1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 261

Substituting eqns. (5.13.15) into eqn. (5.13.17) yields

∑

i

F(c,ik)(∆t)·

[

v(1)i +

1

mi

RFv(μ,∆t)Si(∆t)

∑

l

δμ ̃(1)l F(c,il)(∆t)

+ri(∆t)

(

v(1)ǫ +

1

W

S ̃ǫ(∆t)

∑

j

∑

l

δμ ̃(1)l rj(∆t)·F(c,jl)(∆t)

)]

= 0. (5.13.18)

As we did with eqn. (5.13.8), we can solve eqn. (5.13.18) as a full matrixequation, or
we can make the approximation of independent constraints and iterate to convergence
as in Section 3.9. When the latter procedure is used, eqn. (5.13.18) becomes

∑

i

F(c,il)(∆t)·

[

v(1)i +

1

mi

RFv(μ,∆t)Si(∆t)δμ ̃(1)l F(c,il)(∆t)

+ri(∆t)

(

vǫ(1)+

1

W

S ̃ǫ(∆t)

∑

j

δμ ̃(1)l rj(∆t)·F(c,jl)(∆t)

)]

= 0. (5.13.19)

DenotingF(c,il)·[v(1)i +vǫ(1)ri(∆t)] as ̇σl(∆t), eqn. (5.13.19) can be solved for the mul-

tiplier correctionsδ ̃μ(1)l to yieldδ ̃μ(1)l =−σ ̇l(∆t)/D, whereDis given by

D=

∑

i

1

mi

RFv(λ,∆t)Si(∆t)F(c,il)(∆t)·F(c,il)(∆t)

+

1

W

Sǫ(∆t)

[

∑

i

ri(∆t)·F

(l)

c,i(∆t)

] 2

. (5.13.20)

As in the first part of the ROLL algorithm, once a fully converged setof correction
multipliersδμ ̃(1)l is obtained, we update the pressure virial according to

P(vir)=

1

dV

∑N

i=1

[

ri·Fi+ri·

∑

k

(

μ ̃(1)k +δμ ̃(1)k

)

F(c,ik)

]

. (5.13.21)

We then apply the operators in eqn. (5.13.11) again on the velocities and thevǫ
that emerged from the first part of the ROLL procedure in order to obtain a new
set of ROLL scalars and scaling factors. We cycle through this procedure, obtaining
successively smaller correctionsδμ ̃(ln), until the ROLL scalars stop changing.

5.14 Problems