Phase transition - Knowledge

1312: 479: 114: 342: 420: 699: 1411:, 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. 1325: 5168: 33: 1626:

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

2239:, 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 1461:. 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. 1442:

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

1684:: 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 1414:

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.

1660:, 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 1399:, 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 2289:

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 1962:

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: 2162:

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.

2002:= −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 1708:

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. 3003:

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".

5029:, 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. 1422:

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. 2139: 2028:

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

1823:, 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. 1727:

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.

1931: 2340:(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 2251:, 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, 1419:

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)

1771:. 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".

1403:, 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 3350:

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".

2056:, are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as 2304:

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.

4865: 4726: 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: 2155:

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

3759: 2159:. 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 5077: 3967:

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

1779:. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to 2062: 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. 1497: 546: 121:, showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure. 1936:

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.

2017:< 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: 1520:

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

4849: 4539: 3827: 3736: 2830: 2711: 2535: 4916: 1618:, which may depend on the applied pressure. If the first-order freezing transition occurs over a range of temperatures, and 1874: 1724:

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 1559: 60:

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

4926: 4901: 4893: 2409: 1526: 1342: 1329: 2505: 1990:< 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the 1534: 393: 370: 336: 1311: 5372: 2255: 2003: 1710: 1657: 1505: 5442: 5367: 5098: 5064: 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".

2329:(simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800–1000 °C) 2217: 1464:

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: 5483: 1740:. However, note that order parameters can also be defined for non-symmetry-breaking transitions. 1416: 1408: 859: 1264: 5508: 5498: 5248: 5243: 4982:

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

4766: 1517: 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 3726: 3212: 4995: 2398: 2232:, and cooperative ligand binding to DNA and proteins with the character of phase transition. 2164: 1966: 1776: 1513: 1289: 904: 889: 653: 5427: 5187: 5018: 4958: 4811: 4728:

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: 2316:

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

1830:. 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?".

4311: 4242: 4157:"Determination of membrane lipid phase transition temperature from 13-C NMR intensities" 4043: 3990: 3929: 3876: 3687: 3618: 3549: 3489: 3446: 3385: 3310: 3255: 3204: 3150: 3088: 3027: 2981: 2780: 2578: 2212:

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

5549: 5478: 5312: 4774: 4707: 4681: 4512: 4486: 4450: 4419: 4407: 4388: 4362: 4330: 4297: 4285: 4261: 4228: 4217:"Social interactions dominate speed control in poising natural flocks near criticality" 4216: 4195: 4098: 4055: 4029: 4002: 3976: 3949: 3915: 3888: 3796: 3753: 3707: 3673: 3646: 3604: 3509: 3458: 3432: 3405: 3371: 3332: 3275: 3224: 3190: 3137:

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

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

3158: 3108: 3100: 3051: 3039: 2920: 2902: 2863: 2826: 2804: 2792: 2707: 2685: 2639: 2600: 2553:"Phase diagram for the transition from photonic crystals to dielectric metamaterials" 2531: 2438: 2364: 1991: 1827: 1814: 1772: 1744: 1693: 1677: 1673: 1592: 1477: 1439: 1404: 1229: 1074: 964: 884: 673: 647: 614: 515: 436: 374: 350: 85: 4156: 4102: 4006: 3953: 3892: 3800: 3773:

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

5524: 5377: 5114: 4932: 4792: 4788: 2953: 2493: 2221: 1244: 1194: 1064: 819: 731: 564: 550: 378: 136: 61: 4703: 4603: 2483: – Mathematical theory on behavior of connected clusters in a random graph 2456: – Theory of continuous phase transitions of second order phase transitions 2013:< 1, the heat capacity diverges at the transition temperature (though, since 1834:

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

5322: 5317: 5274: 5207: 5202: 4912: 4873: 4646: 4568: 4051: 3998: 3695: 3557: 3393: 3035: 2883:"The Ehrenfest Classification of Phase Transitions: Introduction and Evolution" 2392: 2248: 1959:. Its actual value depends on the type of phase transition we are considering. 1752: 1509: 1379: 1316: 1294: 1274: 1269: 1224: 1144: 1079: 977: 864: 709: 677: 596: 523: 440: 341: 304: 126: 4938: 4823: 4384: 2526:

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

1736:, 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: 4837: 4802:(1974). "The renormalization group in the theory of critical behavior". 2882: 2859: 2586: 2134:{\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

4408:"Zipf's law and criticality in multivariate data without fine-tuning" 3497: 2432: – Mathematical model of ferromagnetism in statistical mechanics 2377: 2030: 1865: 1501: 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.

1496:. They are characterized by a divergent susceptibility, an infinite 32: 5569: 5397: 4118:"C NMR studies of lipid fatty acyl chains of chloroplast membranes" 3920: 2846:

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

4686: 4491: 4424: 4367: 4302: 4233: 4200: 3678: 3609: 3349: 3079: 2771: 5529: 5417: 5352: 5269: 5264: 2274: 2192:. 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: 1634:

raises the possibility that one can study the interplay between

5138: 5059: 4949:(1974). "The Renormalization Group and the epsilon-Expansion". 4885: 4869:, (Physics Education (India) Volume 32. No. 2, Apr - Jun 2016) 4718: 2525: 1037: 1013: 744: 640: 487: 448: 205: 69: 1808: 1767:

Symmetry-breaking phase transitions play an important role in

575: 5147: 5133: 3064: 2701: 2508: – Field theory involving topological effects in physics 2352: 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

3772: 3526: 3422: 3002: 2386: – Chemical reaction whose product is also its catalyst 4832:

Lectures on Phase Transitions and the Renormalization Group

4670:"Correlations, risk and crisis: From physiology to finance" 4668:

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

3663: 3475: 3858:"Topologically disordered systems at the glass transition" 3578: 2207: 2184:. As a consequence, at a phase transition one may observe 1382:

classified phase transitions based on the behavior of the

5143: 3966: 2259: 1983:, the exponent of the susceptibility) are not identical. 749: 210: 73: 5037:

Introduction to Phase Transitions and Critical Phenomena

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Introduction to Phase Transitions and Critical Phenomena

2825:(Repr ed.). Cambridge: Univ. Pr. pp. 140–141. 2418: – Shift of atomic positions in a crystal structure 385:) are all examples of solid to solid phase transitions. 4406:

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

4988:, 1996. Contains a detailed pedagogical discussion of 3844:

Cosmogenesis, The Development of Order in the Universe

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

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Physica A: Statistical Mechanics and Its Applications

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

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

1508:. Examples of second-order phase transitions are the 5039:(Oxford University Press, Oxford and New York 1971). 3520: 3343: 3132: 3130: 2702:

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

2464:

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2228:, liquid crystal-like transitions in the process of 2200:

of a continuous phase transition split into smaller

1438:. The exact specific heat differed from the earlier 439:. The most well-known is the transition between the 4838:Ivancevic, Vladimir G; Ivancevic, Tijana T (2008), 4736:

Department of Peace Studies, University of Bradford

4405: 3657: 3572: 4529: 3815: 3724: 3058: 2496: – Thin layer of liquid in a superfluid state 2133: 1975: 1925: 606:geometry on coverage and temperature, such as for 4789:Fundamentals of Multiphase Heat Transfer and Flow 4589: 3127: 2996: 2955:Fundamentals of Multiphase Heat Transfer and Flow 2395: – Major stage of a crystallization process 1940:= 0.59 A similar behavior, but with the exponent 5602: 3905: 3180: 1838: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 4724: 4554: 4161:Journal of Biochemical and Biophysical Methods 2752: 2471: – Different known phase of states matter 1512: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 ( 5099: 4472: 4122:Indian Journal of Biochemistry and Biophysics 2640:"Fundamentals of Stable Isotope Geochemistry" 2634: 1350: 400:. As with states of matter, there are also a 173: 4624: 4479:Progress in Biophysics and Molecular Biology 4109: 3856:Ojovan, Michael I.; Lee, William E. (2006). 3758:: CS1 maint: multiple names: authors list ( 2659: 2657: 1716:An example of an order parameter is the net 1600: 1472:mixture of liquid water and vapor bubbles). 1395:are continuous in the first derivative (the 103: 5113: 4944: 4215:Bialek, W; Cavagna, A; Giardina, I (2014). 2550: 2286:points of second-order phase transitions. 2168:correlation length is the essential point. 1809:Critical exponents and universality classes 1374: 377:to another, from a crystalline solid to an 349:, distinguishing between several different 48:, and other related fields like biology, a 5106: 5092: 5047:Statistical Mechanics of Phase Transitions 4352: 3295:"Materials science: Metal turned to glass" 2967: 2697: 2695: 2384:Autocatalytic reactions and order creation 1775:into the U(1) symmetry of the present-day 1696:, which only occurs at low temperatures). 1391:the free energy with respect to pressure. 1357: 1343: 697: 180: 166: 5072:Interactive Phase Transitions on lattices 4771:Basic Notions of Condensed Matter Physics 4685: 4490: 4473:Longo, G.; Montévil, M. 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Berlin: Springer. pp. 176–177. 3292: 3136: 2880: 2848:Industrial & Engineering Chemistry 2845: 2663: 1948:, applies for the correlation length. 1443:able to incorporate such transitions. 369:, whereas in compounds it is known as 84:and the states of matter have uniform 5087: 4753:from the original on 26 November 2022 3241: 2887:Archive for History of Exact Sciences 2726: 2666:Archive for History of Exact Sciences 2355:(measurement of magnetic transitions) 2323:(measurement of magnetic transitions) 2171: 1396: 3865:Journal of Physics: Condensed Matter 3597:Journal of Physics: Condensed Matter 3425:Journal of Physics: Condensed Matter 2816: 2814: 2401: – materials science phenomenon 1955:is positive. This is different with 1605: 4992:'s solution of the 2-D Ising Model. 4906:physik.fu-berlin.de readable online 3183:Annual Review of Physical Chemistry 2530:. Chapman & Hall. p. 286. 2294:Phase transitions in social systems 1699: 1554:. They are continuous but break no 1540:of second-order phase transitions. 1425:supercritical liquid–gas boundaries 108: 24: 4894:World Scientific (Singapore, 1989) 4760: 2412: – Thermoanalytical technique 2036:Several other critical exponents, 1672:Phase transitions often involve a 1651: 1369: 423:A phase diagram showing different 25: 5632: 5053: 5012:, Pergamon Press, 3rd Ed. (1994). 4918:Critical Properties of φ-Theories 2943:, Elsevier, Burlington, MA, 2006, 2811: 2551:Rybin, M.V.; et al. (2015). 2502: – Process in quantum optics 2410:Differential scanning calorimetry 1558:. The most famous example is the 1550:Several transitions are known as 1527:differential scanning calorimetry 668:Phase transitions occur when the 570: 520:liquidus and solidus temperatures 427:in the same crystal structure of 5166: 5058: 5049:, Oxford University Press, 1992. 4878:Gauge Fields in Condensed Matter 4501:10.1016/j.pbiomolbio.2011.03.005 3159:10.1016/j.jnoncrysol.2013.10.016 2789:10.1088/1742-5468/2014/01/P01012 2506:Topological quantum field theory 2477: – Crystal growth technique 2462: – crystal growth technique 1743:Some phase transitions, such as 1732:phase, one must provide the net 1688:, with the exception of certain 1552:infinite-order phase transitions 1324: 1323: 1310: 394:displacive phase transformations 337:Polymorphism (materials science) 286: 284: 255: 236: 4661: 4618: 4583: 4548: 4523: 4466: 4399: 4346: 4277: 4208: 4187: 4148: 4066: 4013: 3960: 3899: 3849: 3836: 3807: 3766: 3718: 3469: 3416: 3286: 3235: 3174: 3165: 2961: 2946: 2931: 2913: 2874: 2839: 2311: 2256:electron paramagnetic resonance 2098: 2004:variational perturbation theory 1476:and Michael Wortis showed that 117:A simplified phase diagram for 4481:. Systems Biology and Cancer. 4442:10.1103/PhysRevLett.113.068102 4355:Journal of Statistical Physics 4286:"Morphogenesis at criticality" 4087:10.1080/07391102.2000.10506578 3938:10.1103/PhysRevLett.115.200601 3627:10.1088/0953-8984/24/38/386004 3097:10.1103/PhysRevLett.100.247003 2958:, Springer, New York, NY, 2020 2746: 2720: 2628: 2611: 2544: 2519: 2125: 2113: 2092: 2080: 1907: 1885: 1585:and other liquids that can be 1572:two-dimensional electron gases 1560:Kosterlitz–Thouless transition 1494:"continuous phase transitions" 1393:Second-order phase transitions 13: 1: 5555:Macroscopic quantum phenomena 5009:Course of Theoretical Physics 3264:10.1126/science.267.5206.1947 2513: 2500:Superradiant phase transition 2448:Kelvin probe force microscope 2416:Diffusionless transformations 2338:Perturbed angular correlation 1692:(e.g. the formation of heavy 1686:spontaneous symmetry breaking 1555: 1487:Second-order phase transition 1455:First-order phase transitions 1388:First-order phase transitions 587:Other phase changes include: 330: 5565:Order and disorder (physics) 4971:10.1016/0370-1573(74)90023-4 4915:; Verena Schulte-Frohlinde. 4864:M.R. 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Oxford: Clarendon Press. 2297: 1812: 910:Spin gapless semiconductor 633:Bose–Einstein condensation 412: 390:martensitic transformation 334: 124: 5517: 5471: 5343: 5257: 5231: 5175: 5164: 5126: 4824:10.1103/revmodphys.46.597 4385:10.1007/s10955-011-0229-4 1799:relational order theories 1601:Characteristic properties 1568:quantum phase transitions 1457:are those that involve a 1384: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 5590:Thermo-dielectric effect 5489:Enthalpy of vaporization 5183:Bose–Einstein condensate 2990:10.1103/physrevb.19.3580 2469:List of states of matter 2342:tantalum hafnium carbide 2306:peace and armed conflict 1781:electroweak baryogenesis 1574:, belong to this class. 1434:, discovered in 1944 by 1375:Ehrenfest classification 760:Bose–Einstein condensate 691:Condensed matter physics 627:Quantum condensation of 5484:Enthalpy of sublimation 4412:Physical Review Letters 4321:10.1073/pnas.1324186111 4252:10.1073/pnas.1324045111 3792:10.1351/pac199466030577 3067:Physical Review Letters 2741:10.1103/PhysRevD.21.446 2218:coil-globule transition 1976:{\displaystyle \gamma } 1579:liquid–glass transition 1562:in the two-dimensional 1409:magnetic susceptibility 5499:Latent internal energy 5249:Color-glass condensate 4592:Human Movement Science 4075:J. Biomol. Struct. Dyn 2327:Mössbauer spectroscopy 2135: 1977: 1927: 1763:Relevance in cosmology 1713:will usually diverge. 1518:Type-II superconductor 1447: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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(2007). 3256:1995Sci...267.1947G 3250:(5206): 1947–1953. 3205:2007ARPC...58..235L 3151:2013JNCS..382...79O 3089:2008PhRvL.100x7003P 3028:2006PhRvB..73r4435K 2982:1979PhRvB..19.3580I 2860:10.1021/ie50275a006 2781:2014JSMTE..01..012M 2587:10.1038/ncomms10102 2579:2015NatCo...610102R 2425:Ehrenfest equations 2333:Neutron diffraction 2244:thylakoid membranes 930:Quantum Hall effect 425:magnetic structures 398:titanium aluminides 188: 86:physical properties 5621:Critical phenomena 5616:Physical phenomena 5550:Leidenfrost effect 5479:Enthalpy of fusion 5244:Quark–gluon plasma 4978:Krieger, Martin H. 4775:Perseus Publishing 2921:Stanley, H. Eugene 2481:Percolation theory 2475:Micro-pulling-down 2348:Raman Spectroscopy 2220:in the process of 2172:Critical phenomena 2131: 1973: 1923: 1832:correlation length 1828:critical exponents 1803:order and disorder 1680:breaks continuous 1498:correlation length 1317:Physics portal 585: 537:transformation. A 500: 433: 415:Magnetic structure 363: 351:crystal structures 347:allotropes of iron 157: 123: 38: 5611:Phase transitions 5598: 5597: 5580:Superheated vapor 5575:Superconductivity 5545:Equation of state 5393:Flash evaporation 5345:Phase transitions 5330:String-net liquid 5223:Photonic molecule 5193:Degenerate matter 5074:with Java applets 5063:Media related to 4890:Phase Transitions 4851:978-3-540-79356-4 4680:(16): 3193–3217. 4627:Physical Review E 4541:978-0-262-61131-2 4296:(10): 3683–3688. 4227:(20): 7212–7217. 4022:Physical Review D 3969:Physical Review B 3829:978-0-674-00342-2 3738:978-3-540-79357-1 3666:Physical Review B 3538:Physical Review B 3364:Physical Review B 3305:(7155): 758–759. 3006:Physical Review B 2854:(11): 1225–1235. 2832:978-0-521-09101-5 2729:Physical Review D 2713:978-0-19-856770-7 2537:978-0-412-53910-7 2439:Jamming (physics) 2365:X-ray diffraction 2182:critical dynamics 1992:lambda transition 1896: 1815:critical exponent 1773:electroweak field 1694:virtual particles 1678:crystalline solid 1674:symmetry breaking 1606:Phase coexistence 1593:quenched disorder 1478:quenched disorder 1405:Curie temperature 1367: 1366: 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 16:(Redirected from 5628: 5535:Compressed fluid 5170: 5115:States of matter 5108: 5101: 5094: 5085: 5084: 5062: 4974: 4936: 4931:. Archived from 4861: 4860: 4858: 4830:Goldenfeld, N., 4827: 4755: 4754: 4752: 4733: 4722: 4716: 4715: 4689: 4665: 4659: 4658: 4622: 4616: 4615: 4587: 4581: 4580: 4552: 4546: 4545: 4527: 4521: 4520: 4494: 4470: 4464: 4463: 4453: 4427: 4403: 4397: 4396: 4370: 4350: 4344: 4343: 4333: 4323: 4305: 4281: 4275: 4274: 4264: 4254: 4236: 4212: 4206: 4205: 4203: 4191: 4185: 4184: 4152: 4146: 4145: 4113: 4107: 4106: 4070: 4064: 4063: 4037: 4017: 4011: 4010: 3984: 3982:cond-mat/0310163 3964: 3958: 3957: 3923: 3903: 3897: 3896: 3862: 3853: 3847: 3840: 3834: 3833: 3821: 3818:Cosmic Evolution 3811: 3805: 3804: 3794: 3770: 3764: 3763: 3757: 3749: 3747: 3745: 3722: 3716: 3715: 3681: 3661: 3655: 3654: 3612: 3576: 3570: 3569: 3524: 3518: 3517: 3498:10.1038/nmat1743 3478:Nature Materials 3473: 3467: 3466: 3440: 3438:cond-mat/0611152 3420: 3414: 3413: 3379: 3377:cond-mat/0012472 3347: 3341: 3340: 3322: 3290: 3284: 3283: 3239: 3233: 3232: 3198: 3196:cond-mat/0607349 3178: 3172: 3169: 3163: 3162: 3134: 3125: 3124: 3082: 3062: 3056: 3055: 3021: 3019:cond-mat/0602627 3000: 2994: 2993: 2976:(7): 3580–3585. 2965: 2959: 2950: 2944: 2935: 2929: 2928: 2917: 2911: 2910: 2878: 2872: 2871: 2843: 2837: 2836: 2818: 2809: 2808: 2774: 2750: 2744: 2743: 2724: 2718: 2717: 2699: 2690: 2689: 2661: 2652: 2651: 2649: 2647: 2632: 2626: 2624:, Elsevier, 2019 2615: 2609: 2608: 2598: 2572: 2548: 2542: 2541: 2523: 2465: 2444: 2435: 2421: 2404: 2389: 2359:Thermogravimetry 2230:DNA condensation 2178:static functions 2140: 2138: 2137: 2132: 2112: 2079: 1982: 1980: 1979: 1974: 1932: 1930: 1929: 1924: 1919: 1918: 1910: 1898: 1897: 1894: 1888: 1864:typically has a 1700:Order parameters 1535:phenomenological 1492:are also called 1489: 1488: 1359: 1352: 1345: 1332: 1327: 1326: 1319: 1315: 1314: 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: 5636: 5635: 5631: 5630: 5629: 5627: 5626: 5625: 5601: 5600: 5599: 5594: 5525:Baryonic matter 5513: 5467: 5438:Saturated fluid 5378:Crystallization 5339: 5313:Antiferromagnet 5253: 5227: 5171: 5162: 5122: 5112: 5056: 4939:readable online 4929: 4913:Kleinert, Hagen 4856: 4854: 4852: 4793:Springer Nature 4763: 4761:Further reading 4758: 4750: 4731: 4723: 4719: 4666: 4662: 4623: 4619: 4588: 4584: 4553: 4549: 4542: 4528: 4524: 4471: 4467: 4404: 4400: 4351: 4347: 4282: 4278: 4213: 4209: 4192: 4188: 4153: 4149: 4114: 4110: 4071: 4067: 4018: 4014: 3965: 3961: 3908:Phys. Rev. Lett 3904: 3900: 3860: 3854: 3850: 3841: 3837: 3830: 3812: 3808: 3771: 3767: 3751: 3750: 3743: 3741: 3739: 3723: 3719: 3662: 3658: 3594: 3590: 3586: 3582: 3577: 3573: 3535: 3531: 3525: 3521: 3484:(11): 881–886. 3474: 3470: 3421: 3417: 3361: 3357: 3353: 3348: 3344: 3320:10.1038/448758a 3291: 3287: 3240: 3236: 3179: 3175: 3170: 3166: 3135: 3128: 3063: 3059: 3001: 2997: 2966: 2962: 2951: 2947: 2936: 2932: 2918: 2914: 2879: 2875: 2844: 2840: 2833: 2819: 2812: 2751: 2747: 2725: 2721: 2714: 2700: 2693: 2662: 2655: 2645: 2643: 2633: 2629: 2616: 2612: 2549: 2545: 2538: 2524: 2520: 2516: 2511: 2494:Superfluid film 2463: 2442: 2433: 2419: 2402: 2387: 2373: 2314: 2302: 2296: 2267:neural networks 2222:protein folding 2216:formation, the 2210: 2174: 2108: 2075: 2064: 2061: 2060: 1968: 1965: 1964: 1911: 1906: 1905: 1893: 1889: 1884: 1876: 1873: 1872: 1859: 1848: 1817: 1811: 1765: 1745:superconducting 1706:order parameter 1702: 1670: 1654: 1652:Critical points 1647: 1640: 1632: 1624: 1617: 1608: 1603: 1529:measurements. 1486: 1485: 1449: 1417:Cornelis Gorter 1397:order parameter 1377: 1372: 1370:Classifications 1363: 1322: 1309: 1308: 1301: 1300: 1299: 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: 5634: 5624: 5623: 5618: 5613: 5596: 5595: 5593: 5592: 5587: 5582: 5577: 5572: 5567: 5562: 5557: 5552: 5547: 5542: 5537: 5532: 5527: 5521: 5519: 5515: 5514: 5512: 5511: 5506: 5504:Trouton's rule 5501: 5496: 5491: 5486: 5481: 5475: 5473: 5469: 5468: 5466: 5465: 5460: 5455: 5450: 5445: 5440: 5435: 5430: 5425: 5420: 5415: 5410: 5405: 5400: 5395: 5390: 5385: 5380: 5375: 5373:Critical point 5370: 5365: 5360: 5355: 5349: 5347: 5341: 5340: 5338: 5337: 5332: 5327: 5326: 5325: 5320: 5315: 5307: 5302: 5297: 5292: 5287: 5282: 5277: 5275:Liquid crystal 5272: 5267: 5261: 5259: 5255: 5254: 5252: 5251: 5246: 5241: 5235: 5233: 5229: 5228: 5226: 5225: 5220: 5215: 5210: 5208:Strange matter 5205: 5203:Rydberg matter 5200: 5195: 5190: 5185: 5179: 5177: 5173: 5172: 5165: 5163: 5161: 5160: 5155: 5150: 5141: 5136: 5130: 5128: 5124: 5123: 5111: 5110: 5103: 5096: 5088: 5082: 5081: 5080:from Sklogwiki 5075: 5055: 5054:External links 5052: 5051: 5050: 5040: 5030: 5016: 5013: 5000:Lifshitz, E.M. 4993: 4975: 4942: 4927: 4909: 4871: 4862: 4850: 4835: 4828: 4810:(4): 597–616. 4804:Rev. Mod. Phys 4796: 4778: 4767:Anderson, P.W. 4762: 4759: 4757: 4756: 4717: 4660: 4617: 4598:(4): 483–493. 4582: 4563:(1): 183–202. 4547: 4540: 4522: 4485:(2): 340–347. 4465: 4398: 4361:(2): 268–302. 4345: 4276: 4207: 4186: 4167:(4): 353–356. 4147: 4128:(6): 177–178. 4108: 4081:(5): 903–911. 4065: 4035:hep-th/9812197 4012: 3975:(17): 174518. 3959: 3914:(20): 200601. 3898: 3848: 3842:David Layzer, 3835: 3828: 3806: 3785:(3): 577–594. 3765: 3737: 3717: 3672:(17): 174413. 3656: 3603:(38): 386004. 3592: 3588: 3584: 3580: 3571: 3533: 3529: 3519: 3468: 3415: 3370:(10): 104416. 3359: 3355: 3351: 3342: 3285: 3234: 3173: 3164: 3126: 3073:(24): 247003. 3057: 3012:(18): 184435. 2995: 2960: 2945: 2930: 2912: 2873: 2838: 2831: 2810: 2745: 2735:(2): 446–453, 2719: 2712: 2691: 2653: 2627: 2610: 2543: 2536: 2517: 2515: 2512: 2510: 2509: 2503: 2497: 2491: 2490: 2489: 2478: 2472: 2466: 2457: 2451: 2445: 2436: 2427: 2422: 2413: 2407: 2406: 2405: 2393:Crystal growth 2390: 2381: 2374: 2372: 2369: 2368: 2367: 2362: 2356: 2350: 2345: 2344:4215 °C.) 2335: 2330: 2324: 2313: 2310: 2295: 2292: 2249:linolenic acid 2209: 2206: 2180:there is also 2173: 2170: 2142: 2141: 2130: 2127: 2124: 2121: 2118: 2115: 2111: 2107: 2104: 2101: 2097: 2094: 2091: 2088: 2085: 2082: 2078: 2074: 2071: 2068: 1972: 1934: 1933: 1922: 1917: 1914: 1909: 1904: 1901: 1892: 1887: 1883: 1880: 1857: 1846: 1813:Main article: 1810: 1807: 1764: 1761: 1711:susceptibility 1701: 1698: 1669: 1666: 1658:critical point 1653: 1650: 1645: 1638: 1630: 1622: 1615: 1607: 1604: 1602: 1599: 1468:, but forms a 1448: 1445: 1380:Paul Ehrenfest 1376: 1373: 1371: 1368: 1365: 1364: 1362: 1361: 1354: 1347: 1339: 1336: 1335: 1334: 1333: 1320: 1303: 1302: 1298: 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: 5633: 5622: 5619: 5617: 5614: 5612: 5609: 5608: 5606: 5591: 5588: 5586: 5583: 5581: 5578: 5576: 5573: 5571: 5568: 5566: 5563: 5561: 5560:Mpemba effect 5558: 5556: 5553: 5551: 5548: 5546: 5543: 5541: 5540:Cooling curve 5538: 5536: 5533: 5531: 5528: 5526: 5523: 5522: 5520: 5516: 5510: 5507: 5505: 5502: 5500: 5497: 5495: 5492: 5490: 5487: 5485: 5482: 5480: 5477: 5476: 5474: 5470: 5464: 5463:Vitrification 5461: 5459: 5456: 5454: 5451: 5449: 5446: 5444: 5441: 5439: 5436: 5434: 5431: 5429: 5428:Recombination 5426: 5424: 5423:Melting point 5421: 5419: 5416: 5414: 5411: 5409: 5406: 5404: 5401: 5399: 5396: 5394: 5391: 5389: 5386: 5384: 5381: 5379: 5376: 5374: 5371: 5369: 5368:Critical line 5366: 5364: 5361: 5359: 5358:Boiling point 5356: 5354: 5351: 5350: 5348: 5346: 5342: 5336: 5333: 5331: 5328: 5324: 5321: 5319: 5316: 5314: 5311: 5310: 5308: 5306: 5303: 5301: 5298: 5296: 5293: 5291: 5290:Exotic matter 5288: 5286: 5283: 5281: 5278: 5276: 5273: 5271: 5268: 5266: 5263: 5262: 5260: 5256: 5250: 5247: 5245: 5242: 5240: 5237: 5236: 5234: 5230: 5224: 5221: 5219: 5216: 5214: 5211: 5209: 5206: 5204: 5201: 5199: 5196: 5194: 5191: 5189: 5186: 5184: 5181: 5180: 5178: 5174: 5169: 5159: 5156: 5154: 5151: 5149: 5145: 5142: 5140: 5137: 5135: 5132: 5131: 5129: 5125: 5120: 5116: 5109: 5104: 5102: 5097: 5095: 5090: 5089: 5086: 5079: 5076: 5073: 5070: 5069: 5068: 5066: 5065:Phase changes 5061: 5048: 5044: 5043:Yeomans J. M. 5041: 5038: 5034: 5033:H. E. Stanley 5031: 5028: 5027:W. H. Freeman 5024: 5020: 5017: 5014: 5011: 5010: 5005: 5001: 4997: 4994: 4991: 4987: 4983: 4979: 4976: 4972: 4968: 4964: 4960: 4957:(2): 75–199. 4956: 4952: 4948: 4943: 4940: 4934: 4930: 4928:981-02-4659-5 4924: 4920: 4919: 4914: 4910: 4907: 4903: 4902:9971-5-0210-0 4899: 4895: 4891: 4887: 4883: 4882:Superfluidity 4879: 4875: 4872: 4870: 4868: 4863: 4853: 4847: 4843: 4842: 4836: 4833: 4829: 4825: 4821: 4817: 4813: 4809: 4805: 4801: 4797: 4794: 4790: 4786: 4782: 4779: 4776: 4772: 4768: 4765: 4764: 4749: 4745: 4741: 4737: 4730: 4729: 4721: 4713: 4709: 4705: 4701: 4697: 4693: 4688: 4683: 4679: 4675: 4671: 4664: 4656: 4652: 4648: 4644: 4640: 4636: 4633:(1): 011920. 4632: 4628: 4621: 4613: 4609: 4605: 4601: 4597: 4593: 4586: 4578: 4574: 4570: 4566: 4562: 4558: 4551: 4543: 4537: 4534:. MIT Press. 4533: 4526: 4518: 4514: 4510: 4506: 4502: 4498: 4493: 4488: 4484: 4480: 4476: 4469: 4461: 4457: 4452: 4447: 4443: 4439: 4435: 4431: 4426: 4421: 4418:(6): 068102. 4417: 4413: 4409: 4402: 4394: 4390: 4386: 4382: 4378: 4374: 4369: 4364: 4360: 4356: 4349: 4341: 4337: 4332: 4327: 4322: 4317: 4313: 4309: 4304: 4299: 4295: 4291: 4287: 4280: 4272: 4268: 4263: 4258: 4253: 4248: 4244: 4240: 4235: 4230: 4226: 4222: 4218: 4211: 4202: 4197: 4190: 4182: 4178: 4174: 4170: 4166: 4162: 4158: 4151: 4143: 4139: 4135: 4131: 4127: 4123: 4119: 4112: 4104: 4100: 4096: 4092: 4088: 4084: 4080: 4076: 4069: 4061: 4057: 4053: 4049: 4045: 4041: 4036: 4031: 4028:(8): 085001. 4027: 4023: 4016: 4008: 4004: 4000: 3996: 3992: 3988: 3983: 3978: 3974: 3970: 3963: 3955: 3951: 3947: 3943: 3939: 3935: 3931: 3927: 3922: 3917: 3913: 3909: 3902: 3894: 3890: 3886: 3882: 3878: 3874: 3870: 3866: 3859: 3852: 3845: 3839: 3831: 3825: 3820: 3819: 3810: 3802: 3798: 3793: 3788: 3784: 3780: 3776: 3769: 3761: 3755: 3740: 3734: 3730: 3729: 3721: 3713: 3709: 3705: 3701: 3697: 3693: 3689: 3685: 3680: 3675: 3671: 3667: 3660: 3652: 3648: 3644: 3640: 3636: 3632: 3628: 3624: 3620: 3616: 3611: 3606: 3602: 3598: 3575: 3567: 3563: 3559: 3555: 3551: 3547: 3544:(1): 012403. 3543: 3539: 3523: 3515: 3511: 3507: 3503: 3499: 3495: 3491: 3487: 3483: 3479: 3472: 3464: 3460: 3456: 3452: 3448: 3444: 3439: 3434: 3430: 3426: 3419: 3411: 3407: 3403: 3399: 3395: 3391: 3387: 3383: 3378: 3373: 3369: 3365: 3346: 3338: 3334: 3330: 3326: 3321: 3316: 3312: 3308: 3304: 3300: 3296: 3289: 3281: 3277: 3273: 3269: 3265: 3261: 3257: 3253: 3249: 3245: 3238: 3230: 3226: 3222: 3218: 3214: 3210: 3206: 3202: 3197: 3192: 3188: 3184: 3177: 3168: 3160: 3156: 3152: 3148: 3144: 3140: 3133: 3131: 3122: 3118: 3114: 3110: 3106: 3102: 3098: 3094: 3090: 3086: 3081: 3076: 3072: 3068: 3061: 3053: 3049: 3045: 3041: 3037: 3033: 3029: 3025: 3020: 3015: 3011: 3007: 2999: 2991: 2987: 2983: 2979: 2975: 2971: 2964: 2957: 2956: 2949: 2942: 2941: 2934: 2926: 2922: 2916: 2908: 2904: 2900: 2896: 2892: 2888: 2884: 2877: 2869: 2865: 2861: 2857: 2853: 2849: 2842: 2834: 2828: 2824: 2817: 2815: 2806: 2802: 2798: 2794: 2790: 2786: 2782: 2778: 2773: 2768: 2765:(1): P01012. 2764: 2760: 2756: 2749: 2742: 2738: 2734: 2730: 2723: 2715: 2709: 2705: 2698: 2696: 2687: 2683: 2679: 2675: 2671: 2667: 2660: 2658: 2641: 2637: 2636:Carol Kendall 2631: 2625: 2622: 2621: 2614: 2606: 2602: 2597: 2592: 2588: 2584: 2580: 2576: 2571: 2566: 2562: 2558: 2554: 2547: 2539: 2533: 2529: 2522: 2518: 2507: 2504: 2501: 2498: 2495: 2492: 2488: 2485: 2484: 2482: 2479: 2476: 2473: 2470: 2467: 2461: 2458: 2455: 2454:Landau theory 2452: 2449: 2446: 2440: 2437: 2431: 2428: 2426: 2423: 2417: 2414: 2411: 2408: 2400: 2397: 2396: 2394: 2391: 2385: 2382: 2379: 2376: 2375: 2366: 2363: 2361:(very common) 2360: 2357: 2354: 2351: 2349: 2346: 2343: 2339: 2336: 2334: 2331: 2328: 2325: 2322: 2319: 2318: 2317: 2309: 2307: 2301: 2291: 2287: 2284: 2280: 2276: 2271: 2268: 2263: 2261: 2257: 2254: 2250: 2245: 2242: 2238: 2233: 2231: 2227: 2223: 2219: 2215: 2214:lipid bilayer 2205: 2203: 2199: 2195: 2191: 2187: 2183: 2179: 2169: 2166: 2160: 2158: 2153: 2151: 2147: 2128: 2122: 2119: 2116: 2109: 2105: 2102: 2099: 2095: 2089: 2086: 2083: 2076: 2072: 2069: 2066: 2059: 2058: 2057: 2055: 2051: 2047: 2043: 2039: 2034: 2032: 2026: 2024: 2020: 2016: 2012: 2007: 2005: 2001: 1997: 1993: 1989: 1984: 1970: 1960: 1958: 1954: 1951:The exponent 1949: 1947: 1943: 1939: 1920: 1915: 1912: 1902: 1899: 1890: 1881: 1878: 1871: 1870: 1869: 1867: 1863: 1856: 1852: 1845: 1841: 1837: 1836:heat capacity 1833: 1829: 1824: 1822: 1816: 1806: 1804: 1800: 1795: 1793: 1789: 1788:Eric Chaisson 1784: 1782: 1778: 1774: 1770: 1760: 1758: 1754: 1749: 1746: 1741: 1739: 1735: 1734:magnetization 1731: 1730:ferromagnetic 1725: 1723: 1722:ferromagnetic 1719: 1718:magnetization 1714: 1712: 1707: 1697: 1695: 1691: 1687: 1683: 1679: 1675: 1665: 1663: 1659: 1649: 1644: 1637: 1629: 1621: 1614: 1598: 1595: 1594: 1588: 1584: 1580: 1575: 1573: 1569: 1565: 1561: 1557: 1553: 1548: 1546: 1541: 1539: 1536: 1532: 1528: 1523: 1519: 1515: 1511: 1510:ferromagnetic 1507: 1503: 1499: 1495: 1491: 1482: 1479: 1475: 1471: 1467: 1462: 1460: 1456: 1452: 1444: 1441: 1437: 1433: 1428: 1426: 1420: 1418: 1412: 1410: 1406: 1402: 1401:magnetization 1398: 1394: 1389: 1385: 1381: 1360: 1355: 1353: 1348: 1346: 1341: 1340: 1338: 1337: 1331: 1321: 1318: 1313: 1307: 1306: 1305: 1304: 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: 18:Phase changes 5585:Superheating 5458:Vaporization 5453:Triple point 5448:Supercooling 5413:Lambda point 5363:Condensation 5280:Time crystal 5258:Other states 5198:Quantum Hall 5057: 5046: 5036: 5025:, New York: 5022: 5007: 5006:, vol. 5 of 5003: 4996:Landau, L.D. 4981: 4954: 4950: 4933:the original 4917: 4896:; Paperback 4886:Vortex lines 4877: 4874:Kleinert, H. 4866: 4855:, retrieved 4840: 4831: 4807: 4803: 4800:Fisher, M.E. 4770: 4727: 4720: 4677: 4673: 4663: 4630: 4626: 4620: 4595: 4591: 4585: 4560: 4556: 4550: 4531: 4525: 4482: 4478: 4468: 4415: 4411: 4401: 4358: 4354: 4348: 4293: 4289: 4279: 4224: 4220: 4210: 4189: 4164: 4160: 4150: 4125: 4121: 4111: 4078: 4074: 4068: 4025: 4021: 4015: 3972: 3968: 3962: 3911: 3907: 3901: 3868: 3864: 3851: 3843: 3838: 3817: 3809: 3782: 3778: 3768: 3742:. Retrieved 3727: 3720: 3669: 3665: 3659: 3600: 3596: 3574: 3541: 3537: 3522: 3481: 3477: 3471: 3431:(49): L605. 3428: 3424: 3418: 3367: 3363: 3345: 3302: 3298: 3288: 3247: 3243: 3237: 3186: 3182: 3176: 3167: 3142: 3138: 3070: 3066: 3060: 3009: 3005: 2998: 2973: 2970:Phys. Rev. B 2969: 2963: 2954: 2948: 2939: 2933: 2924: 2915: 2893:(1): 51–81. 2890: 2886: 2876: 2851: 2847: 2841: 2822: 2762: 2758: 2748: 2732: 2728: 2722: 2703: 2672:(1): 51–81. 2669: 2665: 2644:. 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