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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:
As discussed in the article on phases, phase transitions come
about when the free energy of a system is non-analytic for some choice of thermodynamic variables. This non-analyticity 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.
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
non-analyticity 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. First-order 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 first-order transitions, as the
density, which is the first derivative of the free energy with respect to chemical potential, changes discontinuously across the
transitions. [The pressure must be continuous across the phase boundary in equilibrium.] Second-order 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 applied 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 higher-order 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 "second-order" 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 first-order 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, first-order transitions
are associated with "mixed-phase 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. Mixed-phase 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 second-order
phase transitions. These have no associated latent heat. Examples of second-order phase transitions are the
ferromagnetic transition, the superfluid transition, and Bose-Einstein condensation.
Several transitions are known as the infinite-order phase transitions. They are continuous but break no
symmetries (see Symmetry below). The most famous example is the Berezinsky-Kosterlitz-Thouless transition in the two-dimensional XY model. Many quantum phase transitions in two-dimensional 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 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 (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 symmetry-breaking process. In the fluid-solid
transition, for example, we say that continuous translation symmetry is broken.
The ferromagnetic transition is another example of a symmetry-breaking transition, in this case the symmetry under reversal of
the direction of electric currents and magnetic field lines. This symmetry is referred to as "up-down symmetry" or "time-reversal
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 "time-reversal symmetry" comes from the fact that electric currents reverse direction when the time coordinate is
reversed.
The presence of symmetry-breaking (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 solid-liquid
phase boundary to end in a critical point like the liquid-gas boundary. However, symmetry-breaking transitions can still be
either first or second order.
Typically, the more symmetrical phase is on the high-temperature side of a phase transition, and the less symmetrical phase on
the low-temperature side. This is certainly the case for the solid-fluid and ferromagnetic transitions. This happens because the
Hamiltonian of a system usually exhibits all the possible symmetries of the
system, whereas the low-energy 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 low-energy 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 symmetry-nonbreaking transitions.
Symmetry-breaking 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 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 understanding the asymmetry between the amount
of matter and antimatter in the present-day universe (see electroweak baryogenesis.)
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.
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 Tc. When T is near Tc,
the heat capacity C typically has a power law behaviour:
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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 3-dimensional 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 this 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 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 liquid-gas 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.
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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