1.3 Fluctuations
11
of the ideal quantum photon gas are related by
P =
1
3
U(T , V )
V
.
(1.2.11)
We shall see in Chap. 10 that a photon gas has an internal energy U(T , V ) given by
U(T , V ) =
4σ V
c
T
4 ,
(1.2.12)
in which σ is the Stefan-Boltzmann constant
σ ≡
2π 5 k 4
B
15c 2 h 3 5.67037 × 10
−8 W K
−4 m
−2 .
It will prove useful to write the internal energy expression (1.2.11) for U(T , V )
in terms of the internal energy density u(T ) ≡ U(T , V )/V , i.e.,
U(T , V ) = V u(T ) ,
(1.2.13)
with the internal energy density u(T ) for the photon gas given explicitly as
u(T ) =
4σ
c
T
4 .
(1.2.14)
Substitution of Eq. (1.2.12) into Eq. (1.2.11) gives the photon pressure P as
P (T ) =
4σ
3c
T
4 ,
(1.2.15)
which may be compared with Eq. (1.2.5) for the classical ideal gas, namely,
P V =
2
3 U trans (T ) = Nk B T .
1.3 Fluctuations
We begin by considering a dilute gas of N atoms, all separated by distances that
are on average much larger than the average thermal de Broglie wavelength (more
about this later). Further, we shall assume that the container plus atoms is completely
isolated from the rest of the universe and that it has been there for an extremely long
time. Following Reif [3], we shall attempt to examine what is happening in the
container by examining instantaneous snapshots of the myriad of moving particles.
For reference purposes, as illustrated in Fig. 1.3, we shall place an imaginary
partition that separates the box into two equal-volume parts. For each snapshot,