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Landau Theory
Landau theory (also known as Ginzburg–Landau theory, despite the confusing name) in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. It can also be adapted to systems under externally-applied fields, and used as a quantitative model for discontinuous (i.e., first-order) transitions. Although the theory has now been superseded by the renormalization group and scaling theory formulations, it remains an exceptionally broad and powerful framework for phase transitions, and the associated concept of the Phase transitions#Order parameters, order parameter as a descriptor of the essential character of the transition has proven transformative. Mean-field formulation (no long-range correlation) Landau was motivated to suggest that the free energy of any system should obey two conditions: *Be analytic in the order parameter and its gradients. *Obey the symmetry of the Hamiltonian mechanics, H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rushbrooke Inequality
In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature ''T''. Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_ \frac\log Z_N The magnetization ''M'' per site in the thermodynamic limit, depending on the external magnetic field ''H'' and temperature ''T'' is given by : M(T,H) \ \stackrel\ \lim_ \frac \left( \sum_i \sigma_i \right) where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac \right)_T and : c_H = T \left( \frac \right)_H. Additionally, : c_M = +T \left( \frac \right)_M. Definitions The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stuart–Landau Equation
The Stuart–Landau equation describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, named after John Trevor Stuart and Lev Landau. In 1944, Landau proposed an equation for the evolution of the magnitude of the disturbance, which is now called as the Landau equation, to explain the transition to turbulence based on a phenomenological argument and an attempt to derive this equation from hydrodynamic equations was done by Stuart for plane Poiseuille flow in 1958. The formal derivation to derive the ''Landau equation'' was given by Stuart, Watson and Palm in 1960. The perturbation in the vicinity of bifurcation is governed by the following equation :\frac = \sigma A - \frac A , A, ^2. where *A = , A, e^ is a complex quantity describing the disturbance, *\sigma = \sigma_r + i\sigma_i is the complex growth rate, *l = l_r + i l_i is a complex number and l_r is the ''Landau constant''. The evolution of the actual disturbance is given by the real part of A(t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau–de Gennes Theory
In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969. The phenomonological theory uses the \mathbf tensor as an order parameter in expanding the free energy density. Mathematical description The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the \mathbf tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase. We shall consider a uniaxial \mathbf Q tensor, which is defined by :\mathbf Q = S(\mathbf n\otimes\mathbf n - \tfrac\mathbf I) where S=S(T) is the scalar order parameter and \mathbf n is the director. The \mathbf Q tensor is zero in the isotropic liquid phase since the scalar order parameter S is zero, but becomes non-zero in the nematic phase. Near the NI transition, the (Helmholtz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ginzburg Criterion
Mean field theory gives sensible results as long as one is able to neglect fluctuations in the system under consideration. If \phi is the order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical point. Quantitatively, this means that: : \displaystyle\mathcal \langle(\delta \phi)^2\rangle \quad\quad \langle\phi\rangle^2 The Ginzburg criterion is a restatement of this inequality through measurable quantities, such as the magnetic susceptibility in the Ising model. It also gives the idea of an upper critical dimension, a dimensionality of the system above which mean field theory gives proper results, and the critical exponents predicted by mean field theory match exactly with those obtained by numerical methods. Example: Ising model One can prove that: k_BT\chi \ll \langle M \rangle^2 Where k_B is the Boltzmann constant, T is the system tempera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superconductivity
Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ordinary metallic Electrical conductor, conductor, whose resistance decreases gradually as its temperature is lowered, even down to near absolute zero, a superconductor has a characteristic Phase transition, critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source. The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and Atomic spectral line, atomic spectral lines, superconductivity is a phenomenon which can only be explained by quantum mechanics. It is characterized by the Meissner effect, the complete cancellation of the magnetic field in the interior of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ginzburg–Landau Theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all cuprates. Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two isotopes of helium ( helium-3 and helium-4) when they are liquefied by cooling to cryogenic temperatures. It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity. The theory of superfluidity was developed by Soviet theoretical physicists Lev Landau and Isaak Khalatnikov. Superfluidity often co-occurs with Bose–Einstein condensation, but neither phenomenon is directly related to the other; not all Bose–Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates. Even when superfluidity and condensation co-occur, their magnitudes are not linked: at low temperature, liquid helium has a lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Field Theory
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary). Such models consider many individual components that interact with each other. The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a ''molecular field''. This reduces any many-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost. MFT has since been applied to a wide range of fields outside of physics, including statistical inference, graphical models, neuroscience, artificial intelligence, epidemic models, queueing theory, computer-network ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Critical Dimension , a video game by Marvelous
{{Disambiguation ...
Upper may refer to: * Shoe upper or ''vamp'', the part of a shoe on the top of the foot * Stimulant, drugs which induce temporary improvements in either mental or physical function or both * ''Upper'', the original film title for the 2013 found footage film ''The Upper Footage'' * Dmitri Upper (born 1978), Kazakhstani ice hockey player See also * Uppers (video game) is a Japanese video game developer and publisher, and anime producer. The company was founded in 1997 but formed in its current state in October 2011 by the merger of the original Marvelous Entertainment with AQ Interactive, and Liveware. Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying physical laws (codified in a quantum field theory) as the energy (or mass) scale at which physical processes occur varies. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales ( self-similarity), where under the fixed point of the renormalization group flow the field theory is conformally invariant. As the scale varies, it is as if one is decreasing (as RG is a semi-group and doesn't have a well-defined inverse operation) the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally consist of self- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
