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This is because Landau theory is a mean field theory, and does not include long-range correlations. This theory does not explain non-analyticity at the critical point, but when applied to superfluid and superconductor phase transition, Landau's theory provided inspiration for another theory, the Ginzburg–Landau theory of superconductivity.
Using this in the Landau theory, which is identical to the mean field theory for the Ising model, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean-field results are good and self-consistent. But for dimensions less than 4, the predictions are less accurate.
The classical Landau theory (also known as mean field theory) values of the critical exponents for a scalar field (of which the Ising model is the prototypical example) are given by = ′ =, =, = ′ =, =
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random 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).
In Landau mean-field theory, at temperatures near the superconducting critical temperature , () (/). Up to a factor of 2 {\displaystyle {\sqrt {2}}} , it is equivalent to the characteristic exponent describing a recovery of the order parameter away from a perturbation in the theory of the second order phase transitions.
Based on Landau's previously established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy density of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field () = | | (), where the quantity | | is a measure of the local density of superconducting electrons () analogous to a quantum mechanical wave ...
The theory describes the behavior of many-body systems of particles in which the interactions between particles may be strong. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, [2] and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory ...
While mean field theory is a more reasonable model for ferromagnets at higher magnetic fields, the presence of more than one magnetic domain in real magnets means that especially at low magnetic fields, the experimentally measured macroscopic magnetic field (which is an average over the whole sample) will not be a reasonable way to determine ...