1549380323-Statistical Mechanics Theory and Molecular Simulation

Integrating the MTK equations 253

explicit symmetrization of the pressure tensorPαβ(int). That is, we can simply replace

occurrences ofP
(int)
αβ in eqns. (5.10.2) with
P ̃(int)
αβ = (P

(int)

αβ +P

(int)
βα )/2. This has the
effect of ensuring thatpgandvgare symmetric matrices. If the initial conditions are
chosen such that the angular momentum of the cell is initially zero, then the cell should
not rotate. Both techniques can actually be derived using simple holonomic constraints
and Lagrange undetermined multipliers (see Problem 5.13). When thenumber of de-
grees of freedom in the cell matrix is restricted, factors ofd^2 in eqns. (5.10.3) and
(5.10.4) must be replaced by the correct number of degrees of freedom. If overall cell
rotations are eliminated, then this number isd^2 −d.
The newNPTintegrator can also be applied within the multiple time-step RESPA
framework of Section 3.11. For two time steps,δtand ∆t=nδt, the following contri-
butions to the total Liouville operator are defined as

iL 1 =

∑N

i=1

[

pi

mi

+

pǫ

W

ri

]

·

∂

∂ri

iL(fast) 2 =

∑N

i=1

[

F(fast)i −α

pǫ

W

pi

]

·

∂

∂pi

iL(slow) 2 =

∑N

i=1

F(slow)i ·

∂

∂pi

iLǫ, 1 =

pǫ

W

∂

∂ǫ

iL(fast)ǫ, 2 =G(fast)ǫ

∂

∂pǫ

iL(slow)ǫ, 2 =G(slow)ǫ

∂

∂pǫ

, (5.12.23)

where fast and slow components are designated with superscriptswith

G(fast)ǫ =α

∑

i

p^2 i

mi

+

∑N

i=1

ri·F(fast)i −dV

∂U(fast)

∂V

−dP(fast)V (5.12.24)

G(slow)ǫ =

∑N

i=1

ri·F(slow)i −dV

∂U(slow)

∂V

−dP(slow)V. (5.12.25)

The variablesP(fast)andP(slow)are external pressure components corresponding to the
fast and slow virial contributions and must be chosen such thatP=P(fast)+P(slow).
Although the subdivision of the pressure is arbitrary, a physically meaningful choice
can be made. One possibility is to perform a short calculation with a single time step
and compute the contributions to the pressure from