1549380323-Statistical Mechanics Theory and Molecular Simulation

Anisotropic cells 231

asdh 0 =V−^3 dh. In addition, det(h 0 ) = det(h)/V. Thus, substituting the cell-matrix
transformation into eqn. (5.6.4) yields

∆(N,P,T) =

1

V 0

∫∞

0

dV

∫

dhV−^3 e−βPVQ(N,h,T)δ

(

1

V

det(h)− 1

)

=

1

V 0

∫∞

0

dV

∫

dhV−^3 e−βPVQ(N,h,T)V δ(det(h)−V)

=

1

V 0

∫∞

0

dV

∫

dhV−^2 e−βPVQ(N,h,T)δ(det(h)−V), (5.6.5)

where the dependence ofQonVandh 0 has been expressed as an equivalent depen-
dence only onh. Performing the integration over the volume using theδ-function, we
obtain for the partition function

∆(N,P,T) =

1

V 0

∫

dh[det(h)]−^2 e−βPdet(h)Q(N,h,T). (5.6.6)

In an arbitrary numberdof spatial dimensions, the transformation ish=V^1 /dh 0 ,
and the partition function becomes

∆(N,P,T) =

1

V 0

∫

dh[det(h)]^1 −de−βPdet(h)Q(N,h,T). (5.6.7)

Before describing the generalization of the virial theorems of Section 5.4, we note
that the internal pressure of a canonical ensemble with a fixed cellmatrixhdescribing
an anisotropic system cannot be described by a single scalar quantity as is possible for
an isotropic system. Rather, atensoris needed; this tensor is known as thepressure
tensor,P(int). Since the Helmholtz free energyA=A(N,h,T) depends on the full cell
matrix, the pressure tensor, which is a 3×3 (or rank 2) tensor, has components given
by

Pαβ(int)=−

1

det(h)

∑^3

γ=1

hβγ

(

∂A

∂hαγ

)

N,T

, (5.6.8)

which can be expressed in terms of the canonical partition functionas

Pαβ(int)=

kT

det(h)

∑^3

γ=1

hβγ

(

∂lnQ

∂hαγ

)

N,T

. (5.6.9)

In Section 5.7, an appropriate microscopic estimator for the pressure tensor will be
derived.
If we now consider the average of the pressure tensor in the isothermal-isobaric
ensemble, atensorialversion of virial theorem can be proved for an applied isotropic
external pressureP. The average of the internal pressure tensor is

〈Pαβ(int)〉=

1

∆(N,P,T)

∫

dh[det(h)]−^2 e−βPdet(h)

kTQ(N,h,T)

det(h)

∑^3

γ=1

hβγ

(

∂lnQ

∂hαγ

)

N,T