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75-475: A discrete breather is a breather solution on a nonlinear lattice . The term breather originates from the characteristic that most breathers are localized in space and oscillate ( breathe ) in time. But also the opposite situation: oscillations in space and localized in time, is denoted as a breather. A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon equation and

150-465: A continuum , such as the continuum of space or spacetime . Lattice models originally occurred in the context of condensed matter physics , where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics , for many reasons. Some models are exactly solvable , and thus offer insight into physics beyond what can be learned from perturbation theory . Lattice models are also ideal for study by

225-509: A coset . For example, for the Potts model we have S = Z n {\displaystyle S=\mathbb {Z} _{n}} . In the limit n → ∞ {\displaystyle n\rightarrow \infty } , we obtain the XY model which has S = S O ( 2 ) {\displaystyle S=SO(2)} . Generalising the XY model to higher dimensions gives

300-399: A latent heat . During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or

375-437: 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 the free energy with respect to pressure. Second-order phase transitions are continuous in the first derivative (the order parameter , which

450-479: A divergent susceptibility, an infinite correlation length , and a power law decay of correlations near criticality . Examples of second-order phase transitions are the ferromagnetic transition, superconducting transition (for a Type-I superconductor the phase transition is second-order at zero external field and for a Type-II superconductor the phase transition is second-order for both normal-state–mixed-state and mixed-state–superconducting-state transitions) and

525-415: A finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a logarithmic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior. Several other critical exponents, β , γ , δ , ν , and η , are defined, examining the power law behavior of a measurable physical quantity near

600-550: A liquid phase. A peritectoid reaction is a peritectoid reaction, except involving only solid phases. A monotectic reaction consists of change from a liquid and to a combination of a solid and a second liquid, where the two liquids display a miscibility gap . Separation into multiple phases can occur via spinodal decomposition , in which a single phase is cooled and separates into two different compositions. Non-equilibrium mixtures can occur, such as in supersaturation . Other phase changes include: Phase transitions occur when

675-464: A liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times. No direct experimental evidence supports the existence of these transitions. 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

750-454: A logarithmic divergence at the critical temperature. In the following decades, the Ehrenfest classification was replaced by a simplified classification scheme that is able to incorporate such transitions. In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes: First-order phase transitions are those that involve

825-403: A number of phase transitions involving three phases: a eutectic transformation, in which a two-component single-phase liquid is cooled and transforms into two solid phases. The same process, but beginning with a solid instead of a liquid is called a eutectoid transformation. A peritectic transformation, in which a two-component single-phase solid is heated and transforms into a solid phase and

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900-430: A result of the change of external conditions, such as temperature or pressure . This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point , resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Phase transitions commonly refer to when a substance transforms between one of

975-451: A single compound. While chemically pure compounds exhibit a single temperature melting point between solid and liquid phases, mixtures can either have a single melting point, known as congruent melting , or they have different liquidus and solidus temperatures resulting in a temperature span where solid and liquid coexist in equilibrium. This is often the case in solid solutions , where the two components are isostructural. There are also

1050-456: A spatially varying mean field ⟨ σ ⟩ : R d → ⟨ C ⟩ {\displaystyle \langle \sigma \rangle :\mathbb {R} ^{d}\rightarrow \langle {\mathcal {C}}\rangle } . We relabel ⟨ σ ⟩ {\displaystyle \langle \sigma \rangle } with ϕ {\displaystyle \phi } to bring

1125-482: A system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and are insensitive to the underlying microscopic properties of the system. Again, the divergence of the correlation length is the essential point. There are also other critical phenomena; e.g., besides static functions there is also critical dynamics . As a consequence, at a phase transition one may observe critical slowing down or speeding up . Connected to

1200-425: Is metastable , i.e., less stable than the phase to which the transition would have occurred, but not unstable either. This occurs in superheating and supercooling , for example. Metastable states do not appear on usual phase diagrams. Phase transitions can also occur when a solid changes to a different structure without changing its chemical makeup. In elements, this is known as allotropy , whereas in compounds it

1275-425: 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 susceptibility will usually diverge. An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions,

1350-399: Is almost non-existent. This is associated with the phenomenon of critical opalescence , a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light). Phase transitions often involve a symmetry breaking process. For instance, the cooling of a fluid into a crystalline solid breaks continuous translation symmetry : each point in

1425-410: Is also a metastable 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. Phase transitions can also describe the change between different kinds of magnetic ordering . The most well-known is the transition between

1500-461: Is also a third-order transition, as shown by their specific heat having a sudden change in slope. 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. Cornelis Gorter replied

1575-502: Is known as polymorphism . The change from one crystal structure to another, from a crystalline solid to an amorphous solid , or from one amorphous structure to another ( polyamorphs ) are all examples of solid to solid phase transitions. The martensitic transformation occurs as one of the many phase transformations in carbon steel and stands as a model for displacive phase transformations . Order-disorder transitions such as in alpha- titanium aluminides . As with states of matter, there

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1650-451: Is often difficult due to non-linear interactions between sites. Models with a closed-form expression for the partition function are known as exactly solvable . Examples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, H = 0 , {\displaystyle H=0,} but for dimension d > 2 {\displaystyle d>2} ,

1725-407: Is perhaps the exponent describing the divergence of the thermal correlation length by approaching the transition. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature T of the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperature T c . When T is near T c ,

1800-408: Is periodic in time t and decays exponentially when moving away from x = 0 . The focusing nonlinear Schrödinger equation is the dispersive partial differential equation: with u a complex field as a function of x and t . Further i denotes the imaginary unit . One of the breather solutions (Kuznetsov-Ma breather) is with which gives breathers periodic in space x and approaching

1875-559: Is the QCD lattice model , a discretization of quantum chromodynamics . However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information , aka Holographic principle . More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers. A number of lattice models can be described by

1950-408: Is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter : solid , liquid , and gas , and in rare cases, plasma . A phase of a thermodynamic system and the states of matter have uniform physical properties . During a phase transition of a given medium, certain properties of the medium change as

2025-449: Is the behavior of liquid helium at the lambda transition from a normal state to the superfluid state, for which experiments have found α = −0.013 ± 0.003. At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample. This experimental value of α agrees with theoretical predictions based on variational perturbation theory . For 0 < α < 1,

2100-406: Is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context). The spin-variable space is S = { + 1 , − 1 } = Z 2 {\displaystyle S=\{+1,-1\}=\mathbb {Z} _{2}} . The energy functional is The spin-variable space can often be described as

2175-426: 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 magnetization , which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as

2250-508: The n {\displaystyle n} -vector model which has S = S n = S O ( n + 1 ) / S O ( n ) {\displaystyle S=S^{n}=SO(n+1)/SO(n)} . We specialise to a lattice with a finite number of points, and a finite spin-variable space. This can be achieved by making the lattice periodic, with period n {\displaystyle n} in d {\displaystyle d} dimensions. Then

2325-412: The convex hull of the spin space S {\displaystyle S} , when S {\displaystyle S} has a realisation in terms of a subset of R m {\displaystyle \mathbb {R} ^{m}} . We'll denote this by ⟨ C ⟩ {\displaystyle \langle {\mathcal {C}}\rangle } . This arises as in going to

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2400-583: The ferromagnetic and paramagnetic phases of magnetic materials, which occurs at what is called the Curie point . Another example is the transition between differently ordered, commensurate or incommensurate , magnetic structures, such as in cerium antimonide . A simplified but highly useful model of magnetic phase transitions is provided by the Ising Model Phase transitions involving solutions and mixtures are more complicated than transitions involving

2475-462: The supercritical liquid–gas boundaries . The first example of a phase transition which did not fit into the Ehrenfest classification was the exact solution of the Ising model , discovered in 1944 by Lars Onsager . The exact specific heat differed from the earlier mean-field approximations, which had predicted that it has a simple discontinuity at critical temperature. Instead, the exact specific heat had

2550-506: The superfluid transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature which enables accurate detection using differential scanning calorimetry measurements. Lev Landau gave a phenomenological theory of second-order phase transitions. Apart from isolated, simple phase transitions, there exist transition lines as well as multicritical points , when varying external parameters like

2625-531: The thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables (cf. phases ). This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are small. Phase transitions can occur for non-thermodynamic systems, where temperature is not a parameter. Examples include: quantum phase transitions , dynamic phase transitions, and topological (structural) phase transitions. In these types of systems other parameters take

2700-595: The thermodynamic limit , the saddle point approximation tells us the integral is asymptotically dominated by the value at which f ( ⟨ σ ⟩ ) {\displaystyle f(\langle \sigma \rangle )} is minimised: where ⟨ σ ⟩ 0 {\displaystyle \langle \sigma \rangle _{0}} is the argument minimising f {\displaystyle f} . A simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising

2775-450: The Ising model remains unsolved. Due to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to mean field theory . This mean field may be spatially varying, or global. The configuration space C {\displaystyle {\mathcal {C}}} of functions σ {\displaystyle \sigma } is replaced by

2850-540: The boiling of water (the water does not instantly turn into vapor , but forms a turbulent mixture of liquid water and vapor bubbles). Yoseph Imry and Michael Wortis showed that quenched disorder can broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling. Second-order phase transition s are also called "continuous phase transitions" . They are characterized by

2925-399: The chemical composition of the fluid. More impressively, but understandably from above, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of

3000-514: The configuration space C {\displaystyle {\mathcal {C}}} is also finite. We can define the partition function and there are no issues of convergence (like those which emerge in field theory) since the sum is finite. In theory, this sum can be computed to obtain an expression which is dependent only on the parameters { g i } {\displaystyle \{g_{i}\}} and β {\displaystyle \beta } . In practice, this

3075-413: 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) 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

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3150-467: The crystal positions. This slowing down happens below a glass-formation temperature T g , which may depend on the applied pressure. If the first-order freezing transition occurs over a range of temperatures, and T g falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in

3225-618: The development of order in the universe, as is illustrated by the work of Eric Chaisson and David Layzer . See also relational order theories and order and disorder . Continuous phase transitions are easier to study than first-order transitions due to the absence of latent heat , and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points. Continuous phase transitions can be characterized by parameters known as critical exponents . The most important one

3300-438: The existence of breathers in discrete lattices is that the breather main frequency and all its multipliers are located outside of the phonon spectrum of the lattice. The sine-Gordon equation is the nonlinear dispersive partial differential equation with the field u a function of the spatial coordinate x and time t . An exact solution found by using the inverse scattering transform is: which, for ω < 1 ,

3375-476: The fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to spontaneous symmetry breaking , with the exception of certain accidental symmetries (e.g. the formation of heavy virtual particles , which only occurs at low temperatures). An order parameter

3450-484: The focusing nonlinear Schrödinger equation are examples of one- dimensional partial differential equations that possess breather solutions. Discrete nonlinear Hamiltonian lattices in many cases support breather solutions. Breathers are solitonic structures. There are two types of breathers: standing or traveling ones. Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called oscillons ). A necessary condition for

3525-536: The following data: The Ising model is given by the usual cubic lattice graph G = ( Λ , E ) {\displaystyle G=(\Lambda ,E)} where Λ {\displaystyle \Lambda } is an infinite cubic lattice in R d {\displaystyle \mathbb {R} ^{d}} or a period n {\displaystyle n} cubic lattice in T d {\displaystyle T^{d}} , and E {\displaystyle E}

3600-410: The four states of matter to another. At the phase transition point for a substance, for instance the boiling point , the two phases involved - liquid and vapor , have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the boiling point the gaseous form is the more stable. Common transitions between

3675-491: The heat capacity C typically has a power law behavior: The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent α = 0.59 A similar behavior, but with the exponent ν instead of α , applies for the correlation length. The exponent ν is positive. This is different with α . Its actual value depends on the type of phase transition we are considering. The critical exponents are not necessarily

3750-406: The heat capacity diverges at the transition temperature (though, since α < 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional Ising model for uniaxial magnets, detailed theoretical studies have yielded the exponent α ≈ +0.110. Some model systems do not obey a power-law behavior. For example, mean field theory predicts

3825-528: The magnetic field or composition. Several transitions are known as infinite-order phase transitions . They are continuous but break no symmetries . The most famous example is the Kosterlitz–Thouless transition in the two-dimensional XY model . Many quantum phase transitions , e.g., in two-dimensional electron gases , belong to this class. The liquid–glass transition is observed in many polymers and other liquids that can be supercooled far below

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3900-401: The magnetic fields and temperature differences from the critical value. Phase transitions play many important roles in biological systems. Examples include the lipid bilayer formation, the coil-globule transition in the process of protein folding and DNA melting , liquid crystal-like transitions in the process of DNA condensation , and cooperative ligand binding to DNA and proteins with

3975-543: The mean value of the field, we have σ ↦ ⟨ σ ⟩ := 1 | Λ | ∑ v ∈ Λ σ ( v ) {\displaystyle \sigma \mapsto \langle \sigma \rangle :={\frac {1}{|\Lambda |}}\sum _{v\in \Lambda }\sigma (v)} . As the number of lattice sites N = | Λ | → ∞ {\displaystyle N=|\Lambda |\rightarrow \infty } ,

4050-399: The melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a quenched disorder state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling

4125-492: The methods of computational physics , as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons . Techniques for solving these include the inverse scattering transform and the method of Lax pairs , the Yang–Baxter equation and quantum groups . The solution of these models has given insights into

4200-412: The nature of phase transitions , magnetization and scaling behaviour , as well as insights into the nature of quantum field theory . Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations . An example of a continuum theory that is widely studied by lattice models

4275-414: The notation closer to field theory. This allows the partition function to be written as a path integral where the free energy F [ ϕ ] {\displaystyle F[\phi ]} is a Wick rotated version of the action in quantum field theory . Phase transitions In physics , chemistry , and other related fields like biology, a phase transition (or phase change )

4350-531: The observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition, 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 T g falling within

4425-663: The order parameter is the difference of the densities. From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization , whose direction was spontaneously chosen when the system cooled below the Curie point . However, note that order parameters can also be defined for non-symmetry-breaking transitions. Some phase transitions, such as superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases,

4500-409: The order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition. There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as vortex - or defect lines. Symmetry-breaking phase transitions play an important role in cosmology . As

4575-620: The partition function. Such an approach to the periodic Ising model in d {\displaystyle d} dimensions provides insight into phase transitions . Suppose the continuum limit of the lattice Λ {\displaystyle \Lambda } is R d {\displaystyle \mathbb {R} ^{d}} . Instead of averaging over all of Λ {\displaystyle \Lambda } , we average over neighbourhoods of x ∈ R d {\displaystyle \mathbf {x} \in \mathbb {R} ^{d}} . This gives

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4650-434: The phase transition. Exponents are related by scaling relations, such as It can be shown that there are only two independent exponents, e.g. ν and η . It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as universality . For example, the critical exponents at the liquid–gas critical point have been found to be independent of

4725-438: The place of temperature. For instance, connection probability replaces temperature for percolating networks. Paul Ehrenfest classified phase transitions based on the behavior of the thermodynamic free energy 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. First-order phase transitions exhibit

4800-634: The possible values of ⟨ σ ⟩ {\displaystyle \langle \sigma \rangle } fill out the convex hull of S {\displaystyle S} . By making a suitable approximation, the energy functional becomes a function of the mean field, that is, E ( σ ) ↦ E ( ⟨ σ ⟩ ) . {\displaystyle E(\sigma )\mapsto E(\langle \sigma \rangle ).} The partition function then becomes As N → ∞ {\displaystyle N\rightarrow \infty } , that is, in

4875-425: The previous phenomenon is also the phenomenon of enhanced fluctuations before the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The large static universality classes of a continuous phase transition split into smaller dynamic universality classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of

4950-422: The resolution of outstanding issues in understanding glasses. In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the critical point , at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases

5025-430: 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 γ {\displaystyle \gamma } , the exponent of the susceptibility) are not identical. For −1 < α < 0, the heat capacity has a "kink" at the transition temperature. This

5100-407: The solid, liquid, and gaseous phases of a single component, due to the effects of temperature and/or pressure are identified in the following table: For a single component, the most stable phase at different temperatures and pressures can be shown on a phase diagram . Such a diagram usually depicts states in equilibrium. A phase transition usually occurs when the pressure or temperature changes and

5175-409: The system crosses from one region to another, like water turning from liquid to solid as soon as the temperature drops below the freezing point . In exception to the usual case, it is sometimes possible to change the state of a system diabatically (as opposed to adiabatically ) in such a way that it can be brought past a phase transition point without undergoing a phase transition. The resulting state

5250-521: The temperature is lowered below the Curie temperature . The magnetic susceptibility , the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions. 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

5325-408: The temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into

5400-466: The temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between T g and T c in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable

5475-582: The theory about the mean field ⟨ σ ⟩ {\displaystyle \langle \sigma \rangle } . Writing configurations as σ ( v ) = ⟨ σ ⟩ + Δ σ ( v ) {\displaystyle \sigma (v)=\langle \sigma \rangle +\Delta \sigma (v)} , truncating terms of O ( Δ σ 2 ) {\displaystyle {\mathcal {O}}(\Delta \sigma ^{2})} then summing over configurations allows computation of

5550-462: The uniform value a when moving away from the focus time t = 0. These breathers exist for values of the modulation parameter b less than √ 2 . Note that a limiting case of the breather solution is the Peregrine soliton . Lattice model (physics) In mathematical physics , a lattice model is a mathematical model of a physical system that is defined on a lattice , as opposed to

5625-616: The universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field . This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to electroweak baryogenesis theory. Progressive phase transitions in an expanding universe are implicated in

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