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| [[File:Phase change - en.svg|thumb|right|340px|This diagram shows the nomenclature for the different phase transitions.]]
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| A '''phase transition''' is the transformation of a [[thermodynamics|thermodynamic]] system from one [[phase (matter)|phase]] or [[state of matter]] to another.
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| A phase of a [[thermodynamic system]] and the states of matter have uniform physical properties.
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| During a phase transition of a given medium certain properties of the medium change, often discontinuously, as a result of some external condition, such as temperature, pressure, and others. For example, a liquid may become gas upon heating to the [[boiling point]], resulting in an abrupt change in [[volume]]. The measurement of the external conditions at which the transformation occurs is termed the ''phase transition.
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| Phase transitions are common in nature and used in many technologies.
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| The term is most commonly used to describe transitions between [[solid]], [[liquid]] and [[gas]]eous [[states of matter]], and, in rare cases, [[Plasma (physics)|plasma]]. | |
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| ==Types of phase transition==
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| Examples of phase transitions include:
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| *The transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of [[temperature]] and/or [[pressure]]:
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| {{Table of Phase Transitions}}
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| :* (see also [[vapor pressure]] and [[phase diagram]])
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| [[File:Phase-diag2.svg|thumb|300px|A typical phase diagram. The dotted line gives [[Water (molecule)#Density of water and ice|the anomalous behavior of water]].]]
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| [[File:Argon ice 1.jpg|thumb|right|A small piece of rapidly melting argon ice simultaneously shows the transitions from solid to liquid to gas.]]
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| [[File:Comparison carbon dioxide water phase diagrams.svg|thumb|300px|Comparison of phase diagrams of carbon dioxide (red) and water (blue) explaining their different phase transitions at 1 atmosphere]]
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| * A [[eutectic]] transformation, in which a two component single phase liquid is cooled and transforms into two solid phases. The same process, but beginning with a solid instead of a liquid is called a [[eutectoid]] transformation.
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| * A [[peritectic]] transformation, in which a two component single phase solid is heated and transforms into a solid phase and a liquid phase.
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| * A [[spinodal decomposition]], in which a single phase is cooled and separates into two different compositions of that same phase.
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| * Transition to a [[mesophase]] between solid and liquid, such as one of the "[[liquid crystal]]" phases.
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| * The transition between the [[ferromagnetism|ferromagnetic]] and [[paramagnetism|paramagnetic]] phases of [[magnet]]ic materials at the [[Curie point]].
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| * The transition between differently ordered, [[ANNNI model|commensurate]] or [[commensurability (mathematics)|incommensurate]], magnetic structures, such as in cerium [[antimonide]].
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| * The [[martensitic transformation]] which occurs as one of the many phase transformations in carbon steel and stands as a model for [[displacive phase transformations]].
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| * Changes in the [[crystallographic]] structure such as between [[ferrite (iron)|ferrite]] and [[austenite]] of [[iron]].
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| * Order-disorder transitions such as in alpha-[[titanium aluminide]]s.
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| * The dependence of the [[adsorption]] geometry on coverage and temperature, such as for [[hydrogen]] on [[iron]] (110).
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| * The emergence of [[superconductivity]] in certain [[metal]]s and ceramics when cooled below a critical temperature.
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| * The transition between different molecular structures ([[Polymorphism (materials science)|polymorphs]], [[allotropy|allotropes]] or [[polyamorphism|polyamorphs]]), especially of solids, such as between an [[amorphous solid|amorphous]] structure and a [[crystal]] structure, between two different crystal structures, or between two amorphous structures.
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| * Quantum condensation of [[boson]]ic fluids ([[Bose-Einstein condensate|Bose-Einstein condensation]]). The [[superfluidity|superfluid]] transition in liquid [[helium]] is an example of this.
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| * The [[Symmetry breaking|breaking of symmetries]] in the laws of physics during the early history of the universe as its temperature cooled.
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| Phase transitions occur when the [[thermodynamic free energy]] of a system is [[analytic function|non-analytic]] for some choice of thermodynamic variables (cf. [[phase (matter)|phases]]). This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are too small.
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| At the phase transition point (for instance, [[boiling point]]) the two phases of a substance, [[liquid]] and [[vapor]], have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the gaseous form is preferred.
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| It is sometimes possible to change the state of a system [[diabatic]]ally (as opposed to [[adiabatic invariant|adiabatic]]ally) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is [[metastable]], i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in [[superheating]], [[supercooling]], and [[supersaturation]], for example.
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| ==Classifications==
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| ===Ehrenfest classification===
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| [[Paul Ehrenfest]] classified phase transitions based on the behavior of the [[thermodynamic free energy]] as a function of other thermodynamic variables. Under this scheme, phase transitions were labeled 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 respect to some thermodynamic variable.<ref name = Blundell>{{Cite book | last = Blundell | first = Stephen J. | coauthors = Katherine M. Blundell | title = Concepts in Thermal Physics | publisher = Oxford University Press | year = 2008 | isbn = 978-0-19-856770-7}}</ref> The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the first derivative of the free energy with respect to [[chemical potential]]. ''Second-order phase transitions'' are continuous in the first derivative (the [[order parameter]], which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy.<ref name = Blundell/> These include the ferromagnetic phase transition in materials such as [[iron]], where the [[magnetization]], which is the first derivative of the free energy with respect to 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 classification scheme, there could in principle be third, fourth, and higher-order phase transitions.
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| Though useful, Ehrenfest's classification has been found to be an inaccurate method of classifying phase transitions, for it does not take into account the case where a [[derivative]] of [[Thermodynamic free energy|free energy]] diverges (which is only possible in the [[thermodynamic limit]]). For instance, in the [[ferromagnetic]] transition, the [[heat capacity]] diverges to [[infinity]].
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| ===Modern classifications===
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| In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:
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| 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. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into [[water vapor|vapor]], but forms a [[turbulence|turbulent]] mixture of liquid water and vapor bubbles). Imry and Wortis showed that quenched disorder can broaden a first-order transition in that the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling.<ref>Y. Imry and M. Wortis, Phys. Rev. B 19, 3580 (1979)</ref><ref>K. Kumar et al, Phys. Rev. B 73, 184435 (2006)</ref><ref>G. Pasquini et al, Phys. Rev. Lett 100, 247003 (2008).</ref>
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| Second-order phase transitions are also called ''continuous phase transitions''. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Examples of second-order phase transitions are the ferromagnetic transition, superconducting transition (for a [[Type-I superconductor]] the phase transition is second-order at zero external field and for a [[Type-II superconductor]] the phase transition is second-order for both normal state-mixed state and mixed state-superconducting state transitions) and the [[superfluid]] transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature <ref>M.I. Ojovan. Ordering and structural changes at the glass-liquid transition. J. Non-Cryst. Solids, 382, 79-86 (2013). http://dx.doi.org/10.1016/j.jnoncrysol.2013.10.016</ref> which enable quite exactly to detect it using [[differential scanning calorimetry]] measurements. [[Lev Landau]] gave a [[Phenomenology (science)|phenomenological]] [[Landau theory|theory]] of second order phase transitions.
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| Apart from isolated, simple phase transitions, there exist transition lines as well as [[multicritical point]]s, when varying external parameters like the magnetic field, composition,...
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| Several transitions are known as the ''infinite-order phase transitions''.
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| They are continuous but break no [[#Symmetry|symmetries]]. The most famous example is the [[Kosterlitz–Thouless transition]] in the two-dimensional [[XY model]]. Many [[quantum phase transition]]s, e.g., in two-dimensional [[electron gas]]es, belong to this class.
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| The [[glass transition|liquid-glass transition]] is observed in many [[polymers]] and other liquids that can be [[supercooling|supercooled]] far below the melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a ''[[quenched disorder]]'' state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling a liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times.<ref>Gotze, Wolfgang. "Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory."</ref><ref>Lubchenko, V. Wolynes, P.G. "Theory of Structural Glasses and Supercooled Liquids" Annual Review of Physical Chemistry. 2007, Vol 58. Pg 235.</ref> No direct experimental evidence supports the existence of these transitions.
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| ==Characteristic properties==
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| ===Phase Coexistence===
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| A disorder-broadened first order transition occurs over a finite range of temperatures with 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.<ref>A.L.Greer, Science 267 (1995) 1947</ref> This slowing down happens below a glass-formation temperature Tg, which may depend on the applied pressure.,<ref>G. Tarjus, Nature Nature 448 (2007) 758</ref><ref>M.I. Ojovan. Ordering and structural changes at the glass-liquid transition. J. Non-Cryst. Solids, 382, 79-86 (2013). http://dx.doi.org/10.1016/j.jnoncrysol.2013.10.016</ref> 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.
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| 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,<ref>M.A.Manekar et al, Physical Review B 64 (2001) 104416</ref> such persistent phase coexistence has now been reported across a variety of first order magnetic transitions. These include colossal-magnetoresistance manganite materials,<ref>A.Banerjee et al, J. Phys. Condens. Matter 18 (2006) L605</ref><ref>W. Wu et al, Nature Materials 5 (2006) 881</ref> magnetocaloric materials,<ref>S.B.Roy et al, Physical Review B 74 (2006) 012403</ref> magnetic shape memory materials,<ref>A.Lakhani et al, J. Phys. Condens. Matter 24 (2012) 386004</ref> and other materials.<ref>P.Kushwaha et al, Physical Review B 80 (2009) 174413</ref> 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 field 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.
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| ===Critical points===
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| In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the [[critical point (physics)|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).
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| ===Symmetry===
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| <!--Copyright violation removed
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| [http://books.google.co.uk/books?id=qyK95FevVbIC&pg=PA368&lpg=PA368&dq=Generally,+we+may+speak+of+one+phase+in+a+phase+transition+as+being+more+symmetrical+than+the+other&source=bl&ots=9C-8QBoFft&sig=09jfr9MGV6ea76vEZS8XqN9Z4xc&hl=en&ei=uWEqTajnFOSAhAejs_mWAg&sa=X&oi=book_result&ct=result&resnum=8&ved=0CEgQ6AEwBw#v=onepage&q=Generally%2C%20we%20may%20speak%20of%20one%20phase%20in%20a%20phase%20transition%20as%20being%20more%20symmetrical%20than%20the%20other&f=false]-->
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| ===Order parameters===
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| An order parameter is a measure of the degree of order across the boundaries in a phase transition system; it normally ranges between zero in one phase (usually above the critical point) and nonzero in the other.<ref>{{cite web | title = Compendium of Chemical Terminology (commonly called The Gold Book) | editor = A. D. McNaught and A. Wilkinson | isbn = 0-86542-684-8 | doi = | publisher = [[IUPAC]] | url = http://www.iupac.org/goldbook/O04323.pdf | accessdate = 2007-10-23}}</ref> At the critical point, the order parameter susceptibility will usually diverge.
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| 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.
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| 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.
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| There also exist [[Dualism|dual]] descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as [[Quantum vortex|vortex]]- or [[Topological defect|defect]] lines.
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| ===Relevance in cosmology===
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| Symmetry-breaking phase transitions play an important role in [[physical cosmology|cosmology]]. It has been speculated that, in the [[Big Bang|hot early universe]], the vacuum (i.e. the various [[quantum field theory|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 force|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 [[Baryogenesis|electroweak baryogenesis]].)
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| 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]]<ref>Chaisson, “Cosmic Evolution”, Harvard, 2001</ref> and David Layzer.<ref>David Layzer, Cosmogenesis, The Development of Order in the Universe", Oxford Univ. Press, 1991</ref> See also [[Relational order theories]].
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| {{See also|Order-disorder}}
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| ===Critical exponents and universality classes===
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| {{main|critical exponent}}
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| 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.
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| It turns out that continuous phase transitions can be characterized by parameters known as [[critical exponent]]s. 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 ''T<sub>c</sub>''. When ''T'' is near ''T<sub>c</sub>'', the heat capacity ''C'' typically has a [[power law]] behavior:
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| :<math> C \propto |T_c - T|^{-\alpha}.</math>
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| Such a behaviour has the heat capacity of amorphous materials near the glass transition temperature where the universal critical exponent α = 0.59 <ref>M.I. Ojovan, W.E. Lee. Topologically disordered systems at the glass transition. J. Phys.: Condensed Matter, 18, 11507-11520 (2006). http://eprints.whiterose.ac.uk/1958/</ref> A similar behavior, but with the exponent <math>\nu</math> instead of <math>\alpha</math>, applies for the correlation length.
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| The exponent <math>\nu</math> is positive. This is different with <math>\alpha</math>. Its actual value depends on the type of phase transition we are considering.
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| 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.
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| At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.<ref>{{cite journal | doi=10.1103/PhysRevB.68.174518 | title=Specific heat of liquid helium in zero gravity very near the lambda point | year=2003 | last1=Lipa | first1=J. | last2=Nissen | first2=J. | last3=Stricker | first3=D. | last4=Swanson | first4=D. | last5=Chui | first5=T. | journal=Physical Review B | volume=68 | issue=17|arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L }}</ref> This experimental value of α agrees with theoretical predictions based on [[variational perturbation theory]].<ref>{{cite journal | doi=10.1103/PhysRevD.60.085001 | title=Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions | year=1999 | last1=Kleinert | first1=Hagen | journal=Physical Review D | volume=60 | issue=8|arxiv = hep-th/9812197 |bibcode = 1999PhRvD..60h5001K }}</ref>
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| 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 3-dimensional ferromagnetic phase transition. In the three-dimensional [[Ising model]] for uniaxial magnets, detailed theoretical studies have yielded the exponent α ∼ +0.110.
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| 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 [[logarithm]]ic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior.
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| 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 <math>\beta=\gamma/(\delta-1)</math>, <math>\nu=\gamma/(2-\eta)</math>. It can be shown that there are only two independent exponents, e.g. <math>\nu </math> and <math>\eta</math>.
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| 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, but understandable 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 divergency of the correlation length is the essential point.
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| ===Critical slowing down and other phenomena===
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| 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''. 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.
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| ===Percolation theory===
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| Another phenomenon which shows phase transitions and critical exponents is [[percolation theory|percolation]]. The simplest example is perhaps percolation in a two dimensional square lattice. Sites are randomly occupied with probability p. For small values of p the occupied sites form only small clusters. At a certain threshold p<sub>c</sub> a giant cluster is formed and we have a second order phase transition.<ref>{{cite book |title= Fractals and Disordered Systems|author= Armin Bunde and [[Shlomo Havlin]] |year= 1996|publisher= Springer|url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php}}</ref> The behavior of P<sub>∞</sub> near p<sub>c</sub> is, P<sub>∞</sub>~(p-p<sub>c</sub>)<sup>β</sup>, where β is a critical exponent.
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| <!--==Mean field theory of phase transitions==-->
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| <!--==Fluctuations==-->
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| <!--==Renormalization group theory of phase transitions==-->
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| ===Phase transitions in biological systems===
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| Phase transitions play many important roles in biological systems. Examples include the [[lipid bilayer]] formation, the [[coil-globule]] transition in the process of [[protein folding]] and [[DNA melting]], liquid crystal-like transitions in the process of [[DNA condensation]], and cooperative ligand binding to DNA and proteins with the character of phase transition.<ref>{{cite journal | doi=10.1080/07391102.2000.10506578 | title=Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind | year=2000| author = D.Y. Lando and V.B. Teif| journal=J. Biomol. Struct. Dynam. | volume=17 | issue=5 | pages=903–911 }}</ref>
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| ==See also==
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| {{Condensed matter physics|expanded=States of matter}}
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| * [[Allotropy]]
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| * [[Autocatalytic reactions and order creation]]
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| * [[Crystal growth]]
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| * [[Differential scanning calorimetry]]
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| * [[Diffusionless transformations]]
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| * [[Ehrenfest equations]]
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| * [[Jamming (physics)]]
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| * [[Kelvin probe force microscope]]
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| * [[Lambda transition]] universality class
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| * [[Landau theory]] of second order phase transitions
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| * [[Laser-heated pedestal growth]]
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| * [[List of states of matter]]
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| * [[Micro-Pulling-Down]]
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| * [[Percolation theory]]
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| * [[Continuum percolation theory]]
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| * Phase separation
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| * [[Superfluid film]]
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| * [[Superradiant phase transition]]
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| ==References==
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| {{reflist}}
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| ===Further reading===
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| * [[Philip Warren Anderson|Anderson, P.W.]], ''Basic Notions of Condensed Matter Physics'', [[Perseus Publishing]] (1997).
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| * [[Michael E. Fisher|Fisher, M.E.]], "The renormalization group in the theory of critical behavior", Rev. Mod. Phys. 46, 597–616 (1974).
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| * Goldenfeld, N., ''Lectures on Phase Transitions and the Renormalization Group'', Perseus Publishing (1992).
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| *{{citation |year=2008 |author=Ivancevic, Vladimir G |author2=Ivancevic, Tijana T |title=Chaos, Phase Transitions, Topology Change and Path Integrals |url=http://books.google.com/books?id=wpsPgHgtxEYC&printsec=frontcover&dq=complex+nonlinearity&hl=en&sa=X&ei=YQFCUZ_pMcvYkgXD-YCwBQ&ved=0CC8Q6AEwAA |place=Berlin |publisher=Springer |isbn=978-3-540-79356-4 |accessdate=14 March 2013 |postscript= e-ISBN 978-3-540-79357-1}}
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| * Kogut,J. and [[Kenneth G. Wilson|Wilson,K]], "The Renormalization Group and the epsilon-Expansion," Phys. Rep. 12 (1974), 75.
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| * Krieger, Martin H., ''Constitutions of matter : mathematically modelling the most everyday of physical phenomena'', [[University of Chicago Press]], 1996. Contains a detailed pedagogical discussion of [[Lars Onsager|Onsager]]'s solution of the 2-D Ising Model.
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| * [[Lev Davidovich Landau|Landau, L.D.]] and [[Evgeny Mikhailovich Lifshitz|Lifshitz, E.M.]], ''Statistical Physics Part 1'', vol. 5 of ''[[Course of Theoretical Physics]]'', [[Pergamon Press]], 3rd Ed. (1994).
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| * [[Hagen Kleinert|Kleinert, H.]], ''Gauge Fields in Condensed Matter'', Vol. I, "[[:de:Supraflüssigkeit|Superfluid]] and [[vortex|Vortex lines]]; Disorder Fields, [[Phase Transition]]s,", pp. 1–742, [http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)]; Paperback ISBN 9971-5-0210-0 '' (readable online [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html physik.fu-berlin.de])
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| * [[Hagen Kleinert|Kleinert, H.]] and Verena Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)]; Paperback ISBN 981-02-4659-5'' (readable online [http://www.physik.fu-berlin.de/~kleinert/b8 here]).''
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| * Mussardo G., "Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics", [[Oxford University Press]], 2010.
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| *[[Manfred R. Schroeder|Schroeder, Manfred R.]], ''Fractals, chaos, power laws : minutes from an infinite paradise'', New York: [[W.H. Freeman]], 1991. Very well-written book in "semi-popular" style—not a textbook—aimed at an audience with some training in mathematics and the physical sciences. Explains what scaling in phase transitions is all about, among other things.
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| * Yeomans J. M., ''Statistical Mechanics of Phase Transitions'', Oxford University Press, 1992.
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| * H. E. Stanley, ''Introduction to Phase Transitions and Critical Phenomena'' (Oxford University Press, Oxford and New York 1971).
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| ==External links==
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| {{Commons category|Phase changes}}
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| * [http://www.ibiblio.org/e-notes/Perc/contents.htm Interactive Phase Transitions on lattices] with Java applets
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| {{States of matter}}
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| {{DEFAULTSORT:Phase Transition}}
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| [[Category:Concepts in physics]]
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| [[Category:Phase transitions| ]]
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| [[Category:Critical phenomena]]
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