246 Isobaric ensembles
p ̇i,α=Fi,α−∑
βpg,αβ
Wg
pi,β−1
dNTr [pg]
Wg
pi,α−pη 1
Q 1
pi,αh ̇αβ=∑
γpg,αγhγβ
Wgp ̇g,αβ= det(h)[
P(int)αβ −Pδαβ]
+
1
dN∑
ip^2 i
miδαβ−
pξi
Q′ 1pg,αβ. (5.10.6)Now, the compressibility is given by
κ=∑
i,α[
∂r ̇i,α
∂ri,α+
∂p ̇i,α
∂pi,α]
+
∑
α,β[
∂h ̇αβ
∂hαβ+
∂p ̇g,αβ
∂pg,αβ]
+
∑M
j=1[
∂η ̇j
∂ηj+
∂p ̇ηj
∂pηj+
∂ξ ̇j
∂ξj+
∂p ̇ξj
∂pξj]
. (5.10.7)
Carrying out the differentiation using eqns. (5.10.7) and (5.10.2), wefind that
κ=N∑
α,βpg,αβ
Wgδαβ−N∑
α,βpg,αβ
Wgδαβ−1
dNTr [pg]
WgdN−dNpη 1
Q 1−
∑M
j=2pηj
Qj+dpg,αβ
Wgδαβ−d^2pξ 1
Q′ 1−
∑M
j=2pξj
Q′j=−(1−d)Tr [pg]
Wg−dNpη 1
Q 1−d^2pξ 1
Q′ 1−
∑M
j=2[
pηj
Qj+
pξj
Q′j]
. (5.10.8)
Sincepg/Wg=hh ̇ −^1 ,
Tr [pg]
Wg
= Tr[
hh ̇ −^1]
. (5.10.9)
Using the identity det[h] = exp [Tr(lnh)], we have
d
dtdet[h] = eTr[lnh]Tr[
hh ̇ −^1]
= det[h]Tr[
hh ̇ −^1]
Tr[
hh ̇ −^1]
=
1
det[h]d
dtdet[h] =d
dtln [det(h)]. (5.10.10)Thus, the compressibility becomes
κ=−(1−d)d
dt
ln [det(h)]−dNη ̇ 1 −d^2 ξ ̇ 1 −[
η ̇c+ξ ̇c]
, (5.10.11)
which leads to the metric in eqn. (5.10.1). Assuming that eqn. (5.10.4)is the only
conservation law, then by combining the metric in eqn. (5.10.1) with eqn. (5.10.4) and
inserting these into eqn. (4.9.21) we obtain