Phase transition - Knowledge

1322: 479: 114: 342: 420: 699: 1421:, 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. 1335: 5178: 33: 1636:

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 of

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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.

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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 temperature

2249:, 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 1471:. 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. 1452:

approximations, 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

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exhibit 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

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in 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

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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

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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

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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,

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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

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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. The existence of these fractals proves that proteins function near critical

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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

1694:: 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 1424:

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.

1670:, 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 1409:, 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 the 2299:

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.

303:. 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 the 1972:

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

680:). 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: 2172:

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

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and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.

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to 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.

2012:= −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 on 1718:

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. At the critical point, the order parameter

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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

684:, 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. 3013:

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".

5039:, 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. 1432:

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

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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.

147:, 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. 2149: 2038:

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

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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

1833:, 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. 1737:

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

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transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature which enables accurate detection using

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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

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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.

1941: 2350:(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 2261:, 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, 1429:

replied 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)

1781:. 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 533:

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

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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.

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as a function of other thermodynamic variables. Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition.

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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

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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".

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Tkacik, Gasper; Mora, Thierry; Marre, Olivier; Amodei, Dario; Berry II, Michael J.; Bialek, William (2014). "Thermodynamics for a network of neurons: Signatures of criticality".

1413:, 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 3360:

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

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Kushwaha, Pallavi; Lakhani, Archana; Rawat, R.; Chaddah, P. (2009). "Low-temperature study of field-induced antiferromagnetic-ferromagnetic transition in Pd-doped Fe-Rh".

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Hristovski, R.; Balagué, N. (2010). "Fatigue-induced spontaneous termination point--nonequilibrium phase transitions and critical behavior in quasi-isometric exertion".

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is 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.

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Pasquini, G.; Daroca, D. Pérez; Chiliotte, C.; Lozano, G. S.; Bekeris, V. (2008). "Ordered, Disordered, and Coexistent Stable Vortex Lattices inNbSe2Single Crystals".

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Banerjee, A.; Pramanik, A. K.; Kumar, Kranti; Chaddah, P. (2006). "Coexisting tunable fractions of glassy and equilibrium long-range-order phases in manganites".

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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".

2066:, are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as 2314:

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 of

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in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.

4875: 4736: 100:, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. 179: 2165:

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

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

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

3769: 2169:. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid. 365:

Phase transitions can also occur when a solid changes to a different structure without changing its chemical makeup. In elements, this is known as

88:. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as 5087: 3977:

Lipa, J.; Nissen, J.; Stricker, D.; Swanson, D.; Chui, T. (2003). "Specific heat of liquid helium in zero gravity very near the lambda point".

1789:. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to 2072: 664:), the heavier water isotopes (O and H) become enriched in the liquid phase while the lighter isotopes (O and H) tend toward the vapor phase. 1507: 546: 121:, showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure. 1946:

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent

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of the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperature

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Mussardo G., "Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics", Oxford University Press, 2010.

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

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divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior.

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shows two concurrent phase changes. The transition from solid to liquid, and gas to liquid (shown by the white condensed water vapour).

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hypothetical limit of infinitely long relaxation times. No direct experimental evidence supports the existence of these transitions.

2027:< 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional 315:) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is 172: 1530:

the phase transition is second-order for both normal-state–mixed-state and mixed-state–superconducting-state transitions) and the

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In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the

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Common transitions between the solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/or

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are more complicated than transitions involving a single compound. While chemically pure compounds exhibit a single temperature

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reaction consists of change from a liquid and to a combination of a solid and a second liquid, where the two liquids display a

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transformation, in which a two-component single-phase solid is heated and transforms into a solid phase and a liquid phase. A

4859: 4549: 3837: 3746: 2840: 2721: 2545: 4926: 1628:, which may depend on the applied pressure. If the first-order freezing transition occurs over a range of temperatures, and 1884: 1734:

system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.

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The first example of a phase transition which did not fit into the Ehrenfest classification was the exact solution of the

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Lakhani, Archana; Banerjee, A.; Chaddah, P.; Chen, X.; Ramanujan, R. V. (2012). "Magnetic glass in shape memory alloy: Ni

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before the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The large

165: 319:, i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in 1569: 60:

of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic

4936: 4911: 4903: 2419: 1536: 1352: 1339: 17: 2515: 2000:< 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the 1544: 393: 370: 336: 1321: 5382: 2265: 2013: 1720: 1667: 1515: 5452: 5377: 5108: 5074: 1104: 656:

occurs during a phase transition, the ratio of light to heavy isotopes in the involved molecules changes. When

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Jaeger, Gregg (1 May 1998). "The Ehrenfest Classification of Phase Transitions: Introduction and Evolution".

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Kleinert, Hagen (1999). "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions".

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Leonard, F.; Delamotte, B. (2015). "Critical exponents can be different on the two sides of a transition".

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Moret, Marcelo; Zebende, Gilney (January 2007). "Amino acid hydrophobicity and accessible surface area".

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Lubchenko, V. Wolynes; Wolynes, Peter G. (2007). "Theory of Structural Glasses and Supercooled Liquids".

2339:(simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800–1000 °C) 2227: 1474:

Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into

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resulting in a temperature span where solid and liquid coexist in equilibrium. This is often the case in

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For a single component, the most stable phase at different temperatures and pressures can be shown on a

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Gross, David J. (1980), "Possible third-order phase transition in the large N lattice gauge theory",

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Diedrich, F. J.; Warren, W. H. Jr. (1995). "Why change gaits? Dynamics of the walk-run transition".

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Continuous phase transitions are easier to study than first-order transitions due to the absence of

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It has been proposed that some biological systems might lie near critical points. Examples include

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Imry, Y.; Wortis, M. (1979). "Influence of quenched impurities on first-order phase transitions".

1254: 5493: 1750:. However, note that order parameters can also be defined for non-symmetry-breaking transitions. 1426: 1418: 859: 1264: 5518: 5508: 5258: 5253: 4992:

Constitutions of matter : mathematically modelling the most everyday of physical phenomena

4776: 1527: 1249: 1189: 1159: 1109: 829: 719: 557: 428: 96:. This can be a discontinuous change; for example, a liquid may become gas upon heating to its 3736: 3222: 5005: 2408: 2242:, and cooperative ligand binding to DNA and proteins with the character of phase transition. 2174: 1976: 1786: 1523: 1289: 904: 889: 653: 5437: 5197: 5028: 4968: 4821: 4738:

Complexity Theory and Conflict Transformation: An Exploration of Potential and Implications

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between solid and liquid phases, mixtures can either have a single melting point, known as

503: 307:. In exception to the usual case, it is sometimes possible to change the state of a system 289: 81: 2326:

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

1840:. The most important one is perhaps the exponent describing the divergence of the thermal 650:

in the laws of physics during the early history of the universe as its temperature cooled.

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occurs as one of the many phase transformations in carbon steel and stands as a model for

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Dynamic Patterns: The Self-Organization of Brain and Behavior (Complex Adaptive Systems)

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Mora, Thierry; Bialek, William (2011). "Are biological systems poised at criticality?".

4321: 4252: 4167:"Determination of membrane lipid phase transition temperature from 13-C NMR intensities" 4053: 4000: 3939: 3886: 3697: 3628: 3559: 3499: 3456: 3395: 3320: 3265: 3214: 3160: 3098: 3037: 2991: 2790: 2588: 2222:

Phase transitions play many important roles in biological systems. Examples include the

467:. A simplified but highly useful model of magnetic phase transitions is provided by the 135:

Phase transitions commonly refer to when a substance transforms between one of the four

5559: 5488: 5322: 4784: 4717: 4691: 4522: 4496: 4460: 4429: 4417: 4398: 4372: 4340: 4307: 4295: 4271: 4238: 4227:"Social interactions dominate speed control in poising natural flocks near criticality" 4226: 4205: 4108: 4065: 4039: 4012: 3986: 3959: 3925: 3898: 3806: 3763: 3717: 3683: 3656: 3614: 3519: 3468: 3442: 3415: 3381: 3342: 3285: 3234: 3200: 3147:

Ojovan, M.I. (2013). "Ordering and structural changes at the glass-liquid transition".

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Apart from isolated, simple phase transitions, there exist transition lines as well as

1299: 1259: 1234: 982: 973: 424: 414: 397: 354: 346: 3894: 3464: 2765:"Top eigenvalue of a random matrix: large deviations and third order phase transition" 5589: 5584: 5554: 5513: 5402: 5354: 5339: 5232: 5202: 5042: 4987: 4980: 4956: 4932: 4907: 4855: 4809: 4679: 4660: 4617: 4582: 4545: 4514: 4510: 4465: 4345: 4276: 4186: 4182: 4147: 4143: 4127: 4100: 4069: 3951: 3833: 3826: 3742: 3721: 3709: 3660: 3648: 3640: 3571: 3511: 3407: 3334: 3289: 3277: 3226: 3181:

Gotze, Wolfgang. "Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory."

3168: 3118: 3110: 3061: 3049: 2930: 2912: 2873: 2836: 2814: 2802: 2717: 2695: 2649: 2610: 2563:"Phase diagram for the transition from photonic crystals to dielectric metamaterials" 2541: 2448: 2374: 2001: 1837: 1824: 1782: 1754: 1703: 1687: 1683: 1602: 1487: 1449: 1414: 1229: 1074: 964: 884: 673: 647: 614: 515: 436: 374: 350: 85: 4166: 4112: 4016: 3963: 3902: 3810: 3783:

Clark, J.B.; Hastie, J.W.; Kihlborg, L.H.E.; Metselaar, R.; Thackeray, M.M. (1994).

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Semiconductors and Semimetals. Vol 100. Photonic Crystal Metasurface Optoelectronics

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The characteristic feature of second order phase transitions is the appearance of

5534: 5387: 5124: 4942: 4802: 4798: 2963: 2503: 2231: 1244: 1194: 1064: 819: 731: 564: 550: 378: 136: 61: 4713: 4613: 2493: – Mathematical theory on behavior of connected clusters in a random graph 2466: – Theory of continuous phase transitions of second order phase transitions 2023:< 1, the heat capacity diverges at the transition temperature (though, since 1844:

by approaching the transition. For instance, let us examine the behavior of the

5332: 5327: 5284: 5217: 5212: 4922: 4883: 4656: 4578: 4061: 4008: 3705: 3567: 3403: 3045: 2893:"The Ehrenfest Classification of Phase Transitions: Introduction and Evolution" 2402: 2258: 1969:. Its actual value depends on the type of phase transition we are considering. 1762: 1519: 1389: 1326: 1294: 1274: 1269: 1224: 1144: 1079: 977: 864: 709: 677: 596: 523: 440: 341: 304: 126: 4948: 4833: 4394: 2536:

Askeland, Donald R.; Haddleton, Frank; Green, Phil; Robertson, Howard (1996).

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for uniaxial magnets, detailed theoretical studies have yielded the exponent

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Complex system approach to peace and armed conflict § Phase transitions

1746:, whose direction was spontaneously chosen when the system cooled below the 1199: 139:

to another. At the phase transition point for a substance, for instance the

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Phase transitions can also describe the change between different kinds of

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Continuous phase transitions can be characterized by parameters known as

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reaction is a peritectoid rection, except involving only solid phases. A

542: 495: 468: 456: 452: 89: 4847: 4812:(1974). "The renormalization group in the theory of critical behavior". 2892: 2869: 2596: 2144:{\displaystyle \beta =\gamma /(\delta -1),\quad \nu =\gamma /(2-\eta ).} 617:

in certain metals and ceramics when cooled below a critical temperature.

327:, for example. Metastable states do not appear on usual phase diagrams. 36:

This diagram shows the nomenclature for the different phase transitions.

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properties in artificial photonic media as their parameters are varied.

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There are also a number of phase transitions involving three phases: a

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Elements of classical thermodynamics: for advanced students of physics

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the phase transition is second-order at zero external field and for a

4418:"Zipf's law and criticality in multivariate data without fine-tuning" 3507: 2442: – Mathematical model of ferromagnetism in statistical mechanics 2387: 2040: 1875: 1511: 1204: 1154: 1027: 874: 774: 592: 534: 482:

A binary phase diagram showing the most stable chemical compounds of

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Fractals, chaos, power laws : minutes from an infinite paradise

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Journal of Experimental Psychology. Human Perception and Performance

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It can be shown that there are only two independent exponents, e.g.

1506:. They are characterized by a divergent susceptibility, an infinite 32: 5579: 5407: 4128:"C NMR studies of lipid fatty acyl chains of chloroplast membranes" 3930: 2856:

Austin, J. B. (November 1932). "Heat Capacity of Iron - A Review".

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Ice Phase Transition as a sample of finite system phase transition

4696: 4501: 4434: 4377: 4312: 4243: 4210: 3688: 3619: 3359: 3089: 2781: 5539: 5427: 5362: 5279: 5274: 2284: 2202:. Connected to the previous phenomenon is also the phenomenon of 1084: 1069: 1032: 1023: 1018: 507: 455:. Another example is the transition between differently ordered, 227: 41: 1644:

raises the possibility that one can study the interplay between

5148: 5069: 4959:(1974). "The Renormalization Group and the epsilon-Expansion". 4895: 4879:, (Physics Education (India) Volume 32. No. 2, Apr - Jun 2016) 4728: 2535: 1037: 1013: 744: 640: 487: 448: 205: 69: 1818: 1777:

Symmetry-breaking phase transitions play an important role in

575: 5157: 5143: 3074: 2711: 2518: – Field theory involving topological effects in physics 2362: 1042: 739: 628: 580: 200: 144: 118: 65: 27:

Physical process of transition between basic states of matter

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Chaos, Phase Transitions, Topology Change and Path Integrals

3782: 3536: 3432: 3012: 2396: – Chemical reaction whose product is also its catalyst 4842:

Lectures on Phase Transitions and the Renormalization Group

4680:"Correlations, risk and crisis: From physiology to finance" 4678:

Gorban, A.N.; Smirnova, E.V.; Tyukina, T.A. (August 2010).

3673: 3485: 3868:"Topologically disordered systems at the glass transition" 3588: 2217: 2194:. As a consequence, at a phase transition one may observe 1392:

classified phase transitions based on the behavior of the

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Introduction to Phase Transitions and Critical Phenomena

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Introduction to Phase Transitions and Critical Phenomena

2835:(Repr ed.). Cambridge: Univ. Pr. pp. 140–141. 2428: – Shift of atomic positions in a crystal structure 385:) are all examples of solid to solid phase transitions. 4416:

Schwab, David J; Nemenman, Ilya; Mehta, Pankaj (2014).

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Ivancevic, Vladimir G.; Ivancevic, Tijiana, T. (2008).

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Journal of Statistical Mechanics: Theory and Experiment

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There are also other critical phenomena; e.g., besides

4998:, 1996. Contains a detailed pedagogical discussion of 3854:

Cosmogenesis, The Development of Order in the Universe

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Majumdar, Satya N; Schehr, Grégory (31 January 2014).

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Krotov, D; Dubuis, J O; Gregor, T; Bialek, W (2014).

4224: 4203: 2460: – Noncontact variant of atomic force microscopy 2075: 1979: 1936:{\displaystyle C\propto |T_{\text{c}}-T|^{-\alpha }.} 1887: 1686:

process. For instance, the cooling of a fluid into a

1518:. Examples of second-order phase transitions are the 5049:(Oxford University Press, Oxford and New York 1971). 3530: 3353: 3142: 3140: 2712:

Blundell, Stephen J.; Katherine M. Blundell (2008).

2474:

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2238:, liquid crystal-like transitions in the process of 2210:

of a continuous phase transition split into smaller

1448:. The exact specific heat differed from the earlier 439:. The most well-known is the transition between the 4848:Ivancevic, Vladimir G; Ivancevic, Tijana T (2008), 4746:

Department of Peace Studies, University of Bradford

4415: 3667: 3582: 4539: 3825: 3734: 3068: 2506: – Thin layer of liquid in a superfluid state 2143: 1985: 1935: 606:geometry on coverage and temperature, such as for 4799:Fundamentals of Multiphase Heat Transfer and Flow 4599: 3137: 3006: 2965:Fundamentals of Multiphase Heat Transfer and Flow 2405: – Major stage of a crystallization process 1950:= 0.59 A similar behavior, but with the exponent 5612: 3915: 3190: 1848:near such a transition. We vary the temperature 676:for some choice of thermodynamic variables (cf. 563:Non-equilibrium mixtures can occur, such as in 4734: 4564: 4171:Journal of Biochemical and Biophysical Methods 2762: 2481: – Different known phase of states matter 1522:transition, superconducting transition (for a 595:between solid and liquid, such as one of the " 556:Separation into multiple phases can occur via 526:, where the two components are isostructural. 451:materials, which occurs at what is called the 396:. Order-disorder transitions such as in alpha- 381:, or from one amorphous structure to another ( 5109: 4482: 4132:Indian Journal of Biochemistry and Biophysics 2650:"Fundamentals of Stable Isotope Geochemistry" 2644: 1360: 400:. As with states of matter, there are also a 173: 4634: 4489:Progress in Biophysics and Molecular Biology 4119: 3866:Ojovan, Michael I.; Lee, William E. (2006). 3768:: CS1 maint: multiple names: authors list ( 2669: 2667: 1726:An example of an order parameter is the net 1610: 1482:mixture of liquid water and vapor bubbles). 1405:are continuous in the first derivative (the 103: 5123: 4954: 4225:Bialek, W; Cavagna, A; Giardina, I (2014). 2560: 2296:points of second-order phase transitions. 2178:correlation length is the essential point. 1819:Critical exponents and universality classes 1384: 377:to another, from a crystalline solid to an 349:, distinguishing between several different 48:, and other related fields like biology, a 5116: 5102: 5057:Statistical Mechanics of Phase Transitions 4362: 3305:"Materials science: Metal turned to glass" 2977: 2707: 2705: 2394:Autocatalytic reactions and order creation 1785:into the U(1) symmetry of the present-day 1706:, which only occurs at low temperatures). 1401:the free energy with respect to pressure. 1367: 1353: 697: 180: 166: 5082:Interactive Phase Transitions on lattices 4781:Basic Notions of Condensed Matter Physics 4695: 4500: 4483:Longo, G.; Montévil, M. 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Berlin: Springer. pp. 176–177. 3302: 3146: 2890: 2858:Industrial & Engineering Chemistry 2855: 2673: 1958:, applies for the correlation length. 1453:able to incorporate such transitions. 369:, whereas in compounds it is known as 84:and the states of matter have uniform 5097: 4763:from the original on 26 November 2022 3251: 2897:Archive for History of Exact Sciences 2736: 2676:Archive for History of Exact Sciences 2365:(measurement of magnetic transitions) 2333:(measurement of magnetic transitions) 2181: 1406: 3875:Journal of Physics: Condensed Matter 3607:Journal of Physics: Condensed Matter 3435:Journal of Physics: Condensed Matter 2826: 2824: 2411: – materials science phenomenon 1965:is positive. This is different with 1615: 5002:'s solution of the 2-D Ising Model. 4916:physik.fu-berlin.de readable online 3193:Annual Review of Physical Chemistry 2540:. Chapman & Hall. p. 286. 2304:Phase transitions in social systems 1709: 1564:. They are continuous but break no 1550:of second-order phase transitions. 1435:supercritical liquid–gas boundaries 108: 24: 4904:World Scientific (Singapore, 1989) 4770: 2422: – Thermoanalytical technique 2046:Several other critical exponents, 1682:Phase transitions often involve a 1661: 1379: 423:A phase diagram showing different 25: 5642: 5063: 5022:, Pergamon Press, 3rd Ed. (1994). 4928:Critical Properties of φ-Theories 2953:, Elsevier, Burlington, MA, 2006, 2821: 2561:Rybin, M.V.; et al. (2015). 2512: – Process in quantum optics 2420:Differential scanning calorimetry 1568:. The most famous example is the 1560:Several transitions are known as 1537:differential scanning calorimetry 668:Phase transitions occur when the 570: 520:liquidus and solidus temperatures 427:in the same crystal structure of 5176: 5068: 5059:, Oxford University Press, 1992. 4888:Gauge Fields in Condensed Matter 4511:10.1016/j.pbiomolbio.2011.03.005 3169:10.1016/j.jnoncrysol.2013.10.016 2799:10.1088/1742-5468/2014/01/P01012 2516:Topological quantum field theory 2487: – Crystal growth technique 2472: – crystal growth technique 1753:Some phase transitions, such as 1742:phase, one must provide the net 1698:, with the exception of certain 1562:infinite-order phase transitions 1334: 1333: 1320: 394:displacive phase transformations 337:Polymorphism (materials science) 286: 284: 255: 236: 4671: 4628: 4593: 4558: 4533: 4476: 4409: 4356: 4287: 4218: 4197: 4158: 4076: 4023: 3970: 3909: 3859: 3846: 3817: 3776: 3728: 3479: 3426: 3296: 3245: 3184: 3175: 2971: 2956: 2941: 2923: 2884: 2849: 2321: 2266:electron paramagnetic resonance 2108: 2014:variational perturbation theory 1486:and Michael Wortis showed that 117:A simplified phase diagram for 4491:. Systems Biology and Cancer. 4452:10.1103/PhysRevLett.113.068102 4365:Journal of Statistical Physics 4296:"Morphogenesis at criticality" 4097:10.1080/07391102.2000.10506578 3948:10.1103/PhysRevLett.115.200601 3637:10.1088/0953-8984/24/38/386004 3107:10.1103/PhysRevLett.100.247003 2968:, Springer, New York, NY, 2020 2756: 2730: 2638: 2621: 2554: 2529: 2135: 2123: 2102: 2090: 1917: 1895: 1595:and other liquids that can be 1582:two-dimensional electron gases 1570:Kosterlitz–Thouless transition 1504:"continuous phase transitions" 1403:Second-order phase transitions 13: 1: 5565:Macroscopic quantum phenomena 5019:Course of Theoretical Physics 3274:10.1126/science.267.5206.1947 2523: 2510:Superradiant phase transition 2458:Kelvin probe force microscope 2426:Diffusionless transformations 2348:Perturbed angular correlation 1702:(e.g. the formation of heavy 1696:spontaneous symmetry breaking 1565: 1497:Second-order phase transition 1465:First-order phase transitions 1398:First-order phase transitions 587:Other phase changes include: 330: 5575:Order and disorder (physics) 4981:10.1016/0370-1573(74)90023-4 4925:; Verena Schulte-Frohlinde. 4874:M.R. 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Oxford: Clarendon Press. 2307: 1822: 910:Spin gapless semiconductor 633:Bose–Einstein condensation 412: 390:martensitic transformation 334: 124: 5527: 5481: 5353: 5267: 5241: 5185: 5174: 5136: 4834:10.1103/revmodphys.46.597 4395:10.1007/s10955-011-0229-4 1809:relational order theories 1611:Characteristic properties 1578:quantum phase transitions 1467:are those that involve a 1394:thermodynamic free energy 850:Electronic band structure 682:quantum phase transitions 670:thermodynamic free energy 662:equilibrium fractionation 518:, or they have different 104:Types of phase transition 5600:Thermo-dielectric effect 5499:Enthalpy of vaporization 5193:Bose–Einstein condensate 3000:10.1103/physrevb.19.3580 2479:List of states of matter 2352:tantalum hafnium carbide 2316:peace and armed conflict 1791:electroweak baryogenesis 1584:, belong to this class. 1444:, discovered in 1944 by 1385:Ehrenfest classification 760:Bose–Einstein condensate 691:Condensed matter physics 627:Quantum condensation of 5494:Enthalpy of sublimation 4422:Physical Review Letters 4331:10.1073/pnas.1324186111 4262:10.1073/pnas.1324045111 3802:10.1351/pac199466030577 3077:Physical Review Letters 2751:10.1103/PhysRevD.21.446 2228:coil-globule transition 1986:{\displaystyle \gamma } 1589:liquid–glass transition 1572:in the two-dimensional 1419:magnetic susceptibility 5509:Latent internal energy 5259:Color-glass condensate 4602:Human Movement Science 4085:J. Biomol. Struct. Dyn 2337:Mössbauer spectroscopy 2145: 1987: 1937: 1773:Relevance in cosmology 1723:will usually diverge. 1528:Type-II superconductor 1457:Modern classifications 648:breaking of symmetries 643:is an example of this. 602:The dependence of the 584: 558:spinodal decomposition 499: 432: 429:Manganese monosilicide 373:. 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Solids 2247:biological membranes 2212:dynamic universality 2073: 1977: 1885: 1870:, the heat capacity 1692:translation symmetry 1672:critical opalescence 1591:is observed in many 1555:multicritical points 923:Electronic phenomena 770:Fermionic condensate 82:thermodynamic system 5413:Chemical ionization 5305:Programmable matter 5295:Quantum spin liquid 5163:Supercritical fluid 4973:1974PhR....12...75W 4902:", pp. 1–742, 4898:; Disorder Fields, 4826:1974RvMP...46..597F 4706:2010PhyA..389.3193G 4649:2007PhRvE..75a1920M 4444:2014PhRvL.113f8102S 4387:2011JSP...144..268M 4322:2014PNAS..111.3683K 4253:2014PNAS..111.7212B 4126:Yashroy RC (1987). 4054:1999PhRvD..60h5001K 4001:2003PhRvB..68q4518L 3940:2015PhRvL.115t0601L 3887:2006JPCM...1811507O 3881:(50): 11507–11520. 3698:2009PhRvB..80q4413K 3629:2012JPCM...24L6004L 3560:2006PhRvB..74a2403R 3500:2006NatMa...5..881W 3457:2006JPCM...18L.605B 3396:2001PhRvB..64j4416M 3321:2007Natur.448..758T 3303:Tarjus, G. (2007). 3266:1995Sci...267.1947G 3260:(5206): 1947–1953. 3215:2007ARPC...58..235L 3161:2013JNCS..382...79O 3099:2008PhRvL.100x7003P 3038:2006PhRvB..73r4435K 2992:1979PhRvB..19.3580I 2870:10.1021/ie50275a006 2791:2014JSMTE..01..012M 2597:10.1038/ncomms10102 2589:2015NatCo...610102R 2435:Ehrenfest equations 2343:Neutron diffraction 2254:thylakoid membranes 930:Quantum Hall effect 425:magnetic structures 398:titanium aluminides 188: 86:physical properties 5631:Critical phenomena 5626:Physical phenomena 5560:Leidenfrost effect 5489:Enthalpy of fusion 5254:Quark–gluon plasma 4988:Krieger, Martin H. 4785:Perseus Publishing 2931:Stanley, H. Eugene 2491:Percolation theory 2485:Micro-pulling-down 2358:Raman Spectroscopy 2230:in the process of 2182:Critical phenomena 2141: 1983: 1933: 1842:correlation length 1838:critical exponents 1813:order and disorder 1690:breaks continuous 1508:correlation length 1327:Physics portal 585: 537:transformation. A 500: 433: 415:Magnetic structure 363: 351:crystal structures 347:allotropes of iron 157: 123: 38: 5621:Phase transitions 5608: 5607: 5590:Superheated vapor 5585:Superconductivity 5555:Equation of state 5403:Flash evaporation 5355:Phase transitions 5340:String-net liquid 5233:Photonic molecule 5203:Degenerate matter 5084:with Java applets 5073:Media related to 4900:Phase Transitions 4861:978-3-540-79356-4 4690:(16): 3193–3217. 4637:Physical Review E 4551:978-0-262-61131-2 4306:(10): 3683–3688. 4237:(20): 7212–7217. 4032:Physical Review D 3979:Physical Review B 3839:978-0-674-00342-2 3748:978-3-540-79357-1 3676:Physical Review B 3548:Physical Review B 3374:Physical Review B 3315:(7155): 758–759. 3016:Physical Review B 2864:(11): 1225–1235. 2842:978-0-521-09101-5 2739:Physical Review D 2723:978-0-19-856770-7 2547:978-0-412-53910-7 2449:Jamming (physics) 2375:X-ray diffraction 2192:critical dynamics 2002:lambda transition 1906: 1825:critical exponent 1783:electroweak field 1704:virtual particles 1688:crystalline solid 1684:symmetry breaking 1616:Phase coexistence 1603:quenched disorder 1488:quenched disorder 1415:Curie temperature 1377: 1376: 1075:Granular material 843:Electronic phases 620:The emergence of 615:superconductivity 613:The emergence of 516:congruent melting 437:magnetic ordering 375:crystal structure 297: 296: 159:Phase transitions 18:Phase transitions 16:(Redirected from 5638: 5545:Compressed fluid 5180: 5125:States of matter 5118: 5111: 5104: 5095: 5094: 5072: 4984: 4946: 4941:. Archived from 4871: 4870: 4868: 4840:Goldenfeld, N., 4837: 4765: 4764: 4762: 4743: 4732: 4726: 4725: 4699: 4675: 4669: 4668: 4632: 4626: 4625: 4597: 4591: 4590: 4562: 4556: 4555: 4537: 4531: 4530: 4504: 4480: 4474: 4473: 4463: 4437: 4413: 4407: 4406: 4380: 4360: 4354: 4353: 4343: 4333: 4315: 4291: 4285: 4284: 4274: 4264: 4246: 4222: 4216: 4215: 4213: 4201: 4195: 4194: 4162: 4156: 4155: 4123: 4117: 4116: 4080: 4074: 4073: 4047: 4027: 4021: 4020: 3994: 3992:cond-mat/0310163 3974: 3968: 3967: 3933: 3913: 3907: 3906: 3872: 3863: 3857: 3850: 3844: 3843: 3831: 3828:Cosmic Evolution 3821: 3815: 3814: 3804: 3780: 3774: 3773: 3767: 3759: 3757: 3755: 3732: 3726: 3725: 3691: 3671: 3665: 3664: 3622: 3586: 3580: 3579: 3534: 3528: 3527: 3508:10.1038/nmat1743 3488:Nature Materials 3483: 3477: 3476: 3450: 3448:cond-mat/0611152 3430: 3424: 3423: 3389: 3387:cond-mat/0012472 3357: 3351: 3350: 3332: 3300: 3294: 3293: 3249: 3243: 3242: 3208: 3206:cond-mat/0607349 3188: 3182: 3179: 3173: 3172: 3144: 3135: 3134: 3092: 3072: 3066: 3065: 3031: 3029:cond-mat/0602627 3010: 3004: 3003: 2986:(7): 3580–3585. 2975: 2969: 2960: 2954: 2945: 2939: 2938: 2927: 2921: 2920: 2888: 2882: 2881: 2853: 2847: 2846: 2828: 2819: 2818: 2784: 2760: 2754: 2753: 2734: 2728: 2727: 2709: 2700: 2699: 2671: 2662: 2661: 2659: 2657: 2642: 2636: 2634:, Elsevier, 2019 2625: 2619: 2618: 2608: 2582: 2558: 2552: 2551: 2533: 2475: 2454: 2445: 2431: 2414: 2399: 2369:Thermogravimetry 2240:DNA condensation 2188:static functions 2150: 2148: 2147: 2142: 2122: 2089: 1992: 1990: 1989: 1984: 1942: 1940: 1939: 1934: 1929: 1928: 1920: 1908: 1907: 1904: 1898: 1874:typically has a 1710:Order parameters 1545:phenomenological 1502:are also called 1499: 1498: 1369: 1362: 1355: 1342: 1337: 1336: 1329: 1325: 1324: 935:Spin Hall effect 825:Phase transition 795:Luttinger liquid 732:States of matter 715:Phase transition 701: 687: 686: 591:Transition to a 189: 182: 175: 168: 156: 137:states of matter 109:States of matter 62:states of matter 58:physical process 50:phase transition 21: 5646: 5645: 5641: 5640: 5639: 5637: 5636: 5635: 5611: 5610: 5609: 5604: 5535:Baryonic matter 5523: 5477: 5448:Saturated fluid 5388:Crystallization 5349: 5323:Antiferromagnet 5263: 5237: 5181: 5172: 5132: 5122: 5066: 4949:readable online 4939: 4923:Kleinert, Hagen 4866: 4864: 4862: 4803:Springer Nature 4773: 4771:Further reading 4768: 4760: 4741: 4733: 4729: 4676: 4672: 4633: 4629: 4598: 4594: 4563: 4559: 4552: 4538: 4534: 4481: 4477: 4414: 4410: 4361: 4357: 4292: 4288: 4223: 4219: 4202: 4198: 4163: 4159: 4124: 4120: 4081: 4077: 4028: 4024: 3975: 3971: 3918:Phys. Rev. Lett 3914: 3910: 3870: 3864: 3860: 3851: 3847: 3840: 3822: 3818: 3781: 3777: 3761: 3760: 3753: 3751: 3749: 3733: 3729: 3672: 3668: 3604: 3600: 3596: 3592: 3587: 3583: 3545: 3541: 3535: 3531: 3494:(11): 881–886. 3484: 3480: 3431: 3427: 3371: 3367: 3363: 3358: 3354: 3330:10.1038/448758a 3301: 3297: 3250: 3246: 3189: 3185: 3180: 3176: 3145: 3138: 3073: 3069: 3011: 3007: 2976: 2972: 2961: 2957: 2946: 2942: 2928: 2924: 2889: 2885: 2854: 2850: 2843: 2829: 2822: 2761: 2757: 2735: 2731: 2724: 2710: 2703: 2672: 2665: 2655: 2653: 2643: 2639: 2626: 2622: 2559: 2555: 2548: 2534: 2530: 2526: 2521: 2504:Superfluid film 2473: 2452: 2443: 2429: 2412: 2397: 2383: 2324: 2312: 2306: 2277:neural networks 2232:protein folding 2226:formation, the 2220: 2184: 2118: 2085: 2074: 2071: 2070: 1978: 1975: 1974: 1921: 1916: 1915: 1903: 1899: 1894: 1886: 1883: 1882: 1869: 1858: 1827: 1821: 1775: 1755:superconducting 1716:order parameter 1712: 1680: 1664: 1662:Critical points 1657: 1650: 1642: 1634: 1627: 1618: 1613: 1539:measurements. 1496: 1495: 1459: 1427:Cornelis Gorter 1407:order parameter 1387: 1382: 1380:Classifications 1373: 1332: 1319: 1318: 1311: 1310: 1309: 1099: 1091: 1090: 1089: 1065:Amorphous solid 1059: 1049: 1048: 1047: 1026: 1008: 998: 997: 996: 985: 983:Antiferromagnet 976: 974:Superparamagnet 967: 954: 953:Magnetic phases 946: 945: 944: 924: 916: 915: 914: 844: 836: 835: 834: 820:Order parameter 814: 813:Phase phenomena 806: 805: 804: 734: 724: 672:of a system is 573: 565:supersaturation 551:miscibility gap 524:solid solutions 476: 417: 411: 379:amorphous solid 339: 333: 311:(as opposed to 197: 194: 186: 133: 111: 106: 80:. A phase of a 28: 23: 22: 15: 12: 11: 5: 5644: 5634: 5633: 5628: 5623: 5606: 5605: 5603: 5602: 5597: 5592: 5587: 5582: 5577: 5572: 5567: 5562: 5557: 5552: 5547: 5542: 5537: 5531: 5529: 5525: 5524: 5522: 5521: 5516: 5514:Trouton's rule 5511: 5506: 5501: 5496: 5491: 5485: 5483: 5479: 5478: 5476: 5475: 5470: 5465: 5460: 5455: 5450: 5445: 5440: 5435: 5430: 5425: 5420: 5415: 5410: 5405: 5400: 5395: 5390: 5385: 5383:Critical point 5380: 5375: 5370: 5365: 5359: 5357: 5351: 5350: 5348: 5347: 5342: 5337: 5336: 5335: 5330: 5325: 5317: 5312: 5307: 5302: 5297: 5292: 5287: 5285:Liquid crystal 5282: 5277: 5271: 5269: 5265: 5264: 5262: 5261: 5256: 5251: 5245: 5243: 5239: 5238: 5236: 5235: 5230: 5225: 5220: 5218:Strange matter 5215: 5213:Rydberg matter 5210: 5205: 5200: 5195: 5189: 5187: 5183: 5182: 5175: 5173: 5171: 5170: 5165: 5160: 5151: 5146: 5140: 5138: 5134: 5133: 5121: 5120: 5113: 5106: 5098: 5092: 5091: 5090:from Sklogwiki 5085: 5065: 5064:External links 5062: 5061: 5060: 5050: 5040: 5026: 5023: 5010:Lifshitz, E.M. 5003: 4985: 4952: 4937: 4919: 4881: 4872: 4860: 4845: 4838: 4820:(4): 597–616. 4814:Rev. Mod. Phys 4806: 4788: 4777:Anderson, P.W. 4772: 4769: 4767: 4766: 4727: 4670: 4627: 4608:(4): 483–493. 4592: 4573:(1): 183–202. 4557: 4550: 4532: 4495:(2): 340–347. 4475: 4408: 4371:(2): 268–302. 4355: 4286: 4217: 4196: 4177:(4): 353–356. 4157: 4138:(6): 177–178. 4118: 4091:(5): 903–911. 4075: 4045:hep-th/9812197 4022: 3985:(17): 174518. 3969: 3924:(20): 200601. 3908: 3858: 3852:David Layzer, 3845: 3838: 3816: 3795:(3): 577–594. 3775: 3747: 3727: 3682:(17): 174413. 3666: 3613:(38): 386004. 3602: 3598: 3594: 3590: 3581: 3543: 3539: 3529: 3478: 3425: 3380:(10): 104416. 3369: 3365: 3361: 3352: 3295: 3244: 3183: 3174: 3136: 3083:(24): 247003. 3067: 3022:(18): 184435. 3005: 2970: 2955: 2940: 2922: 2883: 2848: 2841: 2820: 2755: 2745:(2): 446–453, 2729: 2722: 2701: 2663: 2637: 2620: 2553: 2546: 2527: 2525: 2522: 2520: 2519: 2513: 2507: 2501: 2500: 2499: 2488: 2482: 2476: 2467: 2461: 2455: 2446: 2437: 2432: 2423: 2417: 2416: 2415: 2403:Crystal growth 2400: 2391: 2384: 2382: 2379: 2378: 2377: 2372: 2366: 2360: 2355: 2354:4215 °C.) 2345: 2340: 2334: 2323: 2320: 2305: 2302: 2259:linolenic acid 2219: 2216: 2190:there is also 2183: 2180: 2152: 2151: 2140: 2137: 2134: 2131: 2128: 2125: 2121: 2117: 2114: 2111: 2107: 2104: 2101: 2098: 2095: 2092: 2088: 2084: 2081: 2078: 1982: 1944: 1943: 1932: 1927: 1924: 1919: 1914: 1911: 1902: 1897: 1893: 1890: 1867: 1856: 1823:Main article: 1820: 1817: 1774: 1771: 1721:susceptibility 1711: 1708: 1679: 1676: 1668:critical point 1663: 1660: 1655: 1648: 1640: 1632: 1625: 1617: 1614: 1612: 1609: 1478:, but forms a 1458: 1455: 1390:Paul Ehrenfest 1386: 1383: 1381: 1378: 1375: 1374: 1372: 1371: 1364: 1357: 1349: 1346: 1345: 1344: 1343: 1330: 1313: 1312: 1308: 1307: 1302: 1297: 1292: 1287: 1282: 1277: 1272: 1267: 1262: 1257: 1252: 1247: 1242: 1237: 1232: 1227: 1222: 1217: 1212: 1207: 1202: 1197: 1192: 1187: 1182: 1177: 1172: 1167: 1162: 1157: 1152: 1147: 1142: 1137: 1132: 1127: 1122: 1117: 1112: 1107: 1101: 1100: 1097: 1096: 1093: 1092: 1088: 1087: 1082: 1080:Liquid crystal 1077: 1072: 1067: 1061: 1060: 1055: 1054: 1051: 1050: 1046: 1045: 1040: 1035: 1030: 1021: 1016: 1010: 1009: 1006:Quasiparticles 1004: 1003: 1000: 999: 995: 994: 989: 980: 971: 965:Superdiamagnet 962: 956: 955: 952: 951: 948: 947: 943: 942: 937: 932: 926: 925: 922: 921: 918: 917: 913: 912: 907: 902: 897: 892: 890:Thermoelectric 887: 885:Superconductor 882: 877: 872: 867: 865:Mott insulator 862: 857: 852: 846: 845: 842: 841: 838: 837: 833: 832: 827: 822: 816: 815: 812: 811: 808: 807: 803: 802: 797: 792: 787: 782: 777: 772: 767: 762: 757: 752: 747: 742: 736: 735: 730: 729: 726: 725: 723: 722: 717: 712: 706: 703: 702: 694: 693: 666: 665: 660:condenses (an 651: 644: 625: 618: 611: 610:on iron (110). 600: 597:liquid crystal 572: 571:Other examples 569: 475: 472: 461:incommensurate 410: 407: 332: 329: 305:freezing point 295: 294: 292: 287: 285: 283: 279: 278: 273: 271: 266: 261: 257: 256: 254: 249: 247: 242: 238: 237: 235: 230: 225: 223: 219: 218: 213: 208: 203: 198: 195: 192: 185: 184: 177: 170: 162: 127:vapor pressure 110: 107: 105: 102: 26: 9: 6: 4: 3: 2: 5643: 5632: 5629: 5627: 5624: 5622: 5619: 5618: 5616: 5601: 5598: 5596: 5593: 5591: 5588: 5586: 5583: 5581: 5578: 5576: 5573: 5571: 5570:Mpemba effect 5568: 5566: 5563: 5561: 5558: 5556: 5553: 5551: 5550:Cooling curve 5548: 5546: 5543: 5541: 5538: 5536: 5533: 5532: 5530: 5526: 5520: 5517: 5515: 5512: 5510: 5507: 5505: 5502: 5500: 5497: 5495: 5492: 5490: 5487: 5486: 5484: 5480: 5474: 5473:Vitrification 5471: 5469: 5466: 5464: 5461: 5459: 5456: 5454: 5451: 5449: 5446: 5444: 5441: 5439: 5438:Recombination 5436: 5434: 5433:Melting point 5431: 5429: 5426: 5424: 5421: 5419: 5416: 5414: 5411: 5409: 5406: 5404: 5401: 5399: 5396: 5394: 5391: 5389: 5386: 5384: 5381: 5379: 5378:Critical line 5376: 5374: 5371: 5369: 5368:Boiling point 5366: 5364: 5361: 5360: 5358: 5356: 5352: 5346: 5343: 5341: 5338: 5334: 5331: 5329: 5326: 5324: 5321: 5320: 5318: 5316: 5313: 5311: 5308: 5306: 5303: 5301: 5300:Exotic matter 5298: 5296: 5293: 5291: 5288: 5286: 5283: 5281: 5278: 5276: 5273: 5272: 5270: 5266: 5260: 5257: 5255: 5252: 5250: 5247: 5246: 5244: 5240: 5234: 5231: 5229: 5226: 5224: 5221: 5219: 5216: 5214: 5211: 5209: 5206: 5204: 5201: 5199: 5196: 5194: 5191: 5190: 5188: 5184: 5179: 5169: 5166: 5164: 5161: 5159: 5155: 5152: 5150: 5147: 5145: 5142: 5141: 5139: 5135: 5130: 5126: 5119: 5114: 5112: 5107: 5105: 5100: 5099: 5096: 5089: 5086: 5083: 5080: 5079: 5078: 5076: 5075:Phase changes 5071: 5058: 5054: 5053:Yeomans J. M. 5051: 5048: 5044: 5043:H. E. Stanley 5041: 5038: 5037:W. H. Freeman 5034: 5030: 5027: 5024: 5021: 5020: 5015: 5011: 5007: 5004: 5001: 4997: 4993: 4989: 4986: 4982: 4978: 4974: 4970: 4967:(2): 75–199. 4966: 4962: 4958: 4953: 4950: 4944: 4940: 4938:981-02-4659-5 4934: 4930: 4929: 4924: 4920: 4917: 4913: 4912:9971-5-0210-0 4909: 4905: 4901: 4897: 4893: 4892:Superfluidity 4889: 4885: 4882: 4880: 4878: 4873: 4863: 4857: 4853: 4852: 4846: 4843: 4839: 4835: 4831: 4827: 4823: 4819: 4815: 4811: 4807: 4804: 4800: 4796: 4792: 4789: 4786: 4782: 4778: 4775: 4774: 4759: 4755: 4751: 4747: 4740: 4739: 4731: 4723: 4719: 4715: 4711: 4707: 4703: 4698: 4693: 4689: 4685: 4681: 4674: 4666: 4662: 4658: 4654: 4650: 4646: 4643:(1): 011920. 4642: 4638: 4631: 4623: 4619: 4615: 4611: 4607: 4603: 4596: 4588: 4584: 4580: 4576: 4572: 4568: 4561: 4553: 4547: 4544:. MIT Press. 4543: 4536: 4528: 4524: 4520: 4516: 4512: 4508: 4503: 4498: 4494: 4490: 4486: 4479: 4471: 4467: 4462: 4457: 4453: 4449: 4445: 4441: 4436: 4431: 4428:(6): 068102. 4427: 4423: 4419: 4412: 4404: 4400: 4396: 4392: 4388: 4384: 4379: 4374: 4370: 4366: 4359: 4351: 4347: 4342: 4337: 4332: 4327: 4323: 4319: 4314: 4309: 4305: 4301: 4297: 4290: 4282: 4278: 4273: 4268: 4263: 4258: 4254: 4250: 4245: 4240: 4236: 4232: 4228: 4221: 4212: 4207: 4200: 4192: 4188: 4184: 4180: 4176: 4172: 4168: 4161: 4153: 4149: 4145: 4141: 4137: 4133: 4129: 4122: 4114: 4110: 4106: 4102: 4098: 4094: 4090: 4086: 4079: 4071: 4067: 4063: 4059: 4055: 4051: 4046: 4041: 4038:(8): 085001. 4037: 4033: 4026: 4018: 4014: 4010: 4006: 4002: 3998: 3993: 3988: 3984: 3980: 3973: 3965: 3961: 3957: 3953: 3949: 3945: 3941: 3937: 3932: 3927: 3923: 3919: 3912: 3904: 3900: 3896: 3892: 3888: 3884: 3880: 3876: 3869: 3862: 3855: 3849: 3841: 3835: 3830: 3829: 3820: 3812: 3808: 3803: 3798: 3794: 3790: 3786: 3779: 3771: 3765: 3750: 3744: 3740: 3739: 3731: 3723: 3719: 3715: 3711: 3707: 3703: 3699: 3695: 3690: 3685: 3681: 3677: 3670: 3662: 3658: 3654: 3650: 3646: 3642: 3638: 3634: 3630: 3626: 3621: 3616: 3612: 3608: 3585: 3577: 3573: 3569: 3565: 3561: 3557: 3554:(1): 012403. 3553: 3549: 3533: 3525: 3521: 3517: 3513: 3509: 3505: 3501: 3497: 3493: 3489: 3482: 3474: 3470: 3466: 3462: 3458: 3454: 3449: 3444: 3440: 3436: 3429: 3421: 3417: 3413: 3409: 3405: 3401: 3397: 3393: 3388: 3383: 3379: 3375: 3356: 3348: 3344: 3340: 3336: 3331: 3326: 3322: 3318: 3314: 3310: 3306: 3299: 3291: 3287: 3283: 3279: 3275: 3271: 3267: 3263: 3259: 3255: 3248: 3240: 3236: 3232: 3228: 3224: 3220: 3216: 3212: 3207: 3202: 3198: 3194: 3187: 3178: 3170: 3166: 3162: 3158: 3154: 3150: 3143: 3141: 3132: 3128: 3124: 3120: 3116: 3112: 3108: 3104: 3100: 3096: 3091: 3086: 3082: 3078: 3071: 3063: 3059: 3055: 3051: 3047: 3043: 3039: 3035: 3030: 3025: 3021: 3017: 3009: 3001: 2997: 2993: 2989: 2985: 2981: 2974: 2967: 2966: 2959: 2952: 2951: 2944: 2936: 2932: 2926: 2918: 2914: 2910: 2906: 2902: 2898: 2894: 2887: 2879: 2875: 2871: 2867: 2863: 2859: 2852: 2844: 2838: 2834: 2827: 2825: 2816: 2812: 2808: 2804: 2800: 2796: 2792: 2788: 2783: 2778: 2775:(1): P01012. 2774: 2770: 2766: 2759: 2752: 2748: 2744: 2740: 2733: 2725: 2719: 2715: 2708: 2706: 2697: 2693: 2689: 2685: 2681: 2677: 2670: 2668: 2651: 2647: 2646:Carol Kendall 2641: 2635: 2632: 2631: 2624: 2616: 2612: 2607: 2602: 2598: 2594: 2590: 2586: 2581: 2576: 2572: 2568: 2564: 2557: 2549: 2543: 2539: 2532: 2528: 2517: 2514: 2511: 2508: 2505: 2502: 2498: 2495: 2494: 2492: 2489: 2486: 2483: 2480: 2477: 2471: 2468: 2465: 2464:Landau theory 2462: 2459: 2456: 2450: 2447: 2441: 2438: 2436: 2433: 2427: 2424: 2421: 2418: 2410: 2407: 2406: 2404: 2401: 2395: 2392: 2389: 2386: 2385: 2376: 2373: 2371:(very common) 2370: 2367: 2364: 2361: 2359: 2356: 2353: 2349: 2346: 2344: 2341: 2338: 2335: 2332: 2329: 2328: 2327: 2319: 2317: 2311: 2301: 2297: 2294: 2290: 2286: 2281: 2278: 2273: 2271: 2267: 2264: 2260: 2255: 2252: 2248: 2243: 2241: 2237: 2233: 2229: 2225: 2224:lipid bilayer 2215: 2213: 2209: 2205: 2201: 2197: 2193: 2189: 2179: 2176: 2170: 2168: 2163: 2161: 2157: 2138: 2132: 2129: 2126: 2119: 2115: 2112: 2109: 2105: 2099: 2096: 2093: 2086: 2082: 2079: 2076: 2069: 2068: 2067: 2065: 2061: 2057: 2053: 2049: 2044: 2042: 2036: 2034: 2030: 2026: 2022: 2017: 2015: 2011: 2007: 2003: 1999: 1994: 1980: 1970: 1968: 1964: 1961:The exponent 1959: 1957: 1953: 1949: 1930: 1925: 1922: 1912: 1909: 1900: 1891: 1888: 1881: 1880: 1879: 1877: 1873: 1866: 1862: 1855: 1851: 1847: 1846:heat capacity 1843: 1839: 1834: 1832: 1826: 1816: 1814: 1810: 1805: 1803: 1799: 1798:Eric Chaisson 1794: 1792: 1788: 1784: 1780: 1770: 1768: 1764: 1759: 1756: 1751: 1749: 1745: 1744:magnetization 1741: 1740:ferromagnetic 1735: 1733: 1732:ferromagnetic 1729: 1728:magnetization 1724: 1722: 1717: 1707: 1705: 1701: 1697: 1693: 1689: 1685: 1675: 1673: 1669: 1659: 1654: 1647: 1639: 1631: 1624: 1608: 1605: 1604: 1598: 1594: 1590: 1585: 1583: 1579: 1575: 1571: 1567: 1563: 1558: 1556: 1551: 1549: 1546: 1542: 1538: 1533: 1529: 1525: 1521: 1520:ferromagnetic 1517: 1513: 1509: 1505: 1501: 1492: 1489: 1485: 1481: 1477: 1472: 1470: 1466: 1462: 1454: 1451: 1447: 1443: 1438: 1436: 1430: 1428: 1422: 1420: 1416: 1412: 1411:magnetization 1408: 1404: 1399: 1395: 1391: 1370: 1365: 1363: 1358: 1356: 1351: 1350: 1348: 1347: 1341: 1331: 1328: 1323: 1317: 1316: 1315: 1314: 1306: 1303: 1301: 1298: 1296: 1293: 1291: 1288: 1286: 1283: 1281: 1278: 1276: 1273: 1271: 1268: 1266: 1263: 1261: 1258: 1256: 1253: 1251: 1248: 1246: 1243: 1241: 1238: 1236: 1233: 1231: 1228: 1226: 1223: 1221: 1218: 1216: 1213: 1211: 1208: 1206: 1203: 1201: 1198: 1196: 1193: 1191: 1188: 1186: 1183: 1181: 1178: 1176: 1173: 1171: 1168: 1166: 1163: 1161: 1158: 1156: 1153: 1151: 1148: 1146: 1143: 1141: 1138: 1136: 1133: 1131: 1128: 1126: 1123: 1121: 1118: 1116: 1113: 1111: 1108: 1106: 1105:Van der Waals 1103: 1102: 1095: 1094: 1086: 1083: 1081: 1078: 1076: 1073: 1071: 1068: 1066: 1063: 1062: 1058: 1053: 1052: 1044: 1041: 1039: 1036: 1034: 1031: 1029: 1025: 1022: 1020: 1017: 1015: 1012: 1011: 1007: 1002: 1001: 993: 990: 988: 984: 981: 979: 975: 972: 970: 966: 963: 961: 958: 957: 950: 949: 941: 938: 936: 933: 931: 928: 927: 920: 919: 911: 908: 906: 903: 901: 900:Ferroelectric 898: 896: 895:Piezoelectric 893: 891: 888: 886: 883: 881: 878: 876: 873: 871: 870:Semiconductor 868: 866: 863: 861: 858: 856: 853: 851: 848: 847: 840: 839: 831: 828: 826: 823: 821: 818: 817: 810: 809: 801: 798: 796: 793: 791: 790:Superfluidity 788: 786: 783: 781: 778: 776: 773: 771: 768: 766: 763: 761: 758: 756: 753: 751: 748: 746: 743: 741: 738: 737: 733: 728: 727: 721: 718: 716: 713: 711: 708: 707: 705: 704: 700: 696: 695: 692: 689: 688: 685: 683: 679: 675: 671: 663: 659: 655: 652: 649: 645: 642: 638: 634: 630: 626: 623: 619: 616: 612: 609: 605: 601: 598: 594: 590: 589: 588: 582: 577: 568: 566: 561: 559: 554: 552: 548: 544: 540: 536: 532: 527: 525: 521: 517: 513: 512:melting point 509: 505: 497: 493: 492:mixing ratios 490:at different 489: 485: 480: 471: 470: 466: 462: 458: 454: 450: 446: 442: 441:ferromagnetic 438: 430: 426: 421: 416: 406: 403: 399: 395: 391: 386: 384: 380: 376: 372: 368: 360: 357:(α-iron) and 356: 352: 348: 343: 338: 328: 326: 322: 318: 314: 313:adiabatically 310: 306: 302: 301:phase diagram 293: 291: 290:Recombination 288: 281: 280: 277: 274: 272: 270: 267: 265: 262: 259: 258: 253: 250: 248: 246: 243: 240: 239: 234: 231: 229: 226: 224: 221: 220: 217: 214: 212: 209: 207: 204: 202: 199: 191: 190: 183: 178: 176: 171: 169: 164: 163: 160: 155: 153: 148: 146: 142: 141:boiling point 138: 132: 131:phase diagram 128: 120: 115: 101: 99: 98:boiling point 95: 91: 87: 83: 79: 75: 71: 67: 63: 59: 55: 51: 47: 43: 34: 30: 19: 5595:Superheating 5468:Vaporization 5463:Triple point 5458:Supercooling 5423:Lambda point 5373:Condensation 5290:Time crystal 5268:Other states 5208:Quantum Hall 5067: 5056: 5046: 5035:, New York: 5032: 5017: 5016:, vol. 5 of 5013: 5006:Landau, L.D. 4991: 4964: 4960: 4943:the original 4927: 4906:; Paperback 4896:Vortex lines 4887: 4884:Kleinert, H. 4876: 4865:, retrieved 4850: 4841: 4817: 4813: 4810:Fisher, M.E. 4780: 4737: 4730: 4687: 4683: 4673: 4640: 4636: 4630: 4605: 4601: 4595: 4570: 4566: 4560: 4541: 4535: 4492: 4488: 4478: 4425: 4421: 4411: 4368: 4364: 4358: 4303: 4299: 4289: 4234: 4230: 4220: 4199: 4174: 4170: 4160: 4135: 4131: 4121: 4088: 4084: 4078: 4035: 4031: 4025: 3982: 3978: 3972: 3921: 3917: 3911: 3878: 3874: 3861: 3853: 3848: 3827: 3819: 3792: 3788: 3778: 3752:. Retrieved 3737: 3730: 3679: 3675: 3669: 3610: 3606: 3584: 3551: 3547: 3532: 3491: 3487: 3481: 3441:(49): L605. 3438: 3434: 3428: 3377: 3373: 3355: 3312: 3308: 3298: 3257: 3253: 3247: 3196: 3192: 3186: 3177: 3152: 3148: 3080: 3076: 3070: 3019: 3015: 3008: 2983: 2980:Phys. Rev. B 2979: 2973: 2964: 2958: 2949: 2943: 2934: 2925: 2903:(1): 51–81. 2900: 2896: 2886: 2861: 2857: 2851: 2832: 2772: 2768: 2758: 2742: 2738: 2732: 2713: 2682:(1): 51–81. 2679: 2675: 2654:. 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