Chapter 5
Phase Transition I: van der Waals Gas
5.1
General remarks
Phase transitions occur at discontinuities of the equilibrium thermodynamic functions with
varying certain physical parameters such as temperature, density, magnetic field, etc.
For a
finite system, thermodynamic functions such as grand potential are analytical for all
physi
cal
parameters, though it could have singularities when the parameters are taken as imaginary
numbers (by continuation to the complex plane). In the thermodynamic limit, the singularities
approach to the real axis of the physical parameters, and the discontinuity and hence the phase
transition takes place.
5.1.1
Order of phase transition
The energy, free energy, and grand potential of a system are always continuous with varying
the temperature, volume, magnetic field, etc. But their di
ff
erentials may not. Phase transitions
can be classified according to which order of the di
ff
erentials is discontinuous. To be specific,
we consider the grand potential
Ω
. The discussions apply to energy
U
, free energies
F
or
G
similarly.
We consider a macroscopic system with volume
V
→ ∞
or number of particles
N
→ ∞
. So
we would rather consider the potential per unit volume, i,e., the pressure
P
=
−
Ω
/
V
,
(5.1)
which is a function of the density
ρ
(or specific volume
v
≡
V
/
N
=
ρ
−
1
) and the temperature
T
.
A specific physical system is described by its state of equation
P
=
P
(
v
,
T
)
.
(5.2)
For example, the state of equation of an ideal gas is
P
=
k
B
T
v
.
(5.3)
For an ideal gas,
P
is analytical for temperature and volume as real numbers, so there is no
phase transition. When the
n
th order di
ff
erential of
P
regarding
T
or
V
is discontinuous and all
63
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CHAPTER 5. PHASE TRANSITION I: VAN DER WAALS GAS
the lower order di
ff
erentials are continuous, the phase transition is called an
n
th order one. For
example, the
n
th order di
ff
erential of the isothermal curve
[
∂
n
P
(
v
,
T
)
∂
v
n
]
T
(5.4)
is discontinuous for the
n
th order phase transition. Examples of firstorder phase transitions are
melting of ice and condensation of vapor.
5.1.2
Finite systems: No phase transition
With the sole exception of BoseEinstein condensation, all phase transitions are due to inter
action between particles.
1
Let us consider an interacting gas with potential of the following
form
u
(
r
)
=
+
∞
,
r
<
d
>
u
0
,
d
≤
r
<
D
0
,
r
≥
D
.
(5.5)
Such a model simulates atoms of strong repulsion at short distances and shortranged weak
attraction at long distance.
The partition function of a grand canonical ensemble is
Ξ =
∞
∑
N
=
0
z
N
Z
N
,
(5.6)
where
z
≡
exp(
βµ
) and
Z
N
=
Tr
e
−
β
H
(
N
)
,
(5.7)
is the partition function of a canonical ensemble. The interaction potential in Eq. (5.5) means
that two particles cannot be closer than
d
and hence the number of particles in a finite volume
V
cannot exceed a certain number
N
0
∝
V
/
d
3
. Thus the partition function is
2
Ξ
(
V
,
T
)
=
N
0
∑
N
=
0
z
N
Z
N
(
V
,
T
)
.
(5.8)
The pressure is
P
V
=
k
B
T
V
ln
Ξ
(
V
,
N
)
.
(5.9)
When
Ξ
is positive, the pressure is analytical. For a finite system, the partition function is a
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 Spring '10
 DrKwong
 Van der Waals, van der, Critical phenomena, Phase transition, Renormalization group

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