1549380323-Statistical Mechanics Theory and Molecular Simulation

242 Isobaric ensembles

r ̇i=

pi

mi

+

pǫ

W

ri

p ̇i=F ̃i−

(

1 +

d

Nf

)

pǫ

W

pi−

pη 1

Q 1

pi

V ̇=dV pǫ

W

p ̇ǫ=dV(P(int)−P) +

d

Nf

∑N

i=1

p^2 i

mi

−

pξ 1

Q′ 1

pǫ

η ̇j=

pηj

Qj

, ξ ̇j=

pξj

Q′j

p ̇ηj=Gj−

pηj+1

Qj+1

pηj

p ̇ηM=GM

p ̇ξj=G′j−

pξj+1

Q′j+1

pξj

p ̇ξM=G′M, (5.9.5)

where theGjare defined in eqn. (4.11.6). Note that eqns. (5.9.5) possess two Nos ́e–
Hoover chains. One is coupled to the particles and the other to the volume. The reason
for this seemingly baroque scheme is that the particle positions and momenta move on
a considerably faster time scale than the volume. Thus, for practical applications, they
need to be thermalized independently. The volume thermostat forcesG′jare defined
in a manner analogous to the particle thermostat forces:

G′ 1 =

p^2 ǫ

2 W

−kT

G′j=

p^2 ξj− 1

Q′j− 1

−kT. (5.9.6)

Eqns. (5.9.5) are the MTK equations, which have the conserved energy

H′=

∑N

i=1

p^2 i

2 mi

+U(r 1 ,...,rN) +

p^2 ǫ

2 W

+PV

+

∑M

j=1

[

p^2 ηj

2 Qj

+

p^2 ξj

2 Q′j

+kTξj

]

+NfkTη 1 +kT

∑M

j=2

ηj. (5.9.7)

The metric factor associated with these equations is

√

g= exp(dNη 1 +η 2 +···+ηM+
ξ 1 +···ξM). With this metric and eqn. (5.9.7), it is straightforward to prove, using
the techniques of Section 4.9, that these equations do, indeed, generate the correct
isothermal-isobaric phase space distribution (see Problem 5.5). Moreover, they can