowing use of a larger cutoff radius on
the short-range interactions without requiring an increase in system size. When the
diagonal elements ofhare not all identical, it can be shown that the equivalent reset
conditions for the two modular invariant forms occur whenγth 11 /h 22 = 1 and when
γth 11 /h 22 = 1/2.
13.5.2 Other types of flows
Althoughu(r) could describe any type of velocity flow field and eqns. (13.3.6) and
(13.3.7) can describe both the linear and nonlinear regimes, the formal development
of nonlinear response theory is beyond the scope of this book. Therefore, we will
restrict ourselves to examples for whichu(r) depends on linearly onr. This means
that the so-calledstrain-rate tensor, denoted∇u, is a constant dyad. We will also
assume hydrodynamic incompressibility for which∇ ·u(r) = 0. For example, the
velocity profileu(r) = (γy, 0 ,0) that describes planar Couette flow has the associated
strain-rate tensor
∇u=
0 0 0
γ 0 0
0 0 0
. (13.5.19)
Note that the equation of motion for the matrixhis generally given by
h ̇= (∇u)Th, (13.5.20)which reproduces the evolution in eqn. (13.5.1). An example of a different flow satis-
fying the above requirements is planar elongational flow described by the strain-rate
tensor
∇u=
ξ 0 0
0 ξ(b+ 1)/ 2 0
0 0 ξ(b−1)/ 2
(13.5.21)
(Ciccottiet al., 1992), whereξis the elongation rate andbis a parameter such that
0 ≤b≤1 that describes the type of planar elongational flow.