578
10 Quantum Statistics
another. We might thus be tempted to conclude that this relation then means that
(
∂V
∂P ) T is infinite: this would, however, be an incorrect deduction. The proper
interpretation of the vanishing of
∂P
∂V
T
is that the partial derivative
∂V
∂P
T
does not
exist, so that Eq. (4.2.43) does not apply to a thermal photon gas, a point that has
been established quite convincingly by Leff [9]. This conclusion is also supported
by examining expression (4.2.35), which gives N in general in terms of the grand
partition function , V , μ) as
N = k B T
∂ ln
∂μ
T ,V
.
As (T , V , μ) is effectively a function only of T and V for the thermal photon gas,
this expression relating (T , V , μ) to N also does not apply to thermal photons,
and we must therefore consider the behaviour of the number of thermal photons
more carefully.
To examine fluctuations in the number of photons in a thermal photon gas, we
return to the expression for the grand partition function (T , V ), which we may
write explicitly for the photon gas in terms of the occupation numbers, {n s }, and
photon energies, { s }, of the photon energy states {s} as
, V ) =
{n s }
e
−β
s n s s ,
(10.6.48)
from which the mean occupation numbers n s , given by
n s = −k B T
∂ ln
∂∂ s
T ,V
,
are thus obtained as
n s =
∞
n s =0
n s e
−βn s s
∞
n s =0
e
−βn s s
.
(10.6.49)
Straightforward differentiation of expression (10.6.48) for , V ) shows that the
variance in the mean occupation number of photons in photon energy state s is given
by
n 2
s − n
2
s =
1
β 2
∂ 2 ln
∂∂ 2
s
T ,V
.