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Phase Transitions
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Phase transition
In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. The distinguishing characteristic of a phase transition is an abrupt sudden change in one or more physical properties, in particular the heat capacity, with a small change in a thermodynamic variable such as the temperature. Examples of phase transitions are:

The transitions between the solid, liquid, and gaseous phases (boiling, melting, sublimation, etc.)

The transition between the ferromagnetic and paramagnetic phases of magnetic materials at the Curie point.

The emergence of superconductivity in certain metals when cooled below a critical temperature.

Quantum condensation of bosonic fluids, such as BoseEinstein condensation and the superfluid transition in liquid helium.

The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled.
As discussed in the article on phases, phase transitions come about when the free energy of a system is nonanalytic for some choice of thermodynamic variables. This nonanalyticity generally stems from the interactions of an extremely large number of particles in a system, and does not appear in systems that are too small.
Table of contents 
Classification of phase transitions
Ehrenfest classification
The first attempt at classifying phase transitions was the Ehrenfest classification scheme, which grouped phase transitions based on the degree of nonanalyticity involved. Though useful, Ehrenfest's classification is flawed, as we will discuss in the next section.
Under this scheme, phase transitions were labelled by the lowest derivative of the free energy that is discontinuous at the transition. Firstorder phase transitions exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable. The various solid/liquid/gas transitions are classified as firstorder transitions, as the pressure, which is the first derivative of the free energy with volume, changes discontinuously across the transitions. Secondorder phase transitions have a discontinuity in a second derivative of the free energy. These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with the appplied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classication scheme, there could in principle be third, fourth, and higherorder phase transitions.
Modern classification of phase transitions
The Ehrenfest scheme is an inaccurate method of classifying phase transitions, for it is based on the mean field theory of phases (to be described in a later section.) Mean field theory is inaccurate in the vicinity of phase transitions, as it neglects the role of thermodynamic fluctuations. For instance, it predicts a finite discontinuity in the heat capacity at the ferromagnetic transition, which is implied by Ehrenfest's definition of "secondorder" transitions. In real ferromagnets, the heat capacity diverges to infinity at the transition.
In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:
The firstorder phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy. Because energy cannot be instantaneously transferred between the system and its environment, firstorder transitions are associated with "mixedphase regimes" in which some parts of the system have completed the transition and others have not. This phenomenon is familiar to anyone who has boiled a pot of water: the water does not instantly turn into gas, but forms a turbulent mixture of water and water vapor bubbles. Mixedphase systems are difficult to study, because their dynamics are violent and hard to control. However, many important phase transitions fall in this category, including the solid/liquid/gas transitions.
The second class of phase transitions are the continuous phase transitions, also called secondorder phase transitions. These have no associated latent heat. Examples of secondorder phase transitions are the ferromagnetic transition, the superfluid transition, and BoseEinstein condensation.
Several transitions are known as the infiniteorder phase transitions. They are continuous but break no symmetries (see Symmetry below). The most famous example is the BerezinskyKosterlitzThouless transition in the twodimensional XY model. Many quantum phase transitions in twodimensional electron gases belong to this class.
Properties of phase transitions
Critical points
In systems containing liquid and gaseous phases, there exist a special combination of pressure and temperature, known as the critical point, at which the transition between liquid and gas becomes a secondorder transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost nonexistent.
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 (but not always) take place between phases with different symmetry. Consider, for example, the transition between a fluid (i.e. liquid or gas) and a crystalline solid. A fluid, which is composed of atoms arranged in a disordered but homogenous manner, possesses continuous translational symmetry: each point inside the fluid has the same properties as any other point. A crystalline solid, on the other hand, is made up of atoms arranged in a regular lattice. Each point in the solid is not similar to other points, unless those points are displaced by an amount equal to some lattice spacing.
Generally, we may speak of one phase in a phase transition as being more symmetrical than the other. The transition from the more symmetrical phase to the less symmetrical one is a symmetrybreaking process. In the fluidsolid transition, for example, we say that continuous translation symmetry is broken.
The ferromagnetic transition is another example of a symmetrybreaking transition, in this case the symmetry under reversal of the direction of electric currents and magnetic field lines. This symmetry is referred to as "updown symmetry" or "timereversal symmetry". It is broken in the ferromagnetic phase due to the formation of magnetic domains containing aligned magnetic moments. Inside each domain, there is a magnetic field pointing in a fixed direction chosen spontaneously during the phase transition. The name "timereversal symmetry" comes from the fact that electric currents reverse direction when the time coordinate is reversed.
The presence of symmetrybreaking (or nonbreaking) is important to the behavior of phase transitions. It was pointed out by Landau that, given any state of a system, one may unequivocally say whether or not it possesses a given symmetry. Therefore, it cannot be possible to analytically deform a state in one phase into a phase possessing a different symmetry. This means, for example, that it is impossible for the solidliquid phase boundary to end in a critical point like the liquidgas boundary. However, symmetrybreaking transitions can still be either first or second order.
Typically, the more symmetrical phase is on the hightemperature side of a phase transition, and the less symmetrical phase on the lowtemperature side. This is certainly the case for the solidfluid and ferromagnetic transitions. This happens because the Hamiltonian of a system usually exhibits all the possible symmetries of the system, whereas the lowenergy states lack some of these symmetries (this phenomenon is known as spontaneous symmetry breaking.) At low temperatures, the system tends to be confined to the lowenergy states. At higher temperatures, thermal fluctuations allow the system to access states in a broader range of energy, and thus more of the symmetries of the Hamiltonian.
When symmetry is broken, 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. Such variables are instances of order parameters, which will be described later. However, note that order parameters can also be defined for symmetrynonbreaking transitions.
Symmetrybreaking phase transitions play an important role in cosmology. It has been speculated that, in the hot early universe, the vacuum (i.e. the various quantum fields that fill space) possessed a large number of symmetries. As the universe expanded and cooled, the vacuum underwent a series of symmetrybreaking 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 presentday electromagnetic field. This transition is important to understanding the asymmetry between the amount of matter and antimatter in the presentday universe (see electroweak baryogenesis.)
Critical exponents and universality classes
Continuous phase transitions are easier to study than firstorder 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.
It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. 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 T_{c}. When T is near T_{c}, the heat capacity C typically has a power law behaviour:
The constant α is the critical exponent associated with the heat capacity. It is not difficult to see that it must be less than 1 in order for the transition to have no latent heat. Its actual value depends on the type of phase transition we are considering. For 1 < α < 0, 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. For 0 < α < 1, the heat capacity diverges at the transition temperature (though, since α < 1, the divergence is not strong enough to produce a latent heat.) An example of such behavior is the 3dimensional ferromagnetic phase transition. In the threedimensional Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent α ∼ 0.110.
Some model systems do not obey this power law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the twodimensional Ising model has a logarithmic divergence. However, these systems are 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.
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 liquidgas critical point have been found to be independent of the chemical composition of the fluid. More amazingly, 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 is insensitive to the underlying microscopic properties of the system.
Mean field theory of phase transitions
Fluctuations
Renormalization group theory of phase transitions
References

Anderson, P.W., Basic Notions of Condensed Matter Physics, Perseus Publishing (1997).

Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group, Perseus Publishing (1992).

Landau, L.D. and Lifshitz, E.M., Statistical Physics Part 1, vol. 5 of Course of Theoretical Physics, Pargamon, 3rd Ed. (1994).
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