2.2 Landau’s Phenomenology
41
Fig. 2.2 Thermodynamic
potential (Eq. 2.26) as a
function of the order
parameter φ at various
temperatures. T 0 is the
transition temperature of a
second-order transition
T = T0
T > T 0
T < T 0
Since Eq. 2.23 is regarded as thermodynamic potential as a function of φ, its
minimum should characterize the thermodynamic equilibrium. That is,
∂ F(τ , φ)
∂φ
=
∂ΔF(τ , φ)
∂φ
= 0
(2.27)
with
∂
2
ΔF(τ , φ)
∂φ 2
< 0.
(2.28)
Thus, the transition at T 0 is described by setting
a 2 = α(T − T 0 )
(2.29)
with α > 0, because the state with φ = 0 is a (local) minimum of ΔF(φ, τ ) for
T > T 0 while it is a local maximum for T < T 0 . ΔF as a function of φ at various
temperatures (T ≶ T 0 ) are shown in Fig. 2.2. While φ 0 = 0 for T ≥ T 0 , φ 0 for T < T 0
is obtained through
0 =
∂ΔF(τ , φ)
∂φ
= 2φ
α(T − T 0 ) + 2a 4 φ
2
,
(2.30)
giving
φ 0 (τ ) = ±
−
α(T − T 0 )
2a 4
(T < T 0 ).
(2.31)
The appearance of two solutions with a difference only in their sign is a consequence
of the equivalence of two states with ±φ. Equation 2.31 gives φ 0 (τ ) = 0 at T = T 0 .