Phase transition

This diagram shows the nomenclature for the different phase transitions.

Inphysics,chemistry,and other related fields like biology, aphase transition(orphase change) is thephysical processof transition between one state of a medium and another. Commonly the term is used to refer to changes among the basicstates of matter:solid,liquid,andgas,and in rare cases,plasma.A phase of athermodynamic systemand the states of matter have uniformphysical properties.During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such astemperatureorpressure.This can be a discontinuous change; for example, a liquid may become gas upon heating to itsboiling point,resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point.

Types of phase transition

States of matter

A simplified phase diagram forwater,showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure.

See also:vapor pressureandphase diagram

Phase transitions commonly refer to when a substance transforms between one of the fourstates of matterto another. At the phase transition point for a substance, for instance theboiling point,the two phases involved - liquid andvapor,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 boiling point the gaseous form is the more stable.

Common transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/orpressureare identified in the following table:

Phase transitionsof matter ()

To From	Solid	Liquid	Gas	Plasma
Solid		Melting	Sublimation
Liquid	Freezing		Vaporization
Gas	Deposition	Condensation		Ionization
Plasma			Recombination

For a single component, the most stable phase at different temperatures and pressures can be shown on aphase diagram.Such a diagram usually depicts states in equilibrium. A phase transition usually occurs when the pressure or temperature changes and the system crosses from one region to another, like water turning from liquid to solid as soon as the temperature drops below thefreezing point.In exception to the usual case, it is sometimes possible to change the state of a systemdiabatically(as opposed toadiabatically) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state ismetastable,i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs insuperheatingandsupercooling,for example. Metastable states do not appear on usual phase diagrams.

Structural

See also:Polymorphism (materials science)

A phase diagram showing theallotropes of iron,distinguishing between several differentcrystal structuresincludingferrite(α-iron) andaustenite(γ-iron).

Phase transitions can also occur when a solid changes to a different structure without changing its chemical makeup. In elements, this is known asallotropy,whereas in compounds it is known aspolymorphism.The change from onecrystal structureto another, from a crystalline solid to anamorphous solid,or from one amorphous structure to another (polyamorphs) are all examples of solid to solid phase transitions.

Themartensitic transformationoccurs as one of the many phase transformations in carbon steel and stands as a model fordisplacive phase transformations.Order-disorder transitions such as in Alpha -titanium aluminides.As with states of matter, there are also ametastableto equilibrium phase transformation for structural phase transitions. A metastable polymorph which forms rapidly due to lower surface energy will transform to an equilibrium phase given sufficient thermal input to overcome an energetic barrier.

Magnetic

See also:Magnetic structure

A phase diagram showing differentmagnetic structuresin the same crystal structure ofManganese monosilicide.

Phase transitions can also describe the change between different kinds ofmagnetic ordering.The most well-known is the transition between theferromagneticandparamagneticphases ofmagneticmaterials, which occurs at what is called theCurie point.Another example is the transition between differently ordered,commensurateorincommensurate,magnetic structures, such as in ceriumantimonide.A simplified but highly useful model of magnetic phase transitions is provided by theIsing Model

Mixtures

A binary phase diagram showing the most stable chemical compounds oftitaniumandnickelat differentmi xing ratiosandtemperatures.

Phase transitions involvingsolutionsandmixturesare more complicated than transitions involving a single compound. While chemically pure compounds exhibit a single temperaturemelting pointbetween solid and liquid phases, mixtures can either have a single melting point, known ascongruent melting,or they have differentliquidus and solidus temperaturesresulting in a temperature span where solid and liquid coexist in equilibrium. This is often the case insolid solutions,where the two components are isostructural.

There are also a number of phase transitions involving three phases: aeutectictransformation, 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 aeutectoidtransformation. Aperitectictransformation, in which a two-component single-phase solid is heated and transforms into a solid phase and a liquid phase. Aperitectoidreaction is a peritectoid rection, except involving only solid phases. Amonotecticreaction consists of change from a liquid and to a combination of a solid and a second liquid, where the two liquids display amiscibility gap.

Separation into multiple phases can occur viaspinodal decomposition,in which a single phase is cooled and separates into two different compositions.

Non-equilibrium mixtures can occur, such as insupersaturation.

Other examples

A small piece of rapidly melting solidargonshows two concurrent phase changes. The transition from solid to liquid, and gas to liquid (shown by the white condensed water vapour).

Other phase changes include:

• Transition to amesophasebetween solid and liquid, such as one of the "liquid crystal"phases.
• The dependence of theadsorptiongeometry on coverage and temperature, such as forhydrogenon iron (110).
• The emergence ofsuperconductivityin certain metals and ceramics when cooled below a critical temperature.
• The emergence ofmetamaterialproperties in artificial photonic media as their parameters are varied.
• Quantum condensation ofbosonicfluids (Bose–Einstein condensation). Thesuperfluidtransition in liquidheliumis an example of this.
• Thebreaking of symmetriesin the laws of physics during the early history of the universe as its temperature cooled.
• Isotope fractionationoccurs during a phase transition, the ratio of light to heavy isotopes in the involved molecules changes. Whenwater vaporcondenses (anequilibrium fractionation), the heavier water isotopes (¹⁸O and²H) become enriched in the liquid phase while the lighter isotopes (¹⁶O and¹H) tend toward the vapor phase.

Phase transitions occur when thethermodynamic free energyof a system isnon-analyticfor some choice of thermodynamic variables (cf.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 small. Phase transitions can occur for non-thermodynamic systems, where temperature is not a parameter. Examples include:quantum phase transitions,dynamic phase transitions, and topological (structural) phase transitions. In these types of systems other parameters take the place of temperature. For instance, connection probability replaces temperature for percolating networks.

Classifications

Ehrenfest classification

Paul Ehrenfestclassified phase transitions based on the behavior of thethermodynamic free energyas 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 transitionsexhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure.Second-order phase transitionsare continuous in the first derivative (theorder 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.These include the ferromagnetic phase transition in materials such as iron, where themagnetization,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 theCurie temperature.Themagnetic 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. For example, the Gross–Witten–Wadia phase transition in 2-d lattice quantum chromodynamics is a third-order phase transition.The Curie points of many ferromagnetics is also a third-order transition, as shown by their specific heat having a sudden change in slope.

In practice, only the first- and second-order phase transitions are typically observed. The second-order phase transition was for a while controversial, as it seems to require two sheets of the Gibbs free energy to osculate exactly, which is so unlikely as to never occur in practice.Cornelis Gorterreplied the criticism by pointing out that the Gibbs free energy surface might have two sheets on one side, but only one sheet on the other side, creating a forked appearance.(pp. 146--150)

The Ehrenfest classification implicitly allows for continuous phase transformations, where the bonding character of a material changes, but there is no discontinuity in any free energy derivative. An example of this occurs at thesupercritical liquid–gas boundaries.

The first example of a phase transition which did not fit into the Ehrenfest classification was the exact solution of theIsing model,discovered in 1944 byLars Onsager.The exact specific heat differed from the earliermean-fieldapproximations, which had predicted that it has a simple discontinuity at critical temperature. Instead, the exact specific heat had a logarithmic divergence at the critical temperature.In the following decades, the Ehrenfest classification was replaced by a simplified classification scheme that is able to incorporate such transitions.

Modern classifications

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:

First-order phase transitionsare those that involve alatent heat.During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. 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 intovapor,but forms aturbulentmixture of liquid water and vapor bubbles).Yoseph Imryand Michael Wortis showed thatquenched disordercan broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling.

Second-order phase transitionsare also called"continuous phase transitions".They are characterized by a divergent susceptibility, an infinitecorrelation length,and apower lawdecay of correlations nearcriticality.Examples of second-order phase transitions are theferromagnetictransition, superconducting transition (for aType-I superconductorthe phase transition is second-order at zero external field and for aType-II superconductorthe phase transition is second-order for both normal-state–mixed-state and mixed-state–superconducting-state transitions) and thesuperfluidtransition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperaturewhich enables accurate detection usingdifferential scanning calorimetrymeasurements.Lev Landaugave aphenomenologicaltheoryof second-order phase transitions.

Apart from isolated, simple phase transitions, there exist transition lines as well asmulticritical points,when varying external parameters like the magnetic field or composition.

Several transitions are known asinfinite-order phase transitions. They are continuous but break nosymmetries.The most famous example is theKosterlitz–Thouless transitionin the two-dimensionalXY model.Manyquantum phase transitions,e.g., intwo-dimensional electron gases,belong to this class.

Theliquid–glass transitionis observed in manypolymersand other liquids that can besupercooledfar 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 aquenched disorderstate, 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.No direct experimental evidence supports the existence of these transitions.

Characteristic properties

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 temperatureT_g,which may depend on the applied pressure.If the first-order freezing transition occurs over a range of temperatures, andT_gfalls 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 ofT_gfalling 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 betweenT_gandT_cin 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 thecritical 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 ofcritical 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 asymmetry breakingprocess. For instance, the cooling of a fluid into acrystalline solidbreaks continuoustranslation 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 tospontaneous symmetry breaking,with the exception of certainaccidental symmetries(e.g. the formation of heavyvirtual particles,which only occurs at low temperatures).

Order parameters

Anorder parameteris 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.At the critical point, the order parametersusceptibilitywill usually diverge.

An example of an order parameter is the netmagnetizationin aferromagneticsystem 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 theferromagneticphase, one must provide the netmagnetization,whose direction was spontaneously chosen when the system cooled below theCurie point.However, note that order parameters can also be defined for non-symmetry-breaking transitions.

Some phase transitions, such assuperconductingand 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 asvortex- ordefectlines.

Relevance in cosmology

Symmetry-breaking phase transitions play an important role incosmology.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 theelectroweak fieldinto the U(1) symmetry of the present-dayelectromagnetic field.This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according toelectroweak baryogenesistheory.

Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work ofEric ChaissonandDavid Layzer.

See alsorelational order theoriesandorder and disorder.

Critical exponents and universality classes

Main article:critical exponent

Continuous phase transitions are easier to study than first-order transitions due to the absence oflatent 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 ascritical exponents.The most important one is perhaps the exponent describing the divergence of the thermalcorrelation lengthby approaching the transition. For instance, let us examine the behavior of theheat capacitynear such a transition. We vary the temperatureTof the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperatureT_c.WhenTis nearT_c,the heat capacityCtypically has apower lawbehavior:

$C\propto |T_{\text{c}}-T|^{-\ Alpha }.$

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponentα= 0.59A similar behavior, but with the exponentνinstead ofα,applies for the correlation length.

The exponentνis positive. This is different withα.Its actual value depends on the type of phase transition we are considering.

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 −1 <α< 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at thelambda transitionfrom a normal state to thesuperfluidstate, for which experiments have foundα= −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 onvariational perturbation theory.

For 0 <α< 1, the heat capacity diverges at the transition temperature (though, sinceα< 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensionalIsing modelfor uniaxial magnets, detailed theoretical studies have yielded the exponentα≈ +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 alogarithmicdivergence. 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,andη,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.νandη.

It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known asuniversality.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 therenormalization grouptheory 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., besidesstatic functionsthere is alsocritical dynamics.As a consequence, at a phase transition one may observecritical slowing downorspeeding up.Connected to the previous phenomenon is also the phenomenon ofenhanced fluctuationsbefore the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The largestatic universality classesof a continuous phase transition split into smallerdynamic universalityclasses. 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.

Phase transitions in biological systems

Phase transitions play many important roles in biological systems. Examples include thelipid bilayerformation, thecoil-globule transitionin the process ofprotein foldingandDNA melting,liquid crystal-like transitions in the process ofDNA condensation,and cooperative ligand binding to DNA and proteins with the character of phase transition.

Inbiological membranes,gel to liquid crystalline phase transitions play a critical role in physiological functioning of biomembranes. In gel phase, due to low fluidity of membrane lipid fatty-acyl chains, membrane proteins have restricted movement and thus are restrained in exercise of their physiological role. Plants depend critically on photosynthesis bychloroplastthylakoid membraneswhich are exposed cold environmental temperatures. Thylakoid membranes retain innate fluidity even at relatively low temperatures because of high degree of fatty-acyl disorder allowed by their high content oflinolenic acid,18-carbon chain with 3-double bonds.Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence,spin labelelectron paramagnetic resonanceandNMRby recording measurements of the concerned parameter by at series of sample temperatures. A simple method for its determination from 13-C NMR line intensities has also been proposed.

It has been proposed that some biological systems might lie near critical points. Examples includeneural networksin the salamander retina,bird flocks gene expression networks in Drosophila,and protein folding.However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality.It has also been suggested that biological organisms share two key properties of phase transitions: the change of macroscopic behavior and the coherence of a system at a critical point.Phase transitions are prominent feature of motor behavior in biological systems.Spontaneous gait transitions,as well as fatigue-induced motor task disengagements,show typical critical behavior as an intimation of the sudden qualitative change of the previously stable motor behavioral pattern.

The characteristic feature of second order phase transitions is the appearance offractalsin somescale-freeproperties. It has long been known that protein globules are shaped by interactions with water. There are 20 amino acids that form side groups on protein peptide chains range fromhydrophilicto hydrophobic, causing the former to lie near the globular surface, while the latter lie closer to the globular center. Twenty fractals were discovered in solvent associated surface areas of > 5000 protein segments.The existence of these fractals proves that proteins function near critical points of second-order phase transitions.

In groups of organisms in stress (when approaching critical transitions), correlations tend to increase, while at the same time, fluctuations also increase. This effect is supported by many experiments and observations of groups of people, mice, trees, and grassy plants.

Phase transitions in social systems

See also:Complex system approach to peace and armed conflict § Phase transitions

Phase transitions have been hypothesised to occur in social systems viewed as dynamical systems. A hypothesis proposed in the 1990s and 2000s in the context ofpeace and armed conflictis that when a conflict that is non-violent shifts to a phase of armed conflict, this is a phase transition from latent to manifest phases within the dynamical system.

Experimental

A variety of methods are applied for studying the various effects. Selected examples are:

• Thermogravimetry(very common)
• X-ray diffraction
• Neutron diffraction
• Raman Spectroscopy
• SQUID(measurement of magnetic transitions)
• Hall effect(measurement of magnetic transitions)
• Mössbauer spectroscopy(simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800–1000 °C)
• Perturbed angular correlation(simultaneous measurement of magnetic and non-magnetic transitions. No temperature limits. Over 2000 °C already performed, theoretical possible up to the highest crystal material, such astantalum hafnium carbide4215 °C.)

See also

• Allotropy– Property of some chemical elements to exist in two or more different forms
• Autocatalytic reactions and order creation– Chemical reaction whose product is also its catalystPages displaying short descriptions of redirect targets
• Crystal growth– Major stage of a crystallization process
  • Abnormal grain growth– materials science phenomenonPages displaying wikidata descriptions as a fallback
• Differential scanning calorimetry– Thermoanalytical technique
• Diffusionless transformations– Shift of atomic positions in a crystal structurePages displaying short descriptions of redirect targets
• Ehrenfest equations
• Ising Model– Mathematical model of ferromagnetism in statistical mechanicsPages displaying short descriptions of redirect targets
• Jamming (physics)– apparent change of physical statePages displaying wikidata descriptions as a fallback
• Kelvin probe force microscope– Noncontact variant of atomic force microscopy
• Landau theory– Theory of continuous phase transitions of second order phase transitions
• Laser-heated pedestal growth– crystal growth techniquePages displaying wikidata descriptions as a fallback
• List of states of matter– Different known phase of states matter
• Micro-pulling-down– Crystal growth technique
• Percolation theory– Mathematical theory on behavior of connected clusters in a random graph
  • Continuum percolation theory
• Superfluid film– Thin layer of liquid in a superfluid state
• Superradiant phase transition– Process in quantum optics
• Topological quantum field theory– Field theory involving topological effects in physics

Further reading

• Anderson, P.W.,Basic Notions of Condensed Matter Physics,Perseus Publishing(1997).
• Faghri, A.,andZhang, Y.,Fundamentals of Multiphase Heat Transfer and Flow,Springer NatureSwitzerland AG, 2020.
• Fisher, M.E.(1974). "The renormalization group in the theory of critical behavior".Rev. Mod. Phys.46(4): 597–616.Bibcode:1974RvMP...46..597F.doi:10.1103/revmodphys.46.597.
• Goldenfeld, N.,Lectures on Phase Transitions and the Renormalization Group,Perseus Publishing (1992).
• Ivancevic, Vladimir G; Ivancevic, Tijana T (2008),Chaos, Phase Transitions, Topology Change and Path Integrals,Berlin: Springer,ISBN978-3-540-79356-4,retrieved14 March2013
• M.R.Khoshbin-e-Khoshnazar,Ice Phase Transition as a sample of finite system phase transition,(Physics Education(India)Volume 32. No. 2, Apr - Jun 2016)[1]
• Kleinert, H.,Gauge Fields in Condensed Matter,Vol. I, "SuperfluidandVortex lines;Disorder Fields,Phase Transitions",pp. 1–742,World Scientific (Singapore, 1989);PaperbackISBN9971-5-0210-0(readable onlinephysik.fu-berlin.de)
• Kleinert, H.and Verena Schulte-Frohlinde,Critical Properties of φ⁴-Theories,World Scientific (Singapore, 2001);PaperbackISBN981-02-4659-5(readable online here[2]).
• Kogut, J.;Wilson, K(1974). "The Renormalization Group and the epsilon-Expansion".Phys. Rep.12(2): 75–199.Bibcode:1974PhR....12...75W.doi:10.1016/0370-1573(74)90023-4.
• Krieger, Martin H.,Constitutions of matter: mathematically modelling the most everyday of physical phenomena,University of Chicago Press,1996. Contains a detailed pedagogical discussion ofOnsager's solution of the 2-D Ising Model.
• Landau, L.D.andLifshitz, E.M.,Statistical Physics Part 1,vol. 5 ofCourse of Theoretical Physics,Pergamon Press, 3rd Ed. (1994).
• Mussardo G., "Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics", Oxford University Press, 2010.
• 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.
• H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena(Oxford University Press, Oxford and New York 1971).
• Yeomans J. M.,Statistical Mechanics of Phase Transitions,Oxford University Press, 1992.

External links

• Interactive Phase Transitions on latticeswith Java applets
• Universality classesfrom Sklogwiki