scholarly journals Continuum theory of critical phenomena in polymer solutions: Formalism and mean field approximation

1989 ◽  
Vol 90 (12) ◽  
pp. 7448-7460 ◽  
Author(s):  
Raymond E. Goldstein ◽  
Binny J. Cherayil
Keyword(s):  
Critical Phenomena ◽  
Polymer Solutions ◽  
Mean Field ◽  
Continuum Theory ◽  
Field Approximation ◽  
Mean Field Approximation

scholarly journals Superconducting Quantum Critical Phenomena

2018 ◽  
Vol 02 (03) ◽  
pp. 1850009
Author(s):  
Yong Tao
Keyword(s):  
Transition Temperature ◽  
Critical Phenomena ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Superconducting Transition ◽  
Zero Temperature ◽  
Critical Equation ◽  
Quantum Critical Phenomena ◽  
Quantum Critical

When the superconducting transition temperature [Formula: see text] sufficiently approaches zero, quantum fluctuations are expected to be overwhelmingly amplified around zero temperature so that the mean-field approximation may break down. This implies that quantum critical phenomena may emerge in highly underdoped and overdoped regions, where the transition temperature [Formula: see text] is sufficiently low. By using Gor’kov’s Green function method, we propose a superconducting quantum critical equation (SQCE) for describing such critical phenomena. For two-dimensional (2D) overdoped materials, SQCE shows that the transition temperature [Formula: see text] and the zero-temperature superfluid phase stiffness [Formula: see text] will obey a two-class scaling combined by linear and parabolic parts, which agrees with the existing experimental investigation [I. Božović et al., Dependence of the critical temperature in overdoped copper oxides on superfluid density, Nature 536 (2016) 309–311]. For three-dimensional (3D) overdoped materials, SQCE predicts that the two-class scaling will be replaced by the linear scaling. Furthermore, we show that SQCE can be applied into highly underdoped region by using Anderson’s non-Fermi liquid model.

Critical phenomena: General considerations. Mean-field theory (MFT)

2021 ◽  
pp. 324-356
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Critical Temperature ◽  
Critical Phenomena ◽  
Correlation Length ◽  
Particle Physics ◽  
Mean Field ◽  
Lattice Models ◽  
Field Approximation ◽  
Phase Region ◽  
Mean Field Approximation ◽  
The Mean

This chapter is devoted to a brief review of general properties of phase transitions in macroscopic physics and, in particular in lattice models. Some of these lattice models actually appear as lattice regularizations of Euclidean (imaginary time) quantum physics theory (QFT). Most of the transitions considered in this work have the following character: spins on the lattice, or macroscopic particles in the continuum, interact through short-range forces, assumed, for simplicity, to decay exponentially. For simple systems, it is possible to find a local observable, called order parameter, whose expectation values depend on the phase in the several phase region, for example, the spin in ferromagnetic systems. In the disordered phase, the connected two-point function decreases exponentially at large distance, at a rate characterized by the correlation length (the inverse of the smallest physical mass in particle physics). In continuous transitions, the correlation length diverges at the critical temperature. Within the mean-field approximation (consistent with Landau's theory of critical phenomena), it can be shown that the singular behaviour of thermodynamic quantities at the critical temperature is universal. These properties can also be reproduced by calculating correlation functions with a perturbed Gaussian measure. It is then shown that the leading corrections to the mean-field approximation, in Ising-like systems, diverge at the critical temperature for dimensions smaller than or equal to $4$.

A continuum theory of disorder in nematic liquid crystals. V. Anisotropy and order-dependence of the Frank constants

1981 ◽  
Vol 375 (1763) ◽  
pp. 579-597 ◽  
Keyword(s):  
Harmonic Functions ◽  
Mean Field ◽  
Continuum Theory ◽  
Field Approximation ◽  
Relative Magnitude ◽  
Mean Field Approximation ◽  
Nematic Order ◽  
Order Dependence ◽  
The Mean ◽  
Made In

The extent to which the ‘splay ’, ‘bend’ and ‘twist’ constants ( K 1, 2, 3 ) of a nematic liquid crystal differ from one another and the way in which they depend upon the degree of alignment (as characterized by the nematic order parameter S 2 ) are determined by the interaction responsible for alignment, V ij . Priest (1973) has already shown that if V ij is expanded in products of spherical harmonic functions such as Y l i ,m Y l j ,m the contributions made to K 1 , K 2 and K 3 by successive terms in the expansion are additive, and he has discussed the relative magnitude of these contributions for l i = l j = 2 and l i = 2, l j = 4. Priest’s results in the limit S 2 = 1 are here confirmed, and they are extended to the case l i = l j = 4. To obtain results for the range 0.7 > S 2 > 0.4 which is of interest experimentally, however, Priest invoked the mean field approximation, and his conclusion that the contributions he considered are proportional to S 2 2 and S 2 S 4 respectively is invalid for that reason. Methods of analysis developed in previous papers of this series are here used to show that Priest’s S 2 2 should be replaced over the range of interest by say AS n 2 , where both A (≈ 1) and n (≈ 1.35) depend in principle on m and on whether it is K 1 , K 2 , or K 3 that interests us, though the variations are not great in practice. The same expression ( AS n 2 with A ≈ 1) may be used to describe the order-dependence of ( l i = 2, l j = 4)- and ( l i = 4, l j , = 4)-contributions to K 1, 2, 3 , with n ≈ 3.25 ( ± 0.25 say) and n ≈ 4.2 ( ± 0.4 say) respectively. The revised results can be fitted to recent data for nematic 5CB, but it would be premature to draw firm conclusions about the nature of V ij in this substance, because several approximations are still present in the theory. A conjecture made in earlier papers in the series concerning < P 4 (cos β ij )> (alias σ 4 ) is here confirmed.

The Determination of the Magnetoelectric Coupling Coefficient in Ferroelectric–Ferromagnetic Composite Based on PZT–Ferrite

2013 ◽  
Vol 58 (4) ◽  
pp. 1401-1403 ◽  
Author(s):  
J.A. Bartkowska ◽  
R. Zachariasz ◽  
D. Bochenek ◽  
J. Ilczuk
Keyword(s):  
Dielectric Permittivity ◽  
Coupling Coefficient ◽  
Mean Field ◽  
Field Approximation ◽  
Theoretical Description ◽  
Magnetoelectric Coupling ◽  
Mean Field Approximation ◽  
Dielectric Measurements ◽  
Temperature Dependences ◽  
Multiferroic Composite

Abstract In the present work, the magnetoelectric coupling coefficient, from the temperature dependences of the dielectric permittivity for the multiferroic composite was determined. The research material was ferroelectric-ferromagnetic composite on the based PZT and ferrite. We investigated the temperature dependences of the dielectric permittivity (") for the different frequency of measurement’s field. From the dielectric measurements we determined the temperature of phase transition from ferroelectric to paraelectric phase. For the theoretical description of the temperature dependence of the dielectric constant, the Hamiltonian of Alcantara, Gehring and Janssen was used. To investigate the dielectric properties of the multiferroic composite this Hamiltonian was expressed under the mean-field approximation. Based on dielectric measurements and theoretical considerations, the values of the magnetoelectric coupling coefficient were specified.

scholarly journals Hexagonal-Shaped Spin Crossover Nanoparticles Studied by Ising-Like Model Solved by Local Mean Field Approximation

2021 ◽  
Vol 7 (5) ◽  
pp. 69
Author(s):  
Catherine Cazelles ◽  
Jorge Linares ◽  
Mamadou Ndiaye ◽  
Pierre-Richard Dahoo ◽  
Kamel Boukheddaden
Keyword(s):  
Spin Crossover ◽  
Hexagonal Lattice ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Ligand Field ◽  
Order And Disorder ◽  
The Matrix ◽  
Local Mean ◽  
Parameter Values

The properties of spin crossover (SCO) nanoparticles were studied for five 2D hexagonal lattice structures of increasing sizes embedded in a matrix, thus affecting the thermal properties of the SCO region. These effects were modeled using the Ising-like model in the framework of local mean field approximation (LMFA). The systematic combined effect of the different types of couplings, consisting of (i) bulk short- and long-range interactions and (ii) edge and corner interactions at the surface mediated by the matrix environment, were investigated by using parameter values typical of SCO complexes. Gradual two and three hysteretic transition curves from the LS to HS states were obtained. The results were interpreted in terms of the competition between the structure-dependent order and disorder temperatures (TO.D.) of internal coupling origin and the ligand field-dependent equilibrium temperatures (Teq) of external origin.

Geometrical and Transport Properties of Disordered Fibre Networks: Analytical Results

1997 ◽  
Vol 11 (20) ◽  
pp. 867-875 ◽  
Author(s):  
A. A. Rodríaguez ◽  
E. Medina
Keyword(s):  
Probability Distribution ◽  
Transport Properties ◽  
Geometrical Structure ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Fracture Networks ◽  
Simplified Models ◽  
Fibre Density ◽  
Line Network

We study novel geometrical and transport properties of a 2D model of disordered fibre networks. To assess the geometrical structure we determine, analytically, the probability distribution for the number of fibre intersections and resulting segment sizes in the network as a function of fibre density and length. We also determine, numerically, the probability distribution of pore perimeters and areas. We find a non-monotonous behavior of the perimeter distribution whose main features can be explained by solving for two simplified models of the line network. Finally we formulate a mean field approximation to conduction, above the percolation threshold, using the derived results. Relevance of the results to fracture networks will be discussed.

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