1549380323-Statistical Mechanics Theory and Molecular Simulation

244 Isobaric ensembles

0 2 4 6 8 10

q

0

0.5

1

1.5

P

(q

) NumericalAnalytical

0 2 4 6 8 10

L

0

0.1

0.2

0.3

0.4

0.5

P

(L

)

Fig. 5.3Position and box-length distributions for a particle moving in the one-dimensional
potential of eqn. (5.9.8).

whereηc=

∑M

j=2ηjandξc=

∑M

j=2ξj, as required by the partition function in eqn.
(5.6.6).
We begin by defining the 3×3 matrix of box momenta, denotedpg.pgis analogous
topǫin that we letpg/Wg=hh ̇ −^1 whereWgis the time-scale parameter analogous
toWin the isotropic case. Rather than repeat the full development presented for the
isotropic case, here we will simply propose a set of equations of motion that represent a
generalization of eqns. (5.9.5) for fully flexible cells and then prove that they generate
the correct distribution. A proposed set of equations of motion is (Martyna, Tobias
and Klein, 1994)

r ̇i=

pi

mi

+

pg

Wg

ri

p ̇i=F ̃i−

pg

Wg

pi−

1

Nf

Tr [pg]

Wg

pi−

pη 1

Q 1

pi

h ̇=pgh

Wg

p ̇g= det[h](P(int)−IP) +

1

Nf

∑N

i=1

p^2 i

mi

I−

pξ 1

Q′ 1

pg