EQUATION OF STATE
Consider elementary cell in a phase space with a volume
Δ
x
Δ
y
Δ
z
Δ
p
x
Δ
p
y
Δ
p
z
=
h
3
,
(st
.
1)
where
h
= 6
.
63
×
10

27
erg s is the Planck constant, Δ
x
Δ
y
Δ
z
is volume in ordinary space measured
in cm
3
, and Δ
p
x
Δ
p
y
Δ
p
z
is volume in momentum space measured in ( g cm s

1
)
3
. According to
quantum mechanics there is enough room for approximately one particle of any kind within any
elementary cell. More precisely, an average number of particles per cell is given as
n
av
=
g
e
(
E

μ
)
/kT
±
1
,
(st
.
2)
with a ”+” sign for fermions, and a ”
−
” sign for bosons. The corresponding distributions are called
FermiDirac and BoseEinstein, respectively. Particles with a spin of 1
/
2 are called fermions, while
those with a spin 0
,
1
,
2
...
are called bosons. Electrons and protons are fermions, photons are bosons,
while larger nuclei or atoms may be either fermions or bosons, depending on the total spin of such
a composite particle. In the equation (st.2)
E
is the particle energy,
k
= 1
.
38
×
10

16
erg K

1
is
the Boltzman constant,
T
is temperature,
μ
is chemical potential, and
g
is a number of different
quantum states a particle may have within the cell. The meaning of temperature is obvious, while
chemical potential will become more familiar later on. In most cases it will be close to the rest mass
of a particle under consideration. If there are antiparticles present in equilibrium with particles,
and particles have chemical potential
μ
then antiparticles have chemical potential
μ
−
2
m
, where
m
is their rest mass.
For free particles their energy is a function of their momentum only, with the total momentum
p
given as
p
2
=
p
2
x
+
p
2
y
+
p
2
z
.
(st
.
3)
The number density of particle in a unit volume of 1 cm
3
, with momenta between
p
and
p
+
dp
is
given as
n
(
p
)
dp
=
g
e
(
E

μ
)
/kT
±
1
4
πp
2
h
3
dp,
(st
.
4)
because the number of elementary cells within 1 cm
3
and a momentum between
p
and
p
+
dp
, i.e.
within a spherical shell with a surface 4
πp
2
and thickness
dp
is equal to 4
πp
2
dp h

3
. The number
density of particles with all momenta, contained within 1 cm
3
is
n
=
∞
integraldisplay
0
n
(
p
)
dp,
ρ
=
nm
(st
.
5)
where
ρ
is the mass density of gas. Note, that if we know density and temperature, then we can
calculate chemical potential with the eqs. (st.4) and (st.5), provided we know how particle energy
depends on its momentum, i.e. the function
E
(
p
) is known.
In the equation (st.4)
g, k, T, μ, π, h
are all constant, and the energy
E
depends on the momentum
p
only. As our particles may be relativistic as well as nonrelativistic, we have to use a general formula
for the relation between
E
and
p
. We have
E
≡
E
total
=
E
0
+
E
k
,
(st
.
6)
where the rest mass
E
0
=
mc
2
,
c
= 3
×
10
10
cm s

1
is the speed of light, and
E
k
is the kinetic energy
of a particle. For a particle moving with arbitrary velocity there is a special relativistic relation:
st — 1