In chemistry, thermodynamics, and many other related fields, **phase transitions** (or **phase changes**) are the physical processes of transition between the basic states of matter: solid, liquid, and gas, as well as plasma in rare cases.

- Types of phase transition
- Classifications
- Ehrenfest classification
- Modern classifications
- Characteristic properties
- Phase coexistence
- Critical points
- Symmetry
- Order parameters
- Relevance in cosmology
- Critical exponents and universality classes
- Critical slowing down and other phenomena
- Percolation theory
- Phase transitions in biological systems
- Experimental
- See also
- References
- Further reading
- External links

A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change, often discontinuously, as a result of the change of external conditions, such as temperature, pressure, or 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. Phase transitions commonly occur in nature and are used today in many technologies.

Examples of phase transitions include:

- The transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/or pressure:

To | |||||
---|---|---|---|---|---|

Solid | Liquid | Gas | Plasma | ||

From | Solid | Melting | Sublimation | ||

Liquid | Freezing | Vaporization | |||

Gas | Deposition | Condensation | Ionization | ||

Plasma | Recombination |

*See also vapor pressure and phase diagram*

Condensed matter physics |
---|

Phases · Phase transition · QCP |

- 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.
- A metastable to equilibrium phase transformation. 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.
- A peritectic transformation, in which a two-component single-phase solid is heated and transforms into a solid phase and a liquid phase.
- A spinodal decomposition, in which a single phase is cooled and separates into two different compositions of that same phase.
- Transition to a mesophase between solid and liquid, such as one of the "liquid crystal" phases.
- The transition between the ferromagnetic and paramagnetic phases of magnetic materials at the Curie point.
- The transition between differently ordered, commensurate or incommensurate, magnetic structures, such as in cerium antimonide.
- The martensitic transformation which occurs as one of the many phase transformations in carbon steel and stands as a model for displacive phase transformations.
- Changes in the crystallographic structure such as between ferrite and austenite of iron.
- Order-disorder transitions such as in alpha-titanium aluminides.
- The dependence of the adsorption geometry on coverage and temperature, such as for hydrogen on iron (110).
- The emergence of superconductivity in certain metals and ceramics when cooled below a critical temperature.
- The transition between different molecular structures (polymorphs, allotropes or polyamorphs), especially of solids, such as between an amorphous structure and a crystal structure, between two different crystal structures, or between two amorphous structures.
- Quantum condensation of bosonic fluids (Bose–Einstein condensation). The superfluid transition in liquid helium is an example of this.
- The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled.
- Isotope fractionation occurs during a phase transition, the ratio of light to heavy isotopes in the involved molecules changes. When water vapor condenses (an equilibrium fractionation), the heavier water isotopes (
^{18}O and^{2}H) become enriched in the liquid phase while the lighter isotopes (^{16}O and^{1}H) tend toward the vapor phase.^{ [1] }

Phase transitions occur when the thermodynamic free energy of a system is non-analytic for 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 too small. It is important to note that phase transitions can occur and are defined 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.

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.

It is sometimes possible to change the state of a system diabatically (as opposed to adiabatically) 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.

Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy as a function of other thermodynamic variables.^{ [2] } 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.^{ [3] } 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 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.^{ [3] } 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.

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 the supercritical liquid–gas boundaries.

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:^{ [2] }

**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 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.^{ [4] }^{ [5] } Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into vapor, but forms a turbulent mixture of liquid water and vapor bubbles). Imry and Wortis showed that quenched disorder can 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.^{ [6] }^{ [7] }^{ [8] }

**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^{ [9] } which enables accurate detection using differential scanning calorimetry measurements. Lev Landau gave a phenomenological theory of second-order phase transitions.

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

Several transitions are known as *infinite-order phase transitions*. They are continuous but break no symmetries. The most famous example is the Kosterlitz–Thouless transition in the two-dimensional XY model. Many quantum phase transitions, e.g., in two-dimensional electron gases, belong to this class.

The liquid–glass transition is observed in many polymers and other liquids that can be 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.^{ [10] }^{ [11] } No direct experimental evidence supports the existence of these transitions.

The gelation transition of colloidal particles has been shown to be a second-order phase transition under nonequilibrium conditions.^{ [12] }

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.^{ [13] } This slowing down happens below a glass-formation temperature *T*_{g}, which may depend on the applied pressure.^{ [9] }^{ [14] } If the first-order freezing transition occurs over a range of temperatures, and *T*_{g} falls 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,^{ [15] } such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials,^{ [16] }^{ [17] } magnetocaloric materials,^{ [18] } magnetic shape memory materials,^{ [19] } and other materials.^{ [20] } The interesting feature of these observations of *T*_{g} 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 fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between *T*_{g} and *T*_{c} in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.

In any system containing liquid and gaseous phases, there exists 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 involve a symmetry breaking process. For instance, the cooling of a fluid into a crystalline solid breaks continuous translation 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 to spontaneous symmetry breaking, with the exception of certain accidental symmetries (e.g. the formation of heavy virtual particles, which only occurs at low temperatures).^{ [21] }

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.^{ [22] } At the critical point, the order parameter susceptibility will usually diverge.

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.

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.

There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as vortex- or defect lines.

Symmetry-breaking phase transitions play an important role in cosmology. 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 explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to electroweak baryogenesis theory.

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 ^{ [23] } and David Layzer.^{ [24] }

See also relational order theories and order and disorder.

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. 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*_{c}. When *T* is near *T*_{c}, the heat capacity *C* typically has a power law behavior:

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent *α* = 0.59^{ [25] } A similar behavior, but with the exponent *ν* instead of *α*, applies for the correlation length.

The exponent *ν* is positive. This is different with *α*. Its actual value depends on the type of phase transition we are considering.

It is widely believed that the critical exponents are the same above and below the critical temperature. It has now been shown that this is not necessarily true: 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 , the exponent of the susceptibility) are not identical.^{ [26] }

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. At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.^{ [27] } This experimental value of α agrees with theoretical predictions based on variational perturbation theory.^{ [28] }

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 3D 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 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 logarithmic divergence. 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*, 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

It can be shown that there are only two independent exponents, e.g. *ν* and *η*.

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 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 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 divergence of the correlation length is the essential point.

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.

Another phenomenon which shows phase transitions and critical exponents is 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*_{c} a giant cluster is formed, and we have a second-order phase transition.^{ [29] } The behavior of *P*_{∞} near *p*_{c} is *P*_{∞} ~ (*p* − *p*_{c})^{β}, where *β* is a critical exponent. Using percolation theory one can define all critical exponents that appear in phase transitions.^{ [30] }^{ [29] } External fields can be also defined for second order percolation systems^{ [31] } as well as for first order percolation^{ [32] } systems. Percolation has been found useful to study urban traffic and for identifying repetitive bottlenecks.^{ [33] }^{ [34] }

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.^{ [35] }

In *biological 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 by chloroplast thylakoid membranes which 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 of linolenic acid, 18-carbon chain with 3-double bonds.^{ [36] } Gel-to-liquid crystalline phase transition temperature of biological membranes can be determined by many techniques including calorimetry, fluorescence, spin label electron paramagnetic resonance and NMR by 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.^{ [37] }

It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina,^{ [38] } bird flocks^{ [39] } gene expression networks in Drosophila,^{ [40] } and protein folding.^{ [41] } However, it is not clear whether or not alternative reasons could explain some of the phenomena supporting arguments for criticality.^{ [42] } 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.^{ [43] }

The characteristic feature of second order phase transitions is the appearance of fractals in some scale-free properties. 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 from hydrophilic to 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.^{ [44] } 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.^{ [45] }

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 as tantalum hafnium carbide 4215 °C.)

- Allotropy
- Autocatalytic reactions and order creation
- Crystal growth
- Differential scanning calorimetry
- Diffusionless transformations
- Ehrenfest equations
- Jamming (physics)
- Kelvin probe force microscope
- Landau theory of second order phase transitions
- Laser-heated pedestal growth
- List of states of matter
- Micro-pulling-down
- Percolation theory
- Superfluid film
- Superradiant phase transition
- Topological quantum field theory

**Condensed matter physics** is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the subject deals with "condensed" phases of matter: systems of many constituents with strong interactions between them. More exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other theories to develop mathematical models.

**Superconductivity** is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a **superconductor**. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source.

**High-temperature superconductors** are operatively defined as materials that behave as superconductors at temperatures above 77 K, the boiling point of liquid nitrogen, one of the simplest coolants in cryogenics. All materials currently known to conduct at ordinary pressures become superconducting at temperatures far below ambient, and therefore require cooling. The majority of high-temperature superconductors are ceramic materials. On the other hand, *Metallic* superconductors usually work below −200 °C: they are then called *low-temperature superconductors*. Metallic superconductors are also *ordinary superconductors*, since they were discovered and used before the high-temperature ones.

In statistical physics and mathematics, **percolation theory** describes the behavior of a network when nodes or links are removed. This is a geometric type of phase transition, since at a critical fraction of removal the network breaks into significantly smaller connected clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation.

**Spontaneous symmetry breaking** is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

In statistical mechanics, a **universality class** is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in the class.

The **classical XY model** is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's *n*-vector model for *n* = 2.

A **magnon** is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of energy and lattice momentum, and are spin-1, indicating they obey boson behavior.

The **Lambda point** is the temperature at which normal fluid helium makes the transition to superfluid helium II. The lowest pressure at which He-I and He-II can coexist is the vapor−He-I−He-II triple point at 2.1768 K (−270.9732 °C) and 5.048 kPa (0.04982 atm), which is the "saturated vapor pressure" at that temperature. The highest pressure at which He-I and He-II can coexist is the bcc−He-I−He-II triple point with a helium solid at 1.762 K (−271.388 °C), 29.725 atm (3,011.9 kPa).

**Critical exponents** describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on:

**Multiferroics** are defined as materials that exhibit more than one of the primary ferroic properties in the same phase:

In physics, **topological order** is a kind of order in the zero-temperature phase of matter. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders cannot change into each other without a phase transition.

A **quantum critical point** is a point in the phase diagram of a material where a continuous phase transition takes place at absolute zero. A quantum critical point is typically achieved by a continuous suppression of a nonzero temperature phase transition to zero temperature by the application of a pressure, field, or through doping. Conventional phase transitions occur at nonzero temperature when the growth of random thermal fluctuations leads to a change in the physical state of a system. Condensed matter physics research over the past few decades has revealed a new class of phase transitions called quantum phase transitions which take place at absolute zero. In the absence of the thermal fluctuations which trigger conventional phase transitions, quantum phase transitions are driven by the zero point quantum fluctuations associated with Heisenberg's uncertainty principle.

In statistical physics, **directed percolation** (**DP**) refers to a class of models that mimic filtering of fluids through porous materials along a given direction, due to the effect of gravity. Varying the microscopic connectivity of the pores, these models display a phase transition from a macroscopically permeable (percolating) to an impermeable (non-percolating) state. Directed percolation is also used as a simple model for epidemic spreading with a transition between survival and extinction of the disease depending on the infection rate.

The **glass–liquid transition**, or **glass transition**, is the gradual and reversible transition in amorphous materials from a hard and relatively brittle "glassy" state into a viscous or rubbery state as the temperature is increased. An amorphous solid that exhibits a glass transition is called a glass. The reverse transition, achieved by supercooling a viscous liquid into the glass state, is called vitrification.

**Subir Sachdev** is Herchel Smith Professor of Physics at Harvard University specializing in condensed matter. He was elected to the U.S. National Academy of Sciences in 2014, and received the Lars Onsager Prize from the American Physical Society and the Dirac Medal from the ICTP in 2018.

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of *universal* critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.

In condensed matter physics, a **quantum spin liquid** is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

The **Kibble–Zurek mechanism** (**KZM**) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

**Amnon Aharony** is an Israeli Professor (Emeritus) of Physics in the School of Physics and Astronomy at Tel Aviv University, Israel and in the Physics Department of Ben Gurion University of the Negev, Israel. After years of research on statistical physics, his current research focuses on condensed matter theory, especially in mesoscopic physics and spintronics. He is a member of the Israel Academy of Sciences and Humanities, a Foreign Honorary Member of the American Academy of Arts and Sciences and of several other academies. He also received several prizes, including the Rothschild Prize in Physical Sciences, and the Randers prize(no), awarded every other year by the King of Norway.

- ↑ Carol Kendall (2004). "Fundamentals of Stable Isotope Geochemistry". USGS. Retrieved 10 April 2014.
- 1 2 Jaeger, Gregg (1 May 1998). "The Ehrenfest Classification of Phase Transitions: Introduction and Evolution".
*Archive for History of Exact Sciences*.**53**(1): 51–81. doi:10.1007/s004070050021. S2CID 121525126. - 1 2 Blundell, Stephen J.; Katherine M. Blundell (2008).
*Concepts in Thermal Physics*. Oxford University Press. ISBN 978-0-19-856770-7. - ↑ Faghri, A., and Zhang, Y.,
*Transport Phenomena in Multiphase Systems*, Elsevier, Burlington, MA, 2006, - ↑ Faghri, A., and Zhang, Y.,
*Fundamentals of Multiphase Heat Transfer and Flow*, Springer, New York, NY, 2020 - ↑ Imry, Y.; Wortis, M. (1979). "Influence of quenched impurities on first-order phase transitions".
*Phys. Rev. B*.**19**(7): 3580–3585. Bibcode:1979PhRvB..19.3580I. doi:10.1103/physrevb.19.3580. - ↑ Kumar, Kranti; Pramanik, A. K.; Banerjee, A.; Chaddah, P.; Roy, S. B.; Park, S.; Zhang, C. L.; Cheong, S.-W. (2006). "Relating supercooling and glass-like arrest of kinetics for phase separated systems: DopedCeFe2and(La,Pr,Ca)MnO3".
*Physical Review B*.**73**(18): 184435. arXiv: cond-mat/0602627 . Bibcode:2006PhRvB..73r4435K. doi:10.1103/PhysRevB.73.184435. ISSN 1098-0121. S2CID 117080049. - ↑ Pasquini, G.; Daroca, D. Pérez; Chiliotte, C.; Lozano, G. S.; Bekeris, V. (2008). "Ordered, Disordered, and Coexistent Stable Vortex Lattices inNbSe2Single Crystals".
*Physical Review Letters*.**100**(24): 247003. arXiv: 0803.0307 . Bibcode:2008PhRvL.100x7003P. doi:10.1103/PhysRevLett.100.247003. ISSN 0031-9007. PMID 18643617. S2CID 1568288. - 1 2 Ojovan, M.I. (2013). "Ordering and structural changes at the glass-liquid transition".
*J. Non-Cryst. Solids*.**382**: 79–86. Bibcode:2013JNCS..382...79O. doi:10.1016/j.jnoncrysol.2013.10.016. - ↑ Gotze, Wolfgang. "Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory."
- ↑ Lubchenko, V. Wolynes; Wolynes, Peter G. (2007). "Theory of Structural Glasses and Supercooled Liquids".
*Annual Review of Physical Chemistry*.**58**: 235–266. arXiv: cond-mat/0607349 . Bibcode:2007ARPC...58..235L. doi:10.1146/annurev.physchem.58.032806.104653. PMID 17067282. S2CID 46089564. - ↑ Rouwhorst, J; Ness, C.; Soyanov, S.; Zaccone, A.; Schall, P (2020). "Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction".
*Nature Communications*.**11**(1): 3558. arXiv: 2007.10691 . Bibcode:2020NatCo..11.3558R. doi: 10.1038/s41467-020-17353-8 . PMC 7367344 . PMID 32678089. - ↑ Greer, A. L. (1995). "Metallic Glasses".
*Science*.**267**(5206): 1947–1953. Bibcode:1995Sci...267.1947G. doi:10.1126/science.267.5206.1947. PMID 17770105. S2CID 220105648. - ↑ Tarjus, G. (2007). "Materials science: Metal turned to glass".
*Nature*.**448**(7155): 758–759. Bibcode:2007Natur.448..758T. doi: 10.1038/448758a . PMID 17700684. S2CID 4410586. - ↑ Manekar, M. A.; Chaudhary, S.; Chattopadhyay, M. K.; Singh, K. J.; Roy, S. B.; Chaddah, P. (2001). "First-order transition from antiferromagnetism to ferromagnetism inCe(Fe
_{0.96}Al_{0.04})_{2}".*Physical Review B*.**64**(10): 104416. arXiv: cond-mat/0012472 . Bibcode:2001PhRvB..64j4416M. doi:10.1103/PhysRevB.64.104416. ISSN 0163-1829. S2CID 16851501. - ↑ Banerjee, A.; Pramanik, A. K.; Kumar, Kranti; Chaddah, P. (2006). "Coexisting tunable fractions of glassy and equilibrium long-range-order phases in manganites".
*Journal of Physics: Condensed Matter*.**18**(49): L605. arXiv: cond-mat/0611152 . Bibcode:2006JPCM...18L.605B. doi:10.1088/0953-8984/18/49/L02. S2CID 98145553. - ↑ Wu W., Israel C., Hur N., Park S., Cheong S. W., de Lozanne A. (2006). "Magnetic imaging of a supercooling glass transition in a weakly disordered ferromagnet".
*Nature Materials*.**5**(11): 881–886. Bibcode:2006NatMa...5..881W. doi:10.1038/nmat1743. PMID 17028576. S2CID 9036412.CS1 maint: uses authors parameter (link) - ↑ Roy, S. B.; Chattopadhyay, M. K.; Chaddah, P.; Moore, J. D.; Perkins, G. K.; Cohen, L. F.; Gschneidner, K. A.; Pecharsky, V. K. (2006). "Evidence of a magnetic glass state in the magnetocaloric material Gd
_{5}Ge_{4}".*Physical Review B*.**74**(1): 012403. Bibcode:2006PhRvB..74a2403R. doi:10.1103/PhysRevB.74.012403. ISSN 1098-0121. - ↑ Lakhani, Archana; Banerjee, A.; Chaddah, P.; Chen, X.; Ramanujan, R. V. (2012). "Magnetic glass in shape memory alloy: Ni
_{45}Co_{5}Mn_{38}Sn_{12}".*Journal of Physics: Condensed Matter*.**24**(38): 386004. arXiv: 1206.2024 . Bibcode:2012JPCM...24L6004L. doi:10.1088/0953-8984/24/38/386004. ISSN 0953-8984. PMID 22927562. S2CID 206037831. - ↑ Kushwaha, Pallavi; Lakhani, Archana; Rawat, R.; Chaddah, P. (2009). "Low-temperature study of field-induced antiferromagnetic-ferromagnetic transition in Pd-doped Fe-Rh".
*Physical Review B*.**80**(17): 174413. arXiv: 0911.4552 . Bibcode:2009PhRvB..80q4413K. doi:10.1103/PhysRevB.80.174413. ISSN 1098-0121. S2CID 119165221. - ↑ Ivancevic, Vladimir G.; Ivancevic, Tijiana, T. (2008).
*Complex Nonlinearity*. Berlin: Springer. pp. 176–177. ISBN 978-3-540-79357-1 . Retrieved 12 October 2014. - ↑ A. D. McNaught and A. Wilkinson, ed. (1997).
*Compendium of Chemical Terminology*. IUPAC. ISBN 978-0-86542-684-9 . Retrieved 23 October 2007.^{[ permanent dead link ]} - ↑ Chaisson, Eric J. (2001).
*Cosmic Evolution*. Harvard University Press. ISBN 9780674003422. - ↑ David Layzer,
*Cosmogenesis, The Development of Order in the Universe*, Oxford Univ. Press, 1991 - ↑ Ojovan, Michael I.; Lee, William E. (2006). "Topologically disordered systems at the glass transition" (PDF).
*Journal of Physics: Condensed Matter*.**18**(50): 11507–11520. Bibcode:2006JPCM...1811507O. doi:10.1088/0953-8984/18/50/007. - ↑ Leonard, F.; Delamotte, B. (2015). "Critical exponents can be different on the two sides of a transition".
*Phys. Rev. Lett*.**115**(20): 200601. arXiv: 1508.07852 . Bibcode:2015PhRvL.115t0601L. doi:10.1103/PhysRevLett.115.200601. PMID 26613426. S2CID 22181730. - ↑ Lipa, J.; Nissen, J.; Stricker, D.; Swanson, D.; Chui, T. (2003). "Specific heat of liquid helium in zero gravity very near the lambda point".
*Physical Review B*.**68**(17): 174518. arXiv: cond-mat/0310163 . Bibcode:2003PhRvB..68q4518L. doi:10.1103/PhysRevB.68.174518. S2CID 55646571. - ↑ Kleinert, Hagen (1999). "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions".
*Physical Review D*.**60**(8): 085001. arXiv: hep-th/9812197 . Bibcode:1999PhRvD..60h5001K. doi:10.1103/PhysRevD.60.085001. - 1 2 Armin Bunde and Shlomo Havlin (1996).
*Fractals and Disordered Systems*. Springer. - ↑ Stauffer, Dietrich; Aharony, Amnon (1994). "Introduction to Percolation Theory".
*Publ. Math*.**6**: 290–297. ISBN 978-0-7484-0253-3.CS1 maint: multiple names: authors list (link) - ↑ Gaogao Dong, Jingfang Fan, Louis M Shekhtman, Saray Shai, Ruijin Du, Lixin Tian,Xiaosong Chen, H Eugene Stanley, Shlomo Havlin (2018). "Resilience of networks with community structure behaves as if under an external field".
*Proceedings of the National Academy of Sciences*.**115**(25): 6911.CS1 maint: multiple names: authors list (link) - ↑ Bnaya Gross, Hillel Sanhedrai, Louis Shekhtman, Shlomo Havlin (2020). "Interconnections between networks acting like an external field in a first-order percolation transition".
*Physical Review E*.**101**(2): 022316.CS1 maint: multiple names: authors list (link) - ↑ D. Li, B. Fu, Y. Wang, G. Lu, Y. Berezin, H.E. Stanley, S. Havlin (2015). "Percolation transition in dynamical traffic network with evolving critical bottlenecks".
*PNAS*.**112**: 669.CS1 maint: multiple names: authors list (link) - ↑ Guanwen Zeng, Daqing Li, Shengmin Guo, Liang Gao, Ziyou Gao, HEugene Stanley, Shlomo Havlin (2019). "Switch between critical percolation modes in city traffic dynamics".
*Proceedings of the National Academy of Sciences*.**116**(1): 23.CS1 maint: multiple names: authors list (link) - ↑ D.Y. Lando and V.B. Teif (2000). "Long-range interactions between ligands bound to a DNA molecule give rise to adsorption with the character of phase transition of the first kind".
*J. Biomol. Struct. Dyn*.**17**(5): 903–911. doi:10.1080/07391102.2000.10506578. PMID 10798534. S2CID 23837885. - ↑ YashRoy, R.C. (1987). "13-C NMR studies of lipid fatty acyl chains of chloroplast membranes".
*Indian Journal of Biochemistry and Biophysics*.**24**(6): 177–178. - ↑ YashRoy, R C (1990). "Determination of membrane lipid phase transition temperature from 13-C NMR intensities".
*Journal of Biochemical and Biophysical Methods*.**20**(4): 353–356. doi:10.1016/0165-022X(90)90097-V. PMID 2365951. - ↑ Tkacik, Gasper; Mora, Thierry; Marre, Olivier; Amodei, Dario; Berry II, Michael J.; Bialek, William (2014). "Thermodynamics for a network of neurons: Signatures of criticality". arXiv: 1407.5946 [q-bio.NC].
- ↑ Bialek, W; Cavagna, A; Giardina, I (2014). "Social interactions dominate speed control in poising natural flocks near criticality".
*PNAS*.**111**(20): 7212–7217. arXiv: 1307.5563 . Bibcode:2014PNAS..111.7212B. doi:10.1073/pnas.1324045111. PMC 4034227 . PMID 24785504. - ↑ Krotov, D; Dubuis, J O; Gregor, T; Bialek, W (2014). "Morphogenesis at criticality".
*PNAS*.**111**(10): 3683–3688. arXiv: 1309.2614 . Bibcode:2014PNAS..111.3683K. doi:10.1073/pnas.1324186111. PMC 3956198 . PMID 24516161. - ↑ Mora, Thierry; Bialek, William (2011). "Are biological systems poised at criticality?".
*Journal of Statistical Physics*.**144**(2): 268–302. arXiv: 1012.2242 . Bibcode:2011JSP...144..268M. doi:10.1007/s10955-011-0229-4. S2CID 703231. - ↑ Schwab, David J; Nemenman, Ilya; Mehta, Pankaj (2014). "Zipf's law and criticality in multivariate data without fine-tuning".
*Physical Review Letters*.**113**(6): 068102. arXiv: 1310.0448 . Bibcode:2014PhRvL.113f8102S. doi:10.1103/PhysRevLett.113.068102. PMC 5142845 . PMID 25148352. - ↑ Longo, G.; Montévil, M. (1 August 2011). "From physics to biology by extending criticality and symmetry breakings".
*Progress in Biophysics and Molecular Biology*. Systems Biology and Cancer.**106**(2): 340–347. arXiv: 1103.1833 . doi:10.1016/j.pbiomolbio.2011.03.005. PMID 21419157. S2CID 723820. - ↑ Moret, Marcelo; Zebende, Gilney (January 2007). "Amino acid hydrophobicity and accessible surface area".
*Physical Review E*.**75**(1): 011920. Bibcode:2007PhRvE..75a1920M. doi:10.1103/PhysRevE.75.011920. PMID 17358197. - ↑ Gorban, A.N.; Smirnova, E.V.; Tyukina, T.A. (August 2010). "Correlations, risk and crisis: From physiology to finance".
*Physica A: Statistical Mechanics and Its Applications*.**389**(16): 3193–3217. arXiv: 0905.0129 . Bibcode:2010PhyA..389.3193G. doi:10.1016/j.physa.2010.03.035.

- Anderson, P.W.,
*Basic Notions of Condensed Matter Physics*, Perseus Publishing (1997). - Faghri, A., and Zhang, Y., Fundamentals of Multiphase Heat Transfer and Flow, Springer Nature Switzerland 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, ISBN 978-3-540-79356-4 , retrieved 14 March 2013 - 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) - Kleinert, H.,
*Gauge Fields in Condensed Matter*, Vol. I, "Superfluid and Vortex lines; Disorder Fields, Phase Transitions,", pp. 1–742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (readable online physik.fu-berlin.de) - Kleinert, H. and Verena Schulte-Frohlinde,
*Critical Properties of φ*, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4659-5^{4}-Theories*(readable online here).* - 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 of Onsager's solution of the 2-D Ising Model. - Landau, L.D. and Lifshitz, E.M.,
*Statistical Physics Part 1*, vol. 5 of*Course 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.

Wikimedia Commons has media related to . Phase changes |

- Interactive Phase Transitions on lattices with Java applets
- Universality classes from Sklogwiki

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.