1549380323-Statistical Mechanics Theory and Molecular Simulation

238 Isobaric ensembles

but also to promote the volume to a dynamical variable. Moreover, itleads to a force
that is used to propagate the volume. In order to make the volume dynamical, we need
to define a momentumpV conjugate to the volume and introduce a kinetic energy
p^2 V/ 2 Wterm into the Hamiltonian. Here,Wis a mass-like parameter that determines
the time scale of volume motion. Since we already know that the instantaneous pressure
estimator is−∂H/∂V, we seek a Hamiltonian and associated equations of motion that
drive the volume according to the difference between the instantaneous pressure and
the external applied pressureP. The Hamiltonian postulated by Andersen is obtained
from the standard Hamiltonian for anN-particle system by substituting eqn. (5.8.1)
for the coordinates and momenta into the Hamiltonian, adding the volume kinetic
energy and an additional termPV for the action of the imaginary “piston” driving
the volume fluctuations. Andersen’s Hamiltonian is

HA=

∑N

i=1

V−^2 /^3 π^2 i

2 mi

+U(V^1 /^3 s 1 ,...,V^1 /^3 sN) +

p^2 V

2 W

+PV. (5.8.2)

The parameterWis determined by a relation similar to eqn. (4.10.2)

W= (3N+ 1)kTτb^2 , (5.8.3)

whereτbis a time scale for the volume motion. The factor of 3N+ 1 arises because
the barostat scales allNparticles and the volume. Eqn. (5.8.2) is now used to de-
rive equations of motion for generating the isoenthalpic-isobaric ensemble. Applying
Hamilton’s equations, we obtain

s ̇i=

∂HA

∂πi

=

V−^2 /^3 πi

mi

π ̇i=−

∂HA

∂si

=−

∂U

∂(V^1 /^3 si)

V^1 /^3

V ̇ =∂HA

∂pV

=

pV

W

p ̇V=−

∂HA

∂V

=

1

3

V−^5 /^3

∑

i

πi^2

mi

−

1

3

V−^2 /^3

∑

i

∂U

∂(V^1 /^3 si)

·si−P. (5.8.4)

These equations of motion could be integrated numerically using the techniques in-
troduced in Section 3.10 to yield a trajectory in the scaled coordinates. However, it is
not always convenient to work in these coordinates, as they do notcorrespond to the
physical coordinates. Fortunately, eqns. (5.8.4) can be easily transformed back into
the original Cartesian coordinates by inverting the transformation as follows:

si=V−^1 /^3 ri

s ̇i=V−^1 /^3 r ̇i−

1

3

V−^4 /^3 V ̇ri