2.2 Landau’s Phenomenology
39
2.2 Landau’s Phenomenology
The treatments described in this section has been proposed first by L. D. Landau, and
widely known as Landau’s thermodynamic phenomenology of phase transitions. The
phenomenology does not treat microscopic details of substances but gives fruitful
insights [2, 3] as described below. In some respects, it puts a reference framework
for discussing the nature and properties of any phase transitions.
2.2.1 Power Expansion of Thermodynamic Potential
Suppose a thermodynamic system is under the condition characterized by the set
of intensive variables τ . A set of structural variables of the system is denoted by
{r}. The thermodynamic potential of the system is then written as F(τ , {r}). By
introducing the set {r 0 } for the equilibrium state, we can write
F(τ , {r}) = F(τ , {r 0 }) + ΔF(τ , {r}).
(2.23)
According to the variational principle of thermodynamics,
ΔF(τ , {r 0 }) = 0.
(2.24)
Namely, if a parameter φ is assumed to reflect the set {r} concisely, ΔF(τ , {r}) may
be expanded in a power series of (φ − φ 0 (τ )) around φ 0 (τ ), the equilibrium value
of φ under the condition, as
ΔF(τ , φ) =
n≥2
a n (τ )(φ − φ 0 (τ ))
n
(2.25)
with a 2 (τ ) > 0. Note that φ is not restricted to a scalar but can be vector or tensor.
Also, we may consider a parameter independent of φ, say σ, for additional degrees
of freedom.
Since Eq. 2.23 is assumed the thermodynamic potential of the system, it must
be consistent with the full symmetry of the system. This requirement poses some
restrictions on the form of the expansion of ΔF(τ , {r 0 }) (Eq. 2.25). This restriction
appears as the selection of terms retained in the expansion. For example, if the sign
of a scalar φ is not a matter concerning the physical nature of the phase, all odd terms
should vanish. It is emphasized that the form of the expansion crucially depends on
the selection of φ.
39
2.2 Landau’s Phenomenology
The treatments described in this section has been proposed first by L. D. Landau, and
widely known as Landau’s thermodynamic phenomenology of phase transitions. The
phenomenology does not treat microscopic details of substances but gives fruitful
insights [2, 3] as described below. In some respects, it puts a reference framework
for discussing the nature and properties of any phase transitions.
2.2.1 Power Expansion of Thermodynamic Potential
Suppose a thermodynamic system is under the condition characterized by the set
of intensive variables τ . A set of structural variables of the system is denoted by
{r}. The thermodynamic potential of the system is then written as F(τ , {r}). By
introducing the set {r 0 } for the equilibrium state, we can write
F(τ , {r}) = F(τ , {r 0 }) + ΔF(τ , {r}).
(2.23)
According to the variational principle of thermodynamics,
ΔF(τ , {r 0 }) = 0.
(2.24)
Namely, if a parameter φ is assumed to reflect the set {r} concisely, ΔF(τ , {r}) may
be expanded in a power series of (φ − φ 0 (τ )) around φ 0 (τ ), the equilibrium value
of φ under the condition, as
ΔF(τ , φ) =
n≥2
a n (τ )(φ − φ 0 (τ ))
n
(2.25)
with a 2 (τ ) > 0. Note that φ is not restricted to a scalar but can be vector or tensor.
Also, we may consider a parameter independent of φ, say σ, for additional degrees
of freedom.
Since Eq. 2.23 is assumed the thermodynamic potential of the system, it must
be consistent with the full symmetry of the system. This requirement poses some
restrictions on the form of the expansion of ΔF(τ , {r 0 }) (Eq. 2.25). This restriction
appears as the selection of terms retained in the expansion. For example, if the sign
of a scalar φ is not a matter concerning the physical nature of the phase, all odd terms
should vanish. It is emphasized that the form of the expansion crucially depends on
the selection of φ.