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Phase Transitions
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:
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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.
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The emergence of
superconductivity in certain
metals when cooled below a critical temperature.
-
Quantum condensation of
bosonic fluids, such as
Bose-Einstein 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
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.
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
pressure, which is the first derivative of the free energy with volume, changes
discontinuously across the transitions. 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 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 higher-order 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.
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).
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.)
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:
-
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).
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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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Quantum Phase Transitions
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