Phase coexistence A disorder-broadened first-order transition occurs over a finite range of temperatures where the fraction of the low-temperature equilibrium phase grows from zero to one (100%) as the temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into the crystal positions. This slowing down happens below a glass-formation temperature
Tg, which may depend on the applied pressure. If the first-order freezing transition occurs over a range of temperatures, and
Tg falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in the observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition, such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials, magnetocaloric materials, magnetic shape memory materials, and other materials. The interesting feature of these observations of
Tg falling within the temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between
Tg and
Tc in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.
Critical points In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the
critical point, at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost non-existent. This is associated with the phenomenon of
critical opalescence, a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light).
Symmetry Phase transitions often involve a
symmetry breaking process. For instance, the cooling of a fluid into a
crystalline solid breaks continuous
translation symmetry: each point in the fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to
spontaneous symmetry breaking, with the exception of certain
accidental symmetries (e.g. the formation of heavy
virtual particles, which only occurs at low temperatures).
Order parameters An
order parameter is a measure of the degree of order across the boundaries in a phase transition system; it normally takes on the value of zero in one phase (usually above the critical point) and nonzero in the other. At the critical point, the order parameter susceptibility (the change of an
extensive property under variation of an
intensive property) will usually diverge. An example of an order parameter is the net
magnetization in a
ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities. From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the
ferromagnetic phase, one must provide the net
magnetization, whose direction was spontaneously chosen when the system cooled below the
Curie point. However, note that order parameters can also be defined for non-symmetry-breaking transitions. Some phase transitions, such as
superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition. There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as
vortex- or
defect lines.
Relevance in cosmology Symmetry-breaking phase transitions play an important role in
cosmology. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the
electroweak field into the U(1) symmetry of the present-day
electromagnetic field. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to
electroweak baryogenesis theory. Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work of
Eric Chaisson and
David Layzer. See also
relational order theories and
order and disorder.
Critical exponents and universality classes Continuous phase transitions are easier to study than first-order transitions due to the absence of
latent heat, and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points. Continuous phase transitions can be characterized by parameters known as
critical exponents. The most important one is perhaps the exponent describing the divergence of the thermal
correlation length by approaching the transition. For instance, let us examine the behavior of the
heat capacity near such a transition. We vary the temperature
T of the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperature
Tc. When
T is near
Tc, the heat capacity
C typically has a
power law behavior: : C \propto |T_\text{c} - T|^{-\alpha}. The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent
α = 0.59 A similar behavior, but with the exponent
ν instead of
α, applies for the correlation length. The critical exponents are not necessarily the same above and below the critical temperature. When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as \gamma, the exponent of the susceptibility) are not identical. For \alpha , the heat capacity remains differentiable at the transition temperature, although discontinuities appear at higher-order derivatives. For -1 \leq \alpha , the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the
lambda transition from a normal state to the
superfluid state, for which experiments have found
α = −0.013 ± 0.003. At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample. This experimental value of α agrees with theoretical predictions based on
variational perturbation theory. For 0 <
α < 1, the heat capacity diverges at the transition temperature (though, since
α < 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional
Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent
α ≈ +0.110. Some model systems do not obey a power-law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a
logarithmic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior. Several other critical exponents,
β,
γ,
δ,
ν, and
η, are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as : \beta = \gamma / (\delta - 1),\quad \nu = \gamma / (2 - \eta). It can be shown that there are only two independent exponents, e.g.
ν and
η. It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as
universality. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid. More impressively, but understandably from above, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the
renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and are insensitive to the underlying microscopic properties of the system. Again, the divergence of the correlation length is the essential point.
Critical phenomena There are also other critical phenomena; e.g., besides
static functions there is also
critical dynamics. As a consequence, at a phase transition one may observe
critical slowing down or
speeding up. Connected to the previous phenomenon is also the phenomenon of
enhanced fluctuations before the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The large
static universality classes of a continuous phase transition split into smaller
dynamic universality classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of the magnetic fields and temperature differences from the critical value. ==Experimental==