Phase transition - Knowledge

1301: 468: 103: 331: 409: 688: 1400:, 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. 1314: 5157: 22: 1615:

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

2228:, 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 1450:. 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. 1431:

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

1673:: 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 1403:

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.

1649:, 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 1388:, 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 2278:

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.

292:. 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 1951:

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

669:). 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: 2151:

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.

1991:= −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 1697:

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

673:, 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. 2992:

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".

5018:, 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. 1411:

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.

136:, 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. 2128: 2017:

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

1812:, 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. 1716:

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.

1920: 2329:(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 2240:, 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, 1408:

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)

1760:. 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 522:

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".

1392:, 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 3339:

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".

2045:, are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as 2293:

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.

4854: 4715: 89:, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. 168: 2144:

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

3748: 2148:. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid. 354:

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

77:. 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 5066: 3956:

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

1768:. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to 2051: 653:), 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. 1486: 535: 110:, showing whether solid ice, liquid water, or gaseous water vapor is the most stable at different combinations of temperature and pressure. 1925:

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.

2006:< 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional 304:) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state is 161: 1509:

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

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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

154: 308:, i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in 1548: 49:

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

4915: 4890: 4882: 2398: 1515: 1331: 1318: 2494: 1979:< 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the 1523: 382: 359: 325: 1300: 5361: 2244: 1992: 1699: 1646: 1494: 5431: 5356: 5087: 5053: 1093: 645:

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".

2318:(simultaneous measurement of magnetic and non-magnetic transitions. Limited up to about 800–1000 °C) 2206: 1453:

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".

1243: 5472: 1729:. However, note that order parameters can also be defined for non-symmetry-breaking transitions. 1405: 1397: 848: 1253: 5497: 5487: 5237: 5232: 4971:

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

4755: 1506: 1238: 1178: 1148: 1098: 818: 708: 546: 417: 85:. This can be a discontinuous change; for example, a liquid may become gas upon heating to its 3715: 3201: 4984: 2387: 2221:, and cooperative ligand binding to DNA and proteins with the character of phase transition. 2153: 1955: 1765: 1502: 1278: 893: 878: 642: 5416: 5176: 5007: 4947: 4800: 4717:

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

492: 296:. In exception to the usual case, it is sometimes possible to change the state of a system 278: 70: 2305:

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

1819:. The most important one is perhaps the exponent describing the divergence of the thermal 639:

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?".

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

456:. A simplified but highly useful model of magnetic phase transitions is provided by the 124:

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

5538: 5467: 5301: 4763: 4696: 4670: 4501: 4475: 4439: 4408: 4396: 4377: 4351: 4319: 4286: 4274: 4250: 4217: 4206:"Social interactions dominate speed control in poising natural flocks near criticality" 4205: 4184: 4087: 4044: 4018: 3991: 3965: 3938: 3904: 3877: 3785: 3742: 3696: 3662: 3635: 3593: 3498: 3447: 3421: 3394: 3360: 3321: 3264: 3213: 3179: 3126:

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."

3147: 3097: 3089: 3040: 3028: 2909: 2891: 2852: 2815: 2793: 2781: 2696: 2674: 2628: 2589: 2542:"Phase diagram for the transition from photonic crystals to dielectric metamaterials" 2520: 2427: 2353: 1980: 1816: 1803: 1761: 1733: 1682: 1666: 1662: 1581: 1466: 1428: 1393: 1218: 1063: 953: 873: 662: 636: 603: 504: 425: 363: 339: 74: 4145: 4091: 3995: 3942: 3881: 3789: 3762:

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

5513: 5366: 5103: 4921: 4781: 4777: 2942: 2482: 2210: 1233: 1183: 1053: 808: 720: 553: 539: 367: 125: 50: 4692: 4592: 2472: – Mathematical theory on behavior of connected clusters in a random graph 2445: – Theory of continuous phase transitions of second order phase transitions 2002:< 1, the heat capacity diverges at the transition temperature (though, since 1823:

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

5311: 5306: 5263: 5196: 5191: 4901: 4862: 4635: 4557: 4040: 3987: 3684: 3546: 3382: 3024: 2872:"The Ehrenfest Classification of Phase Transitions: Introduction and Evolution" 2381: 2237: 1948:. Its actual value depends on the type of phase transition we are considering. 1741: 1498: 1368: 1305: 1283: 1263: 1258: 1213: 1133: 1068: 966: 853: 698: 666: 585: 512: 429: 330: 293: 115: 4927: 4812: 4373: 2515:

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

1725:, whose direction was spontaneously chosen when the system cooled below the 1188: 128:

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

531: 484: 457: 445: 441: 78: 4826: 4791:(1974). "The renormalization group in the theory of critical behavior". 2871: 2848: 2575: 2123:{\displaystyle \beta =\gamma /(\delta -1),\quad \nu =\gamma /(2-\eta ).} 606:

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

316:, for example. Metastable states do not appear on usual phase diagrams. 25:

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

4397:"Zipf's law and criticality in multivariate data without fine-tuning" 3486: 2421: – Mathematical model of ferromagnetism in statistical mechanics 2366: 2019: 1854: 1490: 1193: 1143: 1016: 863: 763: 581: 523: 471:

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.

1485:. They are characterized by a divergent susceptibility, an infinite 21: 5558: 5386: 4107:"C NMR studies of lipid fatty acyl chains of chloroplast membranes" 3909: 2835:

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

4675: 4480: 4413: 4356: 4291: 4222: 4189: 3667: 3598: 3338: 3068: 2760: 5518: 5406: 5341: 5258: 5253: 2263: 2181:. Connected to the previous phenomenon is also the phenomenon of 1073: 1058: 1021: 1012: 1007: 496: 444:. Another example is the transition between differently ordered, 216: 30: 1623:

raises the possibility that one can study the interplay between

5127: 5048: 4938:(1974). "The Renormalization Group and the epsilon-Expansion". 4874: 4858:, (Physics Education (India) Volume 32. No. 2, Apr - Jun 2016) 4707: 2514: 1026: 1002: 733: 629: 476: 437: 194: 58: 1797: 1756:

Symmetry-breaking phase transitions play an important role in

564: 5136: 5122: 3053: 2690: 2497: – Field theory involving topological effects in physics 2341: 1031: 728: 617: 569: 189: 133: 107: 54: 16:

Physical process of transition between basic states of matter

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Chaos, Phase Transitions, Topology Change and Path Integrals

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Lectures on Phase Transitions and the Renormalization Group

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

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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

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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

4977:, 1996. Contains a detailed pedagogical discussion of 3833:

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

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

1497:. Examples of second-order phase transitions are the 5028:(Oxford University Press, Oxford and New York 1971). 3509: 3332: 3121: 3119: 2691:

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

2453:

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2217:, liquid crystal-like transitions in the process of 2189:

of a continuous phase transition split into smaller

1427:. The exact specific heat differed from the earlier 428:. The most well-known is the transition between the 4827:Ivancevic, Vladimir G; Ivancevic, Tijana T (2008), 4725:

Department of Peace Studies, University of Bradford

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Order-disorder transitions such as in alpha- 370:, or from one amorphous structure to another ( 5088: 4461: 4111:Indian Journal of Biochemistry and Biophysics 2629:"Fundamentals of Stable Isotope Geochemistry" 2623: 1339: 389:. As with states of matter, there are also a 162: 4613: 4468:Progress in Biophysics and Molecular Biology 4098: 3845:Ojovan, Michael I.; Lee, William E. (2006). 3747:: CS1 maint: multiple names: authors list ( 2648: 2646: 1705:An example of an order parameter is the net 1589: 1461:mixture of liquid water and vapor bubbles). 1384:are continuous in the first derivative (the 92: 5102: 4933: 4204:Bialek, W; Cavagna, A; Giardina, I (2014). 2539: 2275:points of second-order phase transitions. 2157:correlation length is the essential point. 1798:Critical exponents and universality classes 1363: 366:to another, from a crystalline solid to an 338:, distinguishing between several different 37:, and other related fields like biology, a 5095: 5081: 5036:Statistical Mechanics of Phase Transitions 4341: 3284:"Materials science: Metal turned to glass" 2956: 2686: 2684: 2373:Autocatalytic reactions and order creation 1764:into the U(1) symmetry of the present-day 1685:, which only occurs at low temperatures). 1380:the free energy with respect to pressure. 1346: 1332: 686: 169: 155: 5061:Interactive Phase Transitions on lattices 4760:Basic Notions of Condensed Matter Physics 4674: 4479: 4462:Longo, G.; Montévil, M. 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They are continuous but break no 1529:of second-order phase transitions. 1414:supercritical liquid–gas boundaries 97: 13: 4883:World Scientific (Singapore, 1989) 4749: 2401: – Thermoanalytical technique 2025:Several other critical exponents, 1661:Phase transitions often involve a 1640: 1358: 412:A phase diagram showing different 14: 5621: 5042: 5001:, Pergamon Press, 3rd Ed. (1994). 4907:Critical Properties of φ-Theories 2932:, Elsevier, Burlington, MA, 2006, 2800: 2540:Rybin, M.V.; et al. (2015). 2491: – Process in quantum optics 2399:Differential scanning calorimetry 1547:. 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Oxford: Clarendon Press. 2286: 1801: 899:Spin gapless semiconductor 622:Bose–Einstein condensation 401: 379:martensitic transformation 323: 113: 5506: 5460: 5332: 5246: 5220: 5164: 5153: 5115: 4813:10.1103/revmodphys.46.597 4374:10.1007/s10955-011-0229-4 1788:relational order theories 1590:Characteristic properties 1557:quantum phase transitions 1446:are those that involve a 1373:thermodynamic free energy 839:Electronic band structure 671:quantum phase transitions 659:thermodynamic free energy 651:equilibrium fractionation 507:, or they have different 93:Types of phase transition 5579:Thermo-dielectric effect 5478:Enthalpy of vaporization 5172:Bose–Einstein condensate 2979:10.1103/physrevb.19.3580 2458:List of states of matter 2331:tantalum hafnium carbide 2295:peace and armed conflict 1770:electroweak baryogenesis 1563:, belong to this class. 1423:, discovered in 1944 by 1364:Ehrenfest classification 749:Bose–Einstein condensate 680:Condensed matter physics 616:Quantum condensation of 5473:Enthalpy of sublimation 4401:Physical Review Letters 4310:10.1073/pnas.1324186111 4241:10.1073/pnas.1324045111 3781:10.1351/pac199466030577 3056:Physical Review Letters 2730:10.1103/PhysRevD.21.446 2207:coil-globule transition 1965:{\displaystyle \gamma } 1568:liquid–glass transition 1551:in the two-dimensional 1398:magnetic susceptibility 5488:Latent internal energy 5238:Color-glass condensate 4581:Human Movement Science 4064:J. 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A 489: 422: 404:Magnetic structure 352: 340:crystal structures 336:allotropes of iron 146: 112: 27: 5600:Phase transitions 5587: 5586: 5569:Superheated vapor 5564:Superconductivity 5534:Equation of state 5382:Flash evaporation 5334:Phase transitions 5319:String-net liquid 5212:Photonic molecule 5182:Degenerate matter 5063:with Java applets 5052:Media related to 4879:Phase Transitions 4840:978-3-540-79356-4 4669:(16): 3193–3217. 4616:Physical Review E 4530:978-0-262-61131-2 4285:(10): 3683–3688. 4216:(20): 7212–7217. 4011:Physical Review D 3958:Physical Review B 3818:978-0-674-00342-2 3727:978-3-540-79357-1 3655:Physical Review B 3527:Physical Review B 3353:Physical Review B 3294:(7155): 758–759. 2995:Physical Review B 2843:(11): 1225–1235. 2821:978-0-521-09101-5 2718:Physical Review D 2702:978-0-19-856770-7 2526:978-0-412-53910-7 2428:Jamming (physics) 2354:X-ray diffraction 2171:critical dynamics 1981:lambda transition 1885: 1804:critical exponent 1762:electroweak field 1683:virtual particles 1667:crystalline solid 1663:symmetry breaking 1595:Phase coexistence 1582:quenched disorder 1467:quenched disorder 1394:Curie temperature 1356: 1355: 1064:Granular material 832:Electronic phases 609:The emergence of 604:superconductivity 602:The emergence of 505:congruent melting 426:magnetic ordering 364:crystal structure 286: 285: 148:Phase transitions 5617: 5524:Compressed fluid 5159: 5104:States of matter 5097: 5090: 5083: 5074: 5073: 5051: 4963: 4925: 4920:. Archived from 4850: 4849: 4847: 4819:Goldenfeld, N., 4816: 4744: 4743: 4741: 4722: 4711: 4705: 4704: 4678: 4654: 4648: 4647: 4611: 4605: 4604: 4576: 4570: 4569: 4541: 4535: 4534: 4516: 4510: 4509: 4483: 4459: 4453: 4452: 4442: 4416: 4392: 4386: 4385: 4359: 4339: 4333: 4332: 4322: 4312: 4294: 4270: 4264: 4263: 4253: 4243: 4225: 4201: 4195: 4194: 4192: 4180: 4174: 4173: 4141: 4135: 4134: 4102: 4096: 4095: 4059: 4053: 4052: 4026: 4006: 4000: 3999: 3973: 3971:cond-mat/0310163 3953: 3947: 3946: 3912: 3892: 3886: 3885: 3851: 3842: 3836: 3829: 3823: 3822: 3810: 3807:Cosmic Evolution 3800: 3794: 3793: 3783: 3759: 3753: 3752: 3746: 3738: 3736: 3734: 3711: 3705: 3704: 3670: 3650: 3644: 3643: 3601: 3565: 3559: 3558: 3513: 3507: 3506: 3487:10.1038/nmat1743 3467:Nature Materials 3462: 3456: 3455: 3429: 3427:cond-mat/0611152 3409: 3403: 3402: 3368: 3366:cond-mat/0012472 3336: 3330: 3329: 3311: 3279: 3273: 3272: 3228: 3222: 3221: 3187: 3185:cond-mat/0607349 3167: 3161: 3158: 3152: 3151: 3123: 3114: 3113: 3071: 3051: 3045: 3044: 3010: 3008:cond-mat/0602627 2989: 2983: 2982: 2965:(7): 3580–3585. 2954: 2948: 2939: 2933: 2924: 2918: 2917: 2906: 2900: 2899: 2867: 2861: 2860: 2832: 2826: 2825: 2807: 2798: 2797: 2763: 2739: 2733: 2732: 2713: 2707: 2706: 2688: 2679: 2678: 2650: 2641: 2640: 2638: 2636: 2621: 2615: 2613:, Elsevier, 2019 2604: 2598: 2597: 2587: 2561: 2537: 2531: 2530: 2512: 2454: 2433: 2424: 2410: 2393: 2378: 2348:Thermogravimetry 2219:DNA condensation 2167:static functions 2129: 2127: 2126: 2121: 2101: 2068: 1971: 1969: 1968: 1963: 1921: 1919: 1918: 1913: 1908: 1907: 1899: 1887: 1886: 1883: 1877: 1853:typically has a 1689:Order parameters 1524:phenomenological 1481:are also called 1478: 1477: 1348: 1341: 1334: 1321: 1316: 1315: 1308: 1304: 1303: 924:Spin Hall effect 814:Phase transition 784:Luttinger liquid 721:States of matter 704:Phase transition 690: 676: 675: 580:Transition to a 178: 171: 164: 157: 145: 126:states of matter 98:States of matter 51:states of matter 47:physical process 39:phase transition 5625: 5624: 5620: 5619: 5618: 5616: 5615: 5614: 5590: 5589: 5588: 5583: 5514:Baryonic matter 5502: 5456: 5427:Saturated fluid 5367:Crystallization 5328: 5302:Antiferromagnet 5242: 5216: 5160: 5151: 5111: 5101: 5045: 4928:readable online 4918: 4902:Kleinert, Hagen 4845: 4843: 4841: 4782:Springer Nature 4752: 4750:Further reading 4747: 4739: 4720: 4712: 4708: 4655: 4651: 4612: 4608: 4577: 4573: 4542: 4538: 4531: 4517: 4513: 4460: 4456: 4393: 4389: 4340: 4336: 4271: 4267: 4202: 4198: 4181: 4177: 4142: 4138: 4103: 4099: 4060: 4056: 4007: 4003: 3954: 3950: 3897:Phys. Rev. Lett 3893: 3889: 3849: 3843: 3839: 3830: 3826: 3819: 3801: 3797: 3760: 3756: 3740: 3739: 3732: 3730: 3728: 3712: 3708: 3651: 3647: 3583: 3579: 3575: 3571: 3566: 3562: 3524: 3520: 3514: 3510: 3473:(11): 881–886. 3463: 3459: 3410: 3406: 3350: 3346: 3342: 3337: 3333: 3309:10.1038/448758a 3280: 3276: 3229: 3225: 3168: 3164: 3159: 3155: 3124: 3117: 3052: 3048: 2990: 2986: 2955: 2951: 2940: 2936: 2925: 2921: 2907: 2903: 2868: 2864: 2833: 2829: 2822: 2808: 2801: 2740: 2736: 2714: 2710: 2703: 2689: 2682: 2651: 2644: 2634: 2632: 2622: 2618: 2605: 2601: 2538: 2534: 2527: 2513: 2509: 2505: 2500: 2483:Superfluid film 2452: 2431: 2422: 2408: 2391: 2376: 2362: 2303: 2291: 2285: 2256:neural networks 2211:protein folding 2205:formation, the 2199: 2163: 2097: 2064: 2053: 2050: 2049: 1957: 1954: 1953: 1900: 1895: 1894: 1882: 1878: 1873: 1865: 1862: 1861: 1848: 1837: 1806: 1800: 1754: 1734:superconducting 1695:order parameter 1691: 1659: 1643: 1641:Critical points 1636: 1629: 1621: 1613: 1606: 1597: 1592: 1518:measurements. 1475: 1474: 1438: 1406:Cornelis Gorter 1386:order parameter 1366: 1361: 1359:Classifications 1352: 1311: 1298: 1297: 1290: 1289: 1288: 1088: 1080: 1079: 1078: 1054:Amorphous solid 1048: 1038: 1037: 1036: 1015: 997: 987: 986: 985: 974: 972:Antiferromagnet 965: 963:Superparamagnet 956: 943: 942:Magnetic phases 935: 934: 933: 913: 905: 904: 903: 833: 825: 824: 823: 809:Order parameter 803: 802:Phase phenomena 795: 794: 793: 723: 713: 661:of a system is 562: 554:supersaturation 540:miscibility gap 513:solid solutions 465: 406: 400: 368:amorphous solid 328: 322: 300:(as opposed to 186: 183: 175: 122: 100: 95: 69:. A phase of a 17: 12: 11: 5: 5623: 5613: 5612: 5607: 5602: 5585: 5584: 5582: 5581: 5576: 5571: 5566: 5561: 5556: 5551: 5546: 5541: 5536: 5531: 5526: 5521: 5516: 5510: 5508: 5504: 5503: 5501: 5500: 5495: 5493:Trouton's rule 5490: 5485: 5480: 5475: 5470: 5464: 5462: 5458: 5457: 5455: 5454: 5449: 5444: 5439: 5434: 5429: 5424: 5419: 5414: 5409: 5404: 5399: 5394: 5389: 5384: 5379: 5374: 5369: 5364: 5362:Critical point 5359: 5354: 5349: 5344: 5338: 5336: 5330: 5329: 5327: 5326: 5321: 5316: 5315: 5314: 5309: 5304: 5296: 5291: 5286: 5281: 5276: 5271: 5266: 5264:Liquid crystal 5261: 5256: 5250: 5248: 5244: 5243: 5241: 5240: 5235: 5230: 5224: 5222: 5218: 5217: 5215: 5214: 5209: 5204: 5199: 5197:Strange matter 5194: 5192:Rydberg matter 5189: 5184: 5179: 5174: 5168: 5166: 5162: 5161: 5154: 5152: 5150: 5149: 5144: 5139: 5130: 5125: 5119: 5117: 5113: 5112: 5100: 5099: 5092: 5085: 5077: 5071: 5070: 5069:from Sklogwiki 5064: 5044: 5043:External links 5041: 5040: 5039: 5029: 5019: 5005: 5002: 4989:Lifshitz, E.M. 4982: 4964: 4931: 4916: 4898: 4860: 4851: 4839: 4824: 4817: 4799:(4): 597–616. 4793:Rev. Mod. Phys 4785: 4767: 4756:Anderson, P.W. 4751: 4748: 4746: 4745: 4706: 4649: 4606: 4587:(4): 483–493. 4571: 4552:(1): 183–202. 4536: 4529: 4511: 4474:(2): 340–347. 4454: 4387: 4350:(2): 268–302. 4334: 4265: 4196: 4175: 4156:(4): 353–356. 4136: 4117:(6): 177–178. 4097: 4070:(5): 903–911. 4054: 4024:hep-th/9812197 4001: 3964:(17): 174518. 3948: 3903:(20): 200601. 3887: 3837: 3831:David Layzer, 3824: 3817: 3795: 3774:(3): 577–594. 3754: 3726: 3706: 3661:(17): 174413. 3645: 3592:(38): 386004. 3581: 3577: 3573: 3569: 3560: 3522: 3518: 3508: 3457: 3404: 3359:(10): 104416. 3348: 3344: 3340: 3331: 3274: 3223: 3162: 3153: 3115: 3062:(24): 247003. 3046: 3001:(18): 184435. 2984: 2949: 2934: 2919: 2901: 2862: 2827: 2820: 2799: 2734: 2724:(2): 446–453, 2708: 2701: 2680: 2642: 2616: 2599: 2532: 2525: 2506: 2504: 2501: 2499: 2498: 2492: 2486: 2480: 2479: 2478: 2467: 2461: 2455: 2446: 2440: 2434: 2425: 2416: 2411: 2402: 2396: 2395: 2394: 2382:Crystal growth 2379: 2370: 2363: 2361: 2358: 2357: 2356: 2351: 2345: 2339: 2334: 2333:4215 °C.) 2324: 2319: 2313: 2302: 2299: 2284: 2281: 2238:linolenic acid 2198: 2195: 2169:there is also 2162: 2159: 2131: 2130: 2119: 2116: 2113: 2110: 2107: 2104: 2100: 2096: 2093: 2090: 2086: 2083: 2080: 2077: 2074: 2071: 2067: 2063: 2060: 2057: 1961: 1923: 1922: 1911: 1906: 1903: 1898: 1893: 1890: 1881: 1876: 1872: 1869: 1846: 1835: 1802:Main article: 1799: 1796: 1753: 1750: 1700:susceptibility 1690: 1687: 1658: 1655: 1647:critical point 1642: 1639: 1634: 1627: 1619: 1611: 1604: 1596: 1593: 1591: 1588: 1457:, but forms a 1437: 1434: 1369:Paul Ehrenfest 1365: 1362: 1360: 1357: 1354: 1353: 1351: 1350: 1343: 1336: 1328: 1325: 1324: 1323: 1322: 1309: 1292: 1291: 1287: 1286: 1281: 1276: 1271: 1266: 1261: 1256: 1251: 1246: 1241: 1236: 1231: 1226: 1221: 1216: 1211: 1206: 1201: 1196: 1191: 1186: 1181: 1176: 1171: 1166: 1161: 1156: 1151: 1146: 1141: 1136: 1131: 1126: 1121: 1116: 1111: 1106: 1101: 1096: 1090: 1089: 1086: 1085: 1082: 1081: 1077: 1076: 1071: 1069:Liquid crystal 1066: 1061: 1056: 1050: 1049: 1044: 1043: 1040: 1039: 1035: 1034: 1029: 1024: 1019: 1010: 1005: 999: 998: 995:Quasiparticles 993: 992: 989: 988: 984: 983: 978: 969: 960: 954:Superdiamagnet 951: 945: 944: 941: 940: 937: 936: 932: 931: 926: 921: 915: 914: 911: 910: 907: 906: 902: 901: 896: 891: 886: 881: 879:Thermoelectric 876: 874:Superconductor 871: 866: 861: 856: 854:Mott insulator 851: 846: 841: 835: 834: 831: 830: 827: 826: 822: 821: 816: 811: 805: 804: 801: 800: 797: 796: 792: 791: 786: 781: 776: 771: 766: 761: 756: 751: 746: 741: 736: 731: 725: 724: 719: 718: 715: 714: 712: 711: 706: 701: 695: 692: 691: 683: 682: 655: 654: 649:condenses (an 640: 633: 614: 607: 600: 599:on iron (110). 589: 586:liquid crystal 561: 560:Other examples 558: 464: 461: 450:incommensurate 399: 396: 321: 318: 294:freezing point 284: 283: 281: 276: 274: 272: 268: 267: 262: 260: 255: 250: 246: 245: 243: 238: 236: 231: 227: 226: 224: 219: 214: 212: 208: 207: 202: 197: 192: 187: 184: 181: 174: 173: 166: 159: 151: 116:vapor pressure 99: 96: 94: 91: 15: 9: 6: 4: 3: 2: 5622: 5611: 5608: 5606: 5603: 5601: 5598: 5597: 5595: 5580: 5577: 5575: 5572: 5570: 5567: 5565: 5562: 5560: 5557: 5555: 5552: 5550: 5549:Mpemba effect 5547: 5545: 5542: 5540: 5537: 5535: 5532: 5530: 5529:Cooling curve 5527: 5525: 5522: 5520: 5517: 5515: 5512: 5511: 5509: 5505: 5499: 5496: 5494: 5491: 5489: 5486: 5484: 5481: 5479: 5476: 5474: 5471: 5469: 5466: 5465: 5463: 5459: 5453: 5452:Vitrification 5450: 5448: 5445: 5443: 5440: 5438: 5435: 5433: 5430: 5428: 5425: 5423: 5420: 5418: 5417:Recombination 5415: 5413: 5412:Melting point 5410: 5408: 5405: 5403: 5400: 5398: 5395: 5393: 5390: 5388: 5385: 5383: 5380: 5378: 5375: 5373: 5370: 5368: 5365: 5363: 5360: 5358: 5357:Critical line 5355: 5353: 5350: 5348: 5347:Boiling point 5345: 5343: 5340: 5339: 5337: 5335: 5331: 5325: 5322: 5320: 5317: 5313: 5310: 5308: 5305: 5303: 5300: 5299: 5297: 5295: 5292: 5290: 5287: 5285: 5282: 5280: 5279:Exotic matter 5277: 5275: 5272: 5270: 5267: 5265: 5262: 5260: 5257: 5255: 5252: 5251: 5249: 5245: 5239: 5236: 5234: 5231: 5229: 5226: 5225: 5223: 5219: 5213: 5210: 5208: 5205: 5203: 5200: 5198: 5195: 5193: 5190: 5188: 5185: 5183: 5180: 5178: 5175: 5173: 5170: 5169: 5167: 5163: 5158: 5148: 5145: 5143: 5140: 5138: 5134: 5131: 5129: 5126: 5124: 5121: 5120: 5118: 5114: 5109: 5105: 5098: 5093: 5091: 5086: 5084: 5079: 5078: 5075: 5068: 5065: 5062: 5059: 5058: 5057: 5055: 5054:Phase changes 5050: 5037: 5033: 5032:Yeomans J. M. 5030: 5027: 5023: 5022:H. E. Stanley 5020: 5017: 5016:W. H. Freeman 5013: 5009: 5006: 5003: 5000: 4999: 4994: 4990: 4986: 4983: 4980: 4976: 4972: 4968: 4965: 4961: 4957: 4953: 4949: 4946:(2): 75–199. 4945: 4941: 4937: 4932: 4929: 4923: 4919: 4917:981-02-4659-5 4913: 4909: 4908: 4903: 4899: 4896: 4892: 4891:9971-5-0210-0 4888: 4884: 4880: 4876: 4872: 4871:Superfluidity 4868: 4864: 4861: 4859: 4857: 4852: 4842: 4836: 4832: 4831: 4825: 4822: 4818: 4814: 4810: 4806: 4802: 4798: 4794: 4790: 4786: 4783: 4779: 4775: 4771: 4768: 4765: 4761: 4757: 4754: 4753: 4738: 4734: 4730: 4726: 4719: 4718: 4710: 4702: 4698: 4694: 4690: 4686: 4682: 4677: 4672: 4668: 4664: 4660: 4653: 4645: 4641: 4637: 4633: 4629: 4625: 4622:(1): 011920. 4621: 4617: 4610: 4602: 4598: 4594: 4590: 4586: 4582: 4575: 4567: 4563: 4559: 4555: 4551: 4547: 4540: 4532: 4526: 4523:. MIT Press. 4522: 4515: 4507: 4503: 4499: 4495: 4491: 4487: 4482: 4477: 4473: 4469: 4465: 4458: 4450: 4446: 4441: 4436: 4432: 4428: 4424: 4420: 4415: 4410: 4407:(6): 068102. 4406: 4402: 4398: 4391: 4383: 4379: 4375: 4371: 4367: 4363: 4358: 4353: 4349: 4345: 4338: 4330: 4326: 4321: 4316: 4311: 4306: 4302: 4298: 4293: 4288: 4284: 4280: 4276: 4269: 4261: 4257: 4252: 4247: 4242: 4237: 4233: 4229: 4224: 4219: 4215: 4211: 4207: 4200: 4191: 4186: 4179: 4171: 4167: 4163: 4159: 4155: 4151: 4147: 4140: 4132: 4128: 4124: 4120: 4116: 4112: 4108: 4101: 4093: 4089: 4085: 4081: 4077: 4073: 4069: 4065: 4058: 4050: 4046: 4042: 4038: 4034: 4030: 4025: 4020: 4017:(8): 085001. 4016: 4012: 4005: 3997: 3993: 3989: 3985: 3981: 3977: 3972: 3967: 3963: 3959: 3952: 3944: 3940: 3936: 3932: 3928: 3924: 3920: 3916: 3911: 3906: 3902: 3898: 3891: 3883: 3879: 3875: 3871: 3867: 3863: 3859: 3855: 3848: 3841: 3834: 3828: 3820: 3814: 3809: 3808: 3799: 3791: 3787: 3782: 3777: 3773: 3769: 3765: 3758: 3750: 3744: 3729: 3723: 3719: 3718: 3710: 3702: 3698: 3694: 3690: 3686: 3682: 3678: 3674: 3669: 3664: 3660: 3656: 3649: 3641: 3637: 3633: 3629: 3625: 3621: 3617: 3613: 3609: 3605: 3600: 3595: 3591: 3587: 3564: 3556: 3552: 3548: 3544: 3540: 3536: 3533:(1): 012403. 3532: 3528: 3512: 3504: 3500: 3496: 3492: 3488: 3484: 3480: 3476: 3472: 3468: 3461: 3453: 3449: 3445: 3441: 3437: 3433: 3428: 3423: 3419: 3415: 3408: 3400: 3396: 3392: 3388: 3384: 3380: 3376: 3372: 3367: 3362: 3358: 3354: 3335: 3327: 3323: 3319: 3315: 3310: 3305: 3301: 3297: 3293: 3289: 3285: 3278: 3270: 3266: 3262: 3258: 3254: 3250: 3246: 3242: 3238: 3234: 3227: 3219: 3215: 3211: 3207: 3203: 3199: 3195: 3191: 3186: 3181: 3177: 3173: 3166: 3157: 3149: 3145: 3141: 3137: 3133: 3129: 3122: 3120: 3111: 3107: 3103: 3099: 3095: 3091: 3087: 3083: 3079: 3075: 3070: 3065: 3061: 3057: 3050: 3042: 3038: 3034: 3030: 3026: 3022: 3018: 3014: 3009: 3004: 3000: 2996: 2988: 2980: 2976: 2972: 2968: 2964: 2960: 2953: 2946: 2945: 2938: 2931: 2930: 2923: 2915: 2911: 2905: 2897: 2893: 2889: 2885: 2881: 2877: 2873: 2866: 2858: 2854: 2850: 2846: 2842: 2838: 2831: 2823: 2817: 2813: 2806: 2804: 2795: 2791: 2787: 2783: 2779: 2775: 2771: 2767: 2762: 2757: 2754:(1): P01012. 2753: 2749: 2745: 2738: 2731: 2727: 2723: 2719: 2712: 2704: 2698: 2694: 2687: 2685: 2676: 2672: 2668: 2664: 2660: 2656: 2649: 2647: 2630: 2626: 2625:Carol Kendall 2620: 2614: 2611: 2610: 2603: 2595: 2591: 2586: 2581: 2577: 2573: 2569: 2565: 2560: 2555: 2551: 2547: 2543: 2536: 2528: 2522: 2518: 2511: 2507: 2496: 2493: 2490: 2487: 2484: 2481: 2477: 2474: 2473: 2471: 2468: 2465: 2462: 2459: 2456: 2450: 2447: 2444: 2443:Landau theory 2441: 2438: 2435: 2429: 2426: 2420: 2417: 2415: 2412: 2406: 2403: 2400: 2397: 2389: 2386: 2385: 2383: 2380: 2374: 2371: 2368: 2365: 2364: 2355: 2352: 2350:(very common) 2349: 2346: 2343: 2340: 2338: 2335: 2332: 2328: 2325: 2323: 2320: 2317: 2314: 2311: 2308: 2307: 2306: 2298: 2296: 2290: 2280: 2276: 2273: 2269: 2265: 2260: 2257: 2252: 2250: 2246: 2243: 2239: 2234: 2231: 2227: 2222: 2220: 2216: 2212: 2208: 2204: 2203:lipid bilayer 2194: 2192: 2188: 2184: 2180: 2176: 2172: 2168: 2158: 2155: 2149: 2147: 2142: 2140: 2136: 2117: 2111: 2108: 2105: 2098: 2094: 2091: 2088: 2084: 2078: 2075: 2072: 2065: 2061: 2058: 2055: 2048: 2047: 2046: 2044: 2040: 2036: 2032: 2028: 2023: 2021: 2015: 2013: 2009: 2005: 2001: 1996: 1994: 1990: 1986: 1982: 1978: 1973: 1959: 1949: 1947: 1943: 1940:The exponent 1938: 1936: 1932: 1928: 1909: 1904: 1901: 1891: 1888: 1879: 1870: 1867: 1860: 1859: 1858: 1856: 1852: 1845: 1841: 1834: 1830: 1826: 1825:heat capacity 1822: 1818: 1813: 1811: 1805: 1795: 1793: 1789: 1784: 1782: 1778: 1777:Eric Chaisson 1773: 1771: 1767: 1763: 1759: 1749: 1747: 1743: 1738: 1735: 1730: 1728: 1724: 1723:magnetization 1720: 1719:ferromagnetic 1714: 1712: 1711:ferromagnetic 1708: 1707:magnetization 1703: 1701: 1696: 1686: 1684: 1680: 1676: 1672: 1668: 1664: 1654: 1652: 1648: 1638: 1633: 1626: 1618: 1610: 1603: 1587: 1584: 1583: 1577: 1573: 1569: 1564: 1562: 1558: 1554: 1550: 1546: 1542: 1537: 1535: 1530: 1528: 1525: 1521: 1517: 1512: 1508: 1504: 1500: 1499:ferromagnetic 1496: 1492: 1488: 1484: 1480: 1471: 1468: 1464: 1460: 1456: 1451: 1449: 1445: 1441: 1433: 1430: 1426: 1422: 1417: 1415: 1409: 1407: 1401: 1399: 1395: 1391: 1390:magnetization 1387: 1383: 1378: 1374: 1370: 1349: 1344: 1342: 1337: 1335: 1330: 1329: 1327: 1326: 1320: 1310: 1307: 1302: 1296: 1295: 1294: 1293: 1285: 1282: 1280: 1277: 1275: 1272: 1270: 1267: 1265: 1262: 1260: 1257: 1255: 1252: 1250: 1247: 1245: 1242: 1240: 1237: 1235: 1232: 1230: 1227: 1225: 1222: 1220: 1217: 1215: 1212: 1210: 1207: 1205: 1202: 1200: 1197: 1195: 1192: 1190: 1187: 1185: 1182: 1180: 1177: 1175: 1172: 1170: 1167: 1165: 1162: 1160: 1157: 1155: 1152: 1150: 1147: 1145: 1142: 1140: 1137: 1135: 1132: 1130: 1127: 1125: 1122: 1120: 1117: 1115: 1112: 1110: 1107: 1105: 1102: 1100: 1097: 1095: 1094:Van der Waals 1092: 1091: 1084: 1083: 1075: 1072: 1070: 1067: 1065: 1062: 1060: 1057: 1055: 1052: 1051: 1047: 1042: 1041: 1033: 1030: 1028: 1025: 1023: 1020: 1018: 1014: 1011: 1009: 1006: 1004: 1001: 1000: 996: 991: 990: 982: 979: 977: 973: 970: 968: 964: 961: 959: 955: 952: 950: 947: 946: 939: 938: 930: 927: 925: 922: 920: 917: 916: 909: 908: 900: 897: 895: 892: 890: 889:Ferroelectric 887: 885: 884:Piezoelectric 882: 880: 877: 875: 872: 870: 867: 865: 862: 860: 859:Semiconductor 857: 855: 852: 850: 847: 845: 842: 840: 837: 836: 829: 828: 820: 817: 815: 812: 810: 807: 806: 799: 798: 790: 787: 785: 782: 780: 779:Superfluidity 777: 775: 772: 770: 767: 765: 762: 760: 757: 755: 752: 750: 747: 745: 742: 740: 737: 735: 732: 730: 727: 726: 722: 717: 716: 710: 707: 705: 702: 700: 697: 696: 694: 693: 689: 685: 684: 681: 678: 677: 674: 672: 668: 664: 660: 652: 648: 644: 641: 638: 634: 631: 627: 623: 619: 615: 612: 608: 605: 601: 598: 594: 590: 587: 583: 579: 578: 577: 571: 566: 557: 555: 550: 548: 543: 541: 537: 533: 529: 525: 521: 516: 514: 510: 506: 502: 501:melting point 498: 494: 486: 482: 481:mixing ratios 479:at different 478: 474: 469: 460: 459: 455: 451: 447: 443: 439: 435: 431: 430:ferromagnetic 427: 419: 415: 410: 405: 395: 392: 388: 384: 380: 375: 373: 369: 365: 361: 357: 349: 346:(α-iron) and 345: 341: 337: 332: 327: 317: 315: 311: 307: 303: 302:adiabatically 299: 295: 291: 290:phase diagram 282: 280: 279:Recombination 277: 270: 269: 266: 263: 261: 259: 256: 254: 251: 248: 247: 242: 239: 237: 235: 232: 229: 228: 223: 220: 218: 215: 213: 210: 209: 206: 203: 201: 198: 196: 193: 191: 188: 180: 179: 172: 167: 165: 160: 158: 153: 152: 149: 144: 142: 137: 135: 131: 130:boiling point 127: 121: 120:phase diagram 117: 109: 104: 90: 88: 87:boiling point 84: 80: 76: 72: 68: 64: 60: 56: 52: 48: 44: 40: 36: 32: 23: 19: 5574:Superheating 5447:Vaporization 5442:Triple point 5437:Supercooling 5402:Lambda point 5352:Condensation 5269:Time crystal 5247:Other states 5187:Quantum Hall 5046: 5035: 5025: 5014:, New York: 5011: 4996: 4995:, vol. 5 of 4992: 4985:Landau, L.D. 4970: 4943: 4939: 4922:the original 4906: 4885:; Paperback 4875:Vortex lines 4866: 4863:Kleinert, H. 4855: 4844:, retrieved 4829: 4820: 4796: 4792: 4789:Fisher, M.E. 4759: 4716: 4709: 4666: 4662: 4652: 4619: 4615: 4609: 4584: 4580: 4574: 4549: 4545: 4539: 4520: 4514: 4471: 4467: 4457: 4404: 4400: 4390: 4347: 4343: 4337: 4282: 4278: 4268: 4213: 4209: 4199: 4178: 4153: 4149: 4139: 4114: 4110: 4100: 4067: 4063: 4057: 4014: 4010: 4004: 3961: 3957: 3951: 3900: 3896: 3890: 3857: 3853: 3840: 3832: 3827: 3806: 3798: 3771: 3767: 3757: 3731:. Retrieved 3716: 3709: 3658: 3654: 3648: 3589: 3585: 3563: 3530: 3526: 3511: 3470: 3466: 3460: 3420:(49): L605. 3417: 3413: 3407: 3356: 3352: 3334: 3291: 3287: 3277: 3236: 3232: 3226: 3175: 3171: 3165: 3156: 3131: 3127: 3059: 3055: 3049: 2998: 2994: 2987: 2962: 2959:Phys. Rev. B 2958: 2952: 2943: 2937: 2928: 2922: 2913: 2904: 2882:(1): 51–81. 2879: 2875: 2865: 2840: 2836: 2830: 2811: 2751: 2747: 2737: 2721: 2717: 2711: 2692: 2661:(1): 51–81. 2658: 2654: 2633:. 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