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Encyclopedia > Phase transition
This diagram shows the nomenclature for the different phase transitions.

In thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another. The distinguishing characteristic of a phase transition is an abrupt change in one or more physical properties, in particular the heat capacity, with a small change in a thermodynamic variable such as the temperature. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For other uses, see Temperature (disambiguation). ...

In the English vernacular, the term is most commonly used to describe transitions between solid, liquid and gaseous states of matter, in rare cases including plasma. For other uses, see Solid (disambiguation). ... For other uses, see Liquid (disambiguation). ... For other uses, see Gas (disambiguation). ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... For other uses, see Plasma. ...

## Types of phase transition GA_googleFillSlot("encyclopedia_square");

A typical phase diagram. The dotted line gives the anomalous behaviour of water
A small piece of rapidly melting argon ice simultaneously shows the transitions from solid to liquid to gas.

Examples of phase transitions include: Image File history File links Phase-diag. ... Image File history File links Phase-diag. ... Image File history File links Download high-resolution version (1829x1632, 569 KB)A small (~2 cm long) piece of rapidly melting argon ice (the liquid is flowing off at the bottom) which has been frozen by allowing a slow stream of the gas to flow into a small graduated cylinder... Image File history File links Download high-resolution version (1829x1632, 569 KB)A small (~2 cm long) piece of rapidly melting argon ice (the liquid is flowing off at the bottom) which has been frozen by allowing a slow stream of the gas to flow into a small graduated cylinder...

To
From Solid Liquid Gas Plasma
Solid Solid-Solid Transformation Melting Sublimation -
Liquid Freezing N/A Boiling/Evaporation -
Gas Deposition Condensation N/A Ionization
Plasma - - Recombination/Deionization N/A
• 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 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.
• 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 emergence of superconductivity in certain metals when cooled below a critical temperature.
• The transition between different molecular structures (polymorphs or allotropes), especially of solids, such as between an amorphous structure and a crystal structure or between two different crystal structures.
• Quantum condensation of bosonic fluids, such as Bose-Einstein condensation and the superfluid transition in liquid helium.
• The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled.
• Phase transitions in intractable computational complexity problems such as NP-complete or PSPACE problems. For example it has been noticed in k-SAT problems that the transition from solvable to unsolvable instances exhibits threshold behavior depending on the ratio of number of clauses to number of variables. Moreover, the amount of computational time required to solve the problem or determine it to be unsolvable increases drastically around the threshold. This line of research comes mostly from investigating similarities between computational complexity and statistical physics.

It is sometimes possible to change the state of a system non-adiabatically in such a way that it can be brought past a phase transition without undergoing a phase transition. The resulting state is metastable i.e. not theoretically stable, but quasistable. See superheating, supercooling and supersaturation. This article covers adiabatic processes in thermodynamics. ... Metastability is the ability of a non-equilibrium state to persist for a long period of time. ... In physics, superheating (sometimes referred to as boiling retardation, boiling delay, or defervescence) is the phenomenon in which a liquid is heated to a temperature higher than its standard boiling point, without actually boiling. ... Supercooling is the process of chilling a liquid below its freezing point, without it becoming solid. ... The term supersaturation refers to a solution that contains more of the dissolved material than could be dissolved by the solvent under normal circumstances. ...

## Classification of phase transitions

### Ehrenfest classification

The first attempt at classifying phase transitions was the Ehrenfest classification scheme, which grouped phase transitions based on the degree of non-analyticity involved. Though useful, Ehrenfest's classification is flawed, as will be discussed in the next section. Paul Ehrenfest Paul Ehrenfest (Vienna, January 18, 1880 â€“ Amsterdam, September 25, 1933) was an Austrian physicist and mathematician, who obtained Dutch citizenship on March 24, 1922. ...

Under this scheme, phase transitions were labelled by the lowest derivative of the free energy that is discontinuous at the transition. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable. The various solid/liquid/gas transitions are classified as first-order transitions 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 have a discontinuity in a second derivative of the free energy. These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classication scheme, there could in principle be third, fourth, and higher-order phase transitions. The thermodynamic free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... For other uses, see Iron (disambiguation). ... The Curie point is a term in physics and materials science, named after Pierre Curie (1859-1906), and refers to a characteristic property of a ferromagnetic material. ... In physics, the susceptibility of a material or substance describes its response to an applied field. ...

### Modern classification of phase transitions

The Ehrenfest scheme is an inaccurate method of classifying phase transitions, for it does not take into account the case where a derivative of free energy diverges (which is only possible in the thermodynamic limit). For instance, in the ferromagnetic transition, the heat capacity diverges to infinity. For a non-technical overview of the subject, see Calculus. ... The thermodynamic free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... For other uses, see Infinity (disambiguation). ...

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

The first-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy. Because energy cannot be instantaneously transferred between the system and its environment, first-order transitions are associated with "mixed-phase regimes" in which some parts of the system have completed the transition and others have not. This phenomenon is familiar to anyone who has boiled a pot of water: the water does not instantly turn into gas, but forms a turbulent mixture of water and water vapor bubbles. Mixed-phase systems are difficult to study, because their dynamics are violent and hard to control. However, many important phase transitions fall in this category, including the solid/liquid/gas transitions and Bose-Einstein condensation. In thermochemistry, latent heat is the amount of energy in the form of heat released or absorbed by a substance during evaporation. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ... Water vapor or water vapour (see spelling differences), also aqueous vapor, is the gas phase of water. ... A Boseâ€“Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero. ...

The second class of phase transitions are the continuous phase transitions, also called second-order phase transitions. These have no associated latent heat. Examples of second-order phase transitions are the ferromagnetic transition and the superfluid transition. Helium II will creep along surfaces in order to find its own level - after a short while, the levels in the two containers will equalize. ...

Several transitions are known as the 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 in two-dimensional electron gases belong to this class. The Kosterlitz-Thouless transition is a special transition seen in the the XY model for interacting spin systems. ... Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... A quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). ... In physics, the free electron model is a possible model for the behaviour of electrons in a crystal structure. ...

## Properties of phase transitions

### Critical points

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. In physics, a critical point is the point of termination of a phase equilibrium curve, which separates two distinct phases. ...

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). Critical Opalescence is a phenomenon in liquids close to their critical point, in which a normally transparent liquid appears milky due to density fluctuations at all possible wavelengths. ...

During a phase change, the temperature of the sample does not change until the phase change is complete.

### Symmetry

Phase transitions often (but not always) take place between phases with different symmetry. Consider, for example, the transition between a fluid (i.e. liquid or gas) and a crystalline solid. A fluid, which is composed of atoms arranged in a disordered but homogeneous manner, possesses continuous translational symmetry: each point inside the fluid has the same properties as any other point. A crystalline solid, on the other hand, is made up of atoms arranged in a regular lattice. Each point in the solid is not similar to other points, unless those points are displaced by an amount equal to some lattice spacing. Sphere symmetry group o. ... For other uses, see Crystal (disambiguation). ... Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

Generally, we may speak of one phase in a phase transition as being more symmetrical than the other. The transition from the more symmetrical phase to the less symmetrical one is a symmetry-breaking process. In the fluid-solid transition, for example, we say that continuous translation symmetry is broken.

The ferromagnetic transition is another example of a symmetry-breaking transition, in this case the symmetry under reversal of the direction of electric currents and magnetic field lines. This symmetry is referred to as "up-down symmetry" or "time-reversal symmetry". It is broken in the ferromagnetic phase due to the formation of magnetic domains containing aligned magnetic moments. Inside each domain, there is a magnetic field pointing in a fixed direction chosen spontaneously during the phase transition. The name "time-reversal symmetry" comes from the fact that electric currents reverse direction when the time coordinate is reversed.

The presence of symmetry-breaking (or nonbreaking) is important to the behavior of phase transitions. It was pointed out by Landau that, given any state of a system, one may unequivocally say whether or not it possesses a given symmetry. Therefore, it cannot be possible to analytically deform a state in one phase into a phase possessing a different symmetry. This means, for example, that it is impossible for the solid-liquid phase boundary to end in a critical point like the liquid-gas boundary. However, symmetry-breaking transitions can still be either first- or second-order. Lev Davidovich Landau (Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ...

Typically, the more symmetrical phase is on the high-temperature side of a phase transition, and the less symmetrical phase on the low-temperature side. This is certainly the case for the solid-fluid and ferromagnetic transitions. This happens because the Hamiltonian of a system usually exhibits all the possible symmetries of the system, whereas the low-energy states lack some of these symmetries (this phenomenon is known as spontaneous symmetry breaking). At low temperatures, the system tends to be confined to the low-energy states. At higher temperatures, thermal fluctuations allow the system to access states in a broader range of energy, and thus more of the symmetries of the Hamiltonian. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ...

### Order parameters

The order parameter is the quantity which is indeterminate at the critical point (the point of the phase transition). For the ferromagnetic case, it is the magnetization. For solid/liquid or liquid/gas transitions, it is the density.

When symmetry is broken, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. Such variables are examples of order parameters. An order parameter is a measure of the degree of order in a system; the extreme values are 0 for total disorder and 1 for complete order.[1] For example, an order parameter can indicate the degree of order in a liquid crystal. However, note that order parameters can also be defined for symmetry-nonbreaking transitions. Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. ... Magnetization is a property of some materials (e. ... The Curie point is a term in physics and materials science, named after Pierre Curie (1859-1906), and refers to a characteristic property of a ferromagnetic material. ... Schlieren texture of Liquid Crystal nematic phase Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid, and those of a solid crystal. ...

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. Look up duality in Wiktionary, the free dictionary. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex (pl. ... Defect is the n00b of the animating world, everybody knows that he cannot and will not animate. ...

### Relevance for cosmology

Symmetry-breaking phase transitions play an important role in cosmology. It has been speculated that, in the hot early universe, the vacuum (i.e. the various quantum fields that fill space) possessed a large number of symmetries. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field. This transition is important to understanding the asymmetry between the amount of matter and antimatter in the present-day universe (see electroweak baryogenesis.) This article is about the physics subject. ... For other uses, see Big Bang (disambiguation). ... Quantum field theory (QFT) is the quantum theory of fields. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...

In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase. ...

### Critical exponents and universality classes

Continuous phase transitions are easier to study than first-order transitions due to the absence of latent heat, and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points.

It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature T of the system while keeping all the other thermodynamic variables fixed, and find that the transition occurs at some critical temperature Tc. When T is near Tc, the heat capacity C typically has a power law behaviour: The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ...

$C propto |T_c - T|^{-alpha}.$

The constant α is the critical exponent associated with the heat capacity. It is not difficult to see that it must be less than 0 in order for the transition to have no latent heat. Its actual value depends on the type of phase transition we are considering. For -1 < α < 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the "lambda transition" from a normal state to the superfluid state, for which experiments have found α = -0.013±0.003. At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample (see here). This experimental value of α agrees with theoretical predictions based on variational perturbation theory (see here). Helium II will creep along surfaces in order to find its own level - after a short while, the levels in the two containers will equalize. ... Mathematical method to convert divergent power series in a small expansion parameter, say , into convergent series in powers , where is a critical exponent. ...

For 0 < α < 1, the heat capacity diverges at the transition temperature (though, since α < 1, the divergence is not strong enough to produce a latent heat.) An example of such behavior is the 3-dimensional ferromagnetic phase transition. In the three-dimensional Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent α ∼ 0.110. The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ...

Some model systems do not obey this power law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a logarithmic divergence. However, these systems are an exception to the rule. Real phase transitions exhibit power law behavior. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

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 β = γ / (δ − 1), ν = γ / (2 − 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 amazingly, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and is insensitive to the underlying microscopic properties of the system. In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

## Phase-change data storage

Several data-storage technologies use chalcogenide glass, which can be "switched" between two states, crystalline or amorphous, with the application of heat. A chalcogenide is a binary compound consisting of a chalcogen and a more electropositive element. ...

### Phase change and optical disc technology

Phase change technology is also used to write to optical discs, such as CD-RW or DVD-RW discs. This is accomplished by including both a read laser and a more powerful write laser inside the drive. The discs are made of a crystalline material that, when hit by a pulse of laser light from the write laser, changes to an amorphous state, thus changing its reflectivity. The read laser is not powerful enough to induce a phase change, but can be used by the drive to tell whether a bit is "on" or "off" based on an area of the disc's reflectivity. â€œOptical mediaâ€ redirects here. ... Compact Disc ReWritable (CD-RW) is a rewritable optical disc format. ... The title given to this article is incorrect due to technical limitations. ... Crystal (disambiguation) Insulin crystals A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... For other uses, see Laser (disambiguation). ... An amorphous solid is a solid in which there is no long-range order of the positions of the atoms. ... In optics, reflectivity is the reflectance (the ratio of reflected power to incident power, generally expressed in decibels or percentage) at the surface of a material so thick that the reflectance does not change with increasing thickness; , the intrinsic reflectance of the surface, irrespective of other parameters such as the... BIT is an acronym for: Bannari amman Institute of Technology Bangalore Institute of Technology Beijing Institute of Technology Benzisothiazolinone Bilateral Investment Treaty Bhilai Institute of Technology - Durg Birla Institute of Technology - Mesra Battles in Time (Doctor Who magazine) BIT International College, formerly the Bohol Institute of Technology in Bohol, Philippines...

### History of phase change optical disc technology

• 1990: LF 7010 by Panasonic, store 472 MB per side.
• 1995: PD (Phasewriter Dual) by Panasonic, store 650 MB.
• 1996: CD-RW (Compact Disc ReWritable) by Philips, Sony, Hewlett-Packard, Mitsubishi Chemical Corp. and Ricoh, store initially 650 MB and later 700 MB.
• 1998: DVD-RAM (DVD-Random Access Memory) by Panasonic, store initially 2.6 GB and later 4.7 GB.
• 199x: DVD±RW (DVD-ReWritable) by supplier consortium, store 4.7 GB.
• 2004: PDD (Professional Disc for Data) by Sony, store 20.5 GB.
• 2004: UDO (Ultra Density Optical) by Plasmon, store 28 GB.
• 2006: BD-RE (Blu-ray Disc Rerecordable) by Sony, store 50 GB.

Panasonic is an international brand name for Japanese electric products manufacturer Matsushita Electric Industrial Co. ... ReBoot character, see Megabyte (ReBoot). ... Phase-Change Dual (PD) is a rewritable optical disc format introduced by Panasonic in 1995. ... Panasonic is an international brand name for Japanese electric products manufacturer Matsushita Electric Industrial Co. ... ReBoot character, see Megabyte (ReBoot). ... Compact Disc ReWritable (CD-RW) is a rewritable optical disc format. ... Philips HQ in Amsterdam Koninklijke Philips Electronics N.V. (Royal Philips Electronics N.V.), usually known as Philips, (Euronext: PHIA, NYSE: PHG) is one of the largest electronics companies in the world, founded and headquartered in the Netherlands. ... Sony Corporation ) is a Japanese multinational corporation and one of the worlds largest media conglomerates with revenue of \$66. ... The Hewlett-Packard Company (NYSE: HPQ), commonly known as HP, is a very large, global company headquartered in Palo Alto, California, United States. ... Ricoh Company, Ltd. ... You can recognize a DVD-RAM immediately because visually there are lots of little rectangles distributed on the surface of the data carrier. ... Panasonic is an international brand name for Japanese electric products manufacturer Matsushita Electric Industrial Co. ... This article is about the unit of measurement. ... The title given to this article is incorrect due to technical limitations. ... PDD, ProDATA or Professional Disc for DATA is a recordable optical disc format which was introduced by Sony in 2003. ... Sony Corporation ) is a Japanese multinational corporation and one of the worlds largest media conglomerates with revenue of \$66. ... Ultra Density Optical (UDO) is a next-generation optical disc format designed for high-density storage of high-definition video and data. ... In physics, the plasmon is the quasiparticle resulting from the quantization of plasma oscillations just as photons and phonons are quantizations of light and sound waves, respectively. ... BD-RE The rewritable form of the blu-ray format. ... Sony Corporation ) is a Japanese multinational corporation and one of the worlds largest media conglomerates with revenue of \$66. ...

### Phase-change memory

Main article: phase-change memory

Phase-change memory (PRAM) is a kind of non-volatile computer memory. Prototype PRAM devices have demonstrated higher density and faster write times than flash memory. Phase-change memory (also known as PCM, PRAM, Ovonic Unified Memory and Chalcogenide RAM [C-RAM]) is a type of non-volatile computer memory. ... Phase-change memory (also known as PCM, PRAM, Ovonic Unified Memory and Chalcogenide RAM [C-RAM]) is a type of non-volatile computer memory. ... It has been suggested that this article or section be merged with Non-volatile memory. ...

PRAM uses chalcogenide glass, the same material utilized in re-writable optical media (such as CD-RW and DVD-RW). The amorphous, high resistance state is used to represent a binary 1, and the crystalline, low resistance state represents a 0. A chalcogenide is a binary compound consisting of a chalcogen and a more electropositive element. ...

Samsung, Intel, and STMicroelectronics demonstrated prototype PRAM devices in 2006, and announced plans for commercial productions.

Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference are measured as a function of temperature. ... // Introduction Background Many macroscopic phenomena may be grouped into a small set of universality classes. ... // Superfluidity and Superconductivity are of inherent interest because they are macroscopic manifestations of quantum mechanics. ...

## References

1. ^ in A. D. McNaught and A. Wilkinson: Compendium of Chemical Terminology (commonly called The Gold Book). IUPAC. ISBN 0-86542-684-8. Retrieved on 2007-10-23.

The Gold Book or Compendium of Chemical Terminology (ISBN 0865426848) is a book published by IUPAC containing internationally accepted definitions for terms in chemistry. ... The International Union of Pure and Applied Chemistry (IUPAC) is an international non-governmental organization devoted to the advancement of chemistry. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 296th day of the year (297th in leap years) in the Gregorian calendar. ...

### General references

• Anderson, P.W., Basic Notions of Condensed Matter Physics, Perseus Publishing (1997).
• Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group, Perseus Publishing (1992).
• 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, 3rd Ed. (1994).
• Kleinert, H., Critical Properties of φ4-Theories, World Scientific (Singapore, 2001); Paperback ISBN 9810246595 (readable online here).
• Kleinert, H. and Verena Schulte-Frohlinde, Gauge Fields in Condensed Matter, Vol. I, "SUPERFLOW AND VORTEX LINES; Disorder Fields, Phase Transitions,", pp. 1--742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (readable online here)
• 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.

Philip Warren Anderson (born December 13, 1923) is one of the most influential theoretical physicists of the 20th century. ... Lev Davidovich Landau (Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... Evgeny Mikhailovich Lifshitz (Ð•Ð²Ð³ÐµÐ½Ð¸Ð¹ ÐœÐ¸Ñ…Ð°Ð¹Ð»Ð¾Ð²Ð¸Ñ‡ Ð›Ð¸Ñ„ÑˆÐ¸Ñ†) (February 21, 1915 â€“ October 29, 1985) was a Russian physicist. ... Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany, and Honorary Member of the Russian Academy of Creative Endeavors. ... Hagen Kleinert, Photo taken in 2006 Hagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany, and Honorary Member of the Russian Academy of Creative Endeavors. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex (pl. ... This diagram shows the nomenclature for the different phase transitions. ...

Results from FactBites:

 Phase transition - definition of Phase transition in Encyclopedia (1934 words) The distinguishing characteristic of a phase transition is an abrupt sudden change in one or more physical properties, in particular the heat capacity, with a small change in a thermodynamic variable such as the temperature. The Ehrenfest scheme is an inaccurate method of classifying phase transitions, for it is based on the mean field theory of phases (to be described in a later section.) Mean field theory is inaccurate in the vicinity of phase transitions, as it neglects the role of thermodynamic fluctuations. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and is insensitive to the underlying microscopic properties of the system.
 Phase (matter) - Wikipedia, the free encyclopedia (2719 words) The transition from a liquid phase to a solid phase is called freezing (or "melting" if we go in the opposite direction), and it is a type of phenomenon known as a phase transition. In contrast, in the plasma phase the atoms are dissociated, i.e. Phases are emergent phenomena produced by the self-organization of a macroscopic number of particles.
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