48
2 Phase Transitions
assumed that the order parameter not only characterizes the low-symmetry phase
but also drives the phase transition. If another quantity, such as u in the previous
paragraph, possesses the property of order parameters that are null in the highsymmetry phase but finite in the low-symmetry phase, this is a subsidiary order
parameter. On the other hand, plural degrees of freedom, each of which has an order
parameter, may possess potential instability to a respective symmetry-broken state.
In this case, their virtual transition points (expected without coupling with others)
would be different from each other. If there is no coupling, each degree of freedom
undergoes an intrinsic phase transition. When the difference in the transition point is
small with some coupling, however, transitions merge into a single phase transition
[18, 19]. Furthermore, the merged transition may be of first-order even if each of two
degrees of freedom has individually a continuous instability [19],
9 i.e., the respective
expansion being the form of Eq. 2.26. In short, the joint instability of many order
parameters potentially brings about a first-order phase transition.
2.3 Critical Phenomena and Universality
In the vicinity of a transition temperature of a continuous transition, many physical
quantities exhibit singularity. Equations 2.40 and 2.43 are examples of susceptibilities
in Landau’s phenomenology. Overall behaviors near a transition point of a continuous phase transition are generally termed as critical phenomena. In this context, the
transition point is often denoted as a critical point. The critical phenomenon is the
central issue in the physical study of phase transitions and discussed in many specialized monographs [20–22]. The description in this section is limited to issues having
a possibility to encounter in the context of general studies of molecular systems.
Some of the physical quantities exhibiting singularities such as heat capacity C
and susceptibility χ are related to fluctuation. It is easy to verify
10
C =
1
k B T 2 (E − −E)
2
(2.59)
χ ∝
1
k B T
(σ − −σ)
2
(2.60)
where E is (internal) energy and ·· means the average over the system and σ is
(instantaneous) order parameter (i.e., φ = =σ). If these quantities diverge at the
critical point, the fluctuation of the system also diverges, accordingly. It is known
that the role of fluctuations in critical phenomena is crucial. Theories of critical
phenomena neglecting the effect of fluctuations are classified as mean-field theory.
9 Although the reference [19] considers only one sign of the difference in the transition point, the
other sign can also bring about a first-order transition.
10 While considering the possibility of difference in the definition of susceptibility, the formula is
written as the proportionality.
2 Phase Transitions
assumed that the order parameter not only characterizes the low-symmetry phase
but also drives the phase transition. If another quantity, such as u in the previous
paragraph, possesses the property of order parameters that are null in the highsymmetry phase but finite in the low-symmetry phase, this is a subsidiary order
parameter. On the other hand, plural degrees of freedom, each of which has an order
parameter, may possess potential instability to a respective symmetry-broken state.
In this case, their virtual transition points (expected without coupling with others)
would be different from each other. If there is no coupling, each degree of freedom
undergoes an intrinsic phase transition. When the difference in the transition point is
small with some coupling, however, transitions merge into a single phase transition
[18, 19]. Furthermore, the merged transition may be of first-order even if each of two
degrees of freedom has individually a continuous instability [19],
9 i.e., the respective
expansion being the form of Eq. 2.26. In short, the joint instability of many order
parameters potentially brings about a first-order phase transition.
2.3 Critical Phenomena and Universality
In the vicinity of a transition temperature of a continuous transition, many physical
quantities exhibit singularity. Equations 2.40 and 2.43 are examples of susceptibilities
in Landau’s phenomenology. Overall behaviors near a transition point of a continuous phase transition are generally termed as critical phenomena. In this context, the
transition point is often denoted as a critical point. The critical phenomenon is the
central issue in the physical study of phase transitions and discussed in many specialized monographs [20–22]. The description in this section is limited to issues having
a possibility to encounter in the context of general studies of molecular systems.
Some of the physical quantities exhibiting singularities such as heat capacity C
and susceptibility χ are related to fluctuation. It is easy to verify
10
C =
1
k B T 2 (E − −E)
2
(2.59)
χ ∝
1
k B T
(σ − −σ)
2
(2.60)
where E is (internal) energy and ·· means the average over the system and σ is
(instantaneous) order parameter (i.e., φ = =σ). If these quantities diverge at the
critical point, the fluctuation of the system also diverges, accordingly. It is known
that the role of fluctuations in critical phenomena is crucial. Theories of critical
phenomena neglecting the effect of fluctuations are classified as mean-field theory.
9 Although the reference [19] considers only one sign of the difference in the transition point, the
other sign can also bring about a first-order transition.
10 While considering the possibility of difference in the definition of susceptibility, the formula is
written as the proportionality.