10.10 Fluctuations
503
dS =
1
T
dE +
p
T
dV .
(10.246)
The coefficients of the differentials should be considered functions of the extrinsic
variables E and V . The usual form of the equations of state appear as
E = Nc V T
(10.247)
pV = Nk B T
(10.248)
so
∂S
∂E
=
1
T
=
Nc V
E
(10.249)
∂S
∂V
=
p
T
=
Nk B
V
(10.250)
and
∂ 2 S
∂E 2 = −
Nc V
E 2
(10.251)
∂ 2 S
∂E∂V
= 0
(10.252)
∂ 2 S
∂V 2 = −
Nk B
V 2 .
(10.253)
From these we have
β =
Nc V
k B
E −2 0
0
NV −2
.
(10.254)
It follows from Eq. (10.235) that the simplest transformation of the variables to bring
β to a unit matrix is achieved by
a =
⎛
⎝
k B
Nc V
E 0
0
1
N V
⎞
⎠ .
(10.255)
The root-mean-square fluctuations in E will be the square root of
(δE)
2
=
l
a
2
1l
(10.256)
=
k B
Nc V
E
2 .
(10.257)
503
dS =
1
T
dE +
p
T
dV .
(10.246)
The coefficients of the differentials should be considered functions of the extrinsic
variables E and V . The usual form of the equations of state appear as
E = Nc V T
(10.247)
pV = Nk B T
(10.248)
so
∂S
∂E
=
1
T
=
Nc V
E
(10.249)
∂S
∂V
=
p
T
=
Nk B
V
(10.250)
and
∂ 2 S
∂E 2 = −
Nc V
E 2
(10.251)
∂ 2 S
∂E∂V
= 0
(10.252)
∂ 2 S
∂V 2 = −
Nk B
V 2 .
(10.253)
From these we have
β =
Nc V
k B
E −2 0
0
NV −2
.
(10.254)
It follows from Eq. (10.235) that the simplest transformation of the variables to bring
β to a unit matrix is achieved by
a =
⎛
⎝
k B
Nc V
E 0
0
1
N V
⎞
⎠ .
(10.255)
The root-mean-square fluctuations in E will be the square root of
(δE)
2
=
l
a
2
1l
(10.256)
=
k B
Nc V
E
2 .
(10.257)