On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

We present new estimates on the two-dimensional Ginzburg–Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg–Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg–Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg–Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, which is related to a conjecture by Xing-Bin Pan.     

Extensive Properties of the Complex Ginzburg–Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in R^d, d <3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q_L of side L. We cover this set by a (minimal) number N_QL(l) of balls of radius l in $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, H_\epsilon =\lim_{L\to\infty} L^{-d} \logtwo N_{Q_L}(\epsilon)< /FORMULA > exists. In particular, we bound < FORMULA FORM=ÏNLINE» exists. In particular, we bound H_\epsilon< /FORMULA > by < FORMULA FORM=ÏNLINE» by \OO\bigl(\logtwo(1/\epsilon )\bigr)< /FORMULA > , which shows that the attracting set is < SMALL > smaller < /SMALL > than the set of bounded analytic functions in a strip. We finally give a positive lower bound: < FORMULA FORM=ÏNLINE», which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H_\epsilon>\OO\bigl (\logtwo(1/\epsilon)\bigr)$.     

On the nature of the phase transition triggered by vortex-like defects in the 2D Ginzburg–Landau model

The two-dimensional lattice Ginzburg–Landau Hamiltonian is simulated numerically for different values of the coherence length ξ in units of the lattice spacing a, a parameter which controls amplitude fluctuations. The phase diagram on the plane T–ξ is measured. Amplitude fluctuations change dramatically the nature of the phase transition: for values of ξ/a≃1, instead of the smooth Kosterlitz–Thouless transition there is a first-order transition with a discontinuity in the vortex density v and a sharper drop in the helicity modulus Γ. Both observables v and Γ are analyzed in detail at the crossover region between first and second order which occurs for intermediate values of ξ/a.     

Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg–Landau Polynomials

The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(β _n ,K _n ) for appropriate sequences (β _n ,K _n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (β _n ,K _n ) converges to one of these points (β,K), . In this formula γ is a positive constant, and is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg–Landau polynomial because of its close connection with the Ginzburg–Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg–Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which (β _n ,K _n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg–Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume–Capel model.     

Time-dependent Ginzburg–Landau equation in the Nambu–Jona-Lasinio model

We apply the closed time-path Green function formalism in the Nambu–Jona-Lasinio model. First of all, we use this formalism to obtain the well-known gap equation for the quark condensate in a stationary homogeneous system. We have also used this formalism to obtain the Ginzburg–Landau (GL) equation and the time-dependent Ginzburg–Landau (TDGL) equation for the chiral order parameter in an inhomogeneous system. In our derived GL and TDGL equations, there is no other parameters except for those in the original NJL model.     

Stability of the Landau–Fermi Liquid Theory

The stability of the Landau–Fermi liquid theory is investigated. It has been shown that if the interaction function of the Fermi system is a finite function of the angle between the momenta of two particles at the Fermi surface, then the liquid can be stable. We have shown that the absolute value of the expansion coefficients of the interaction functions in Legendre polynomials are decreasing function of the coefficients indices. We solve the stability condition for one photon exchange (OPE) in an electron gas. The results show that we must use the massive boson propagator (higher order corrections to the photon propagator). Similar to previous works (Abrikosov et al. in Method of Quantum Field Theory in Statistical Physics, Pergamon, Elmsford, 1965), our result is proportional to g ². The density and temperature dependence of results is occulted in the effective mass of the system.     

Analysis of Energy Eigenvalue in Complex Ginzburg–Landau Equation

In this paper, we consider the two-dimensional complex Ginzburg–Landau equation(CGLE) as the spatiotemporal model, and an expression of energy eigenvalue is derived by using the phase-amplitude representation and the basic ideas from quantum mechanics. By numerical simulation, we find the energy eigenvalue in the CGLE system can be divided into two parts, corresponding to spiral wave and bulk oscillation. The energy eigenvalue of spiral wave is positive, which shows that it propagates outwardly; while the energy eigenvalue of spiral wave is negative, which shows that it propagates inwardly. There is a necessary condition for generating a spiral wave that the energy eigenvalue of spiral wave is greater than bulk oscillation. A wave with larger energy eigenvalue dominates when it competes with another wave with smaller energy eigenvalue in the space of the CGLE system. At the end of this study, a tentative discussion of the relationship between wave propagation and energy transmission is given.     

Inviscid Limits¶of the Complex Ginzburg–Landau Equation

In the inviscid limit the generalized complex Ginzburg–Landau equation reduces to the nonlinear Schr?dinger equation. This limit is proved rigorously with H ¹ data in the whole space for the Cauchy problem and in the torus with periodic boundary conditions. The results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of L ² in the focusing case, in any spatial dimension. Furthermore, optimal convergence rates are proved. The proofs are based on estimates of the Schr?dinger energy functional and on Gagliardo–Nirenberg inequalities. Received: 2 April 1999 / Accepted: 29 March 2000     

Effect of Quantum Jumpers on the Critical Supercurrent in Dirty S–I–S Junctions

It has been shown that the presence of narrowband quantum jumpers in “dirty” (low concentrations of identical nonmagnetic impurities in the insulator (I) layer) S–I–S (S is a superconductor) junctions at the temperature T = 0 significantly reduces the critical supercurrent (Josephson current) as compared to the value given by the known the Ambegaokar–Baratoff relation. The performed estimates have shown the possibility of the experimental manifestation of this effect.     

Quantum critical properties in the topological Ginzburg–Landau theory of self-dual Josephson junction arrays

This paper examines the multicritical behavior of a generalized U(_N1)×U(_N2)$U\left(N_1\right)\times U\left(N_2\right)$ Ginzburg–Landau theory containing two multicomponent complex fields which couple differently to two gauge fields described by two Maxwell terms and one mixed-Chern–Simons term. This model is relevant to the dynamics of Cooper pairs and vortices in a self-dual Josephson junction array system near its superconductor–insulator transition. We develop a renormalization group flow at fixed dimension and obtain the beta functions at one loop when both disorder fields are critical. Two sets of infrared-stable charged fixed points solutions are found for N>_Nc$N>N_c$: partially charged solutions with respect to the gauge fields exist with _Nc=35.6$N_c=35.6$, and fully charged solutions exist with _Nc=12.16$N_c=12.16$. We show that fine tuning the ratio of the two energy scales in the model has the effect of reducing the critical number _Nc$N_c$ and thus enlarges the region where the quantum phase transition is continuous. It is also found that the decoupled fixed point which is stable in the neutral case is no longer attainable in the presence of fluctuating gauge fields. We probe the conductivity at the critical point and show that it has a universal character determined by the renormalization group infrared-stable fixed-point values of the gauge couplings.     

Eigenstates of the linearlized Ginzburg–Landau equation for mesoscopic multiply-connected superconductor

Multiply-connected mesoscopic superconductors have rich structures of vortex systems that result from interference of order parameter. We studied magnetic field dependence of transition temperatures and vortex arrangements of finite sized honeycomb superconducting networks with 6-fold rotational symmetries. Near and above the lower critical field, vortices locate at center of the network. As increasing the field, vortices form a hexagon or hexagonal multi-shell structure. In higher field, order parameter damps exponentially from the central point of the network to the edge of the network.     

On the Theory of the Mandelstam–Brillouin Scattering in a Plasma Layer

Optics and Spectroscopy - The evolution of a perturbation from a local source upon the Mandelstam–Brillouin scattering in a plasma layer of a finite thickness and infinite length is examined...     

Dynamic Statistical Scaling in the Landau–de Gennes Theory of Nematic Liquid Crystals

In this article, we investigate the long time behaviour of a correlation function $c_{\mu _{0}}$ which is associated with a nematic liquid crystal system that is undergoing an isotropic-nematic phase transition. Within the setting of Landau–de Gennes theory, we confirm a hypothesis in the condensed matter physics literature on the average self-similar behaviour of this correlation function in the asymptotic regime at time infinity, namely $$\begin{aligned} \left\| c_{\mu _{0}}(r, t)-e^{-\frac{|r|^{2}}{8t}}\right\| _{L^{\infty }(\mathbb {R}^{3}, \,dr)}=\mathcal {O}(t^{-\frac{1}{2}}) \quad \mathrm as \quad t\longrightarrow \infty . \end{aligned}$$ In the final sections, we also pass comment on other scaling regimes of the correlation function.     

Results of the Experiments on the Crystallization of a Ge–Si–Sb Solid Solution under the Conditions of Microgravity on the Soyuz–Apollo Spacecraft

Physics of the Solid State - Objective factors that allow the experiment on growing crystals of a Ge–Si–Sb solid solution on the Soyuz–Apollo spacecraft to take a special place...     

On the Effect of Stochastic Fluctuations in the Dynamics of the Lifshitz–Slyozov–Wagner Model

In this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed. Corrections to the Lifshitz–Slyozov–Wagner (LSW) model that take into account the stochastic character of the problem are computed. It is proved, under suitable smallness assumptions for the volume fraction filled by the particles, that the effect of these corrections does not modify much the dynamics of the self-similar solutions of the LSW system of equations.     

Anisotropy of Critical Fields in MgB2： Two-Band Ginzburg-Landau Theory for Layered Superconductors

The temperature dependence of the anisotropy parameter of upper critical field γHc2(T)=H_{c2}^||(T)/H_{c2}^{\bot}(T) and London penetration depth γ_λ(T)=λ_{\perp} (T)/λ_{\bot} (T) are calculated using two-band Ginzburg-Landau theory for layered superconductors. It is shown that, with decreasing temperature the anisotropy parameter γ_{H_{c2}}(T) is increased, while theLondon penetration depth anisotropy γλ(T) reveals n opposite behavior.Results of our calculations are in agreement with experimental datafor single crystal MgB₂ nd with other calculations. Results of an analysis of magnetic field H_c1 in a single vortex between superconducting layers are also presented.     

The Ginzburg–Landau theory and the surface energy of a colour superconductor

We apply the Ginzburg–Landau theory to the colour superconducting phase of a lump of dense quark matter. We calculate the surface energy of a domain wall separating the normal phase from the super phase with the bulk equilibrium maintained by a critical external magnetic field. Because of the symmetry of the problem, we are able to simplify the Ginzburg–Landau equations and express them in terms of two components of the di-quark condensate and one component of the gauge potential. The equations also contain two dimensionless parameters: the Ginzburg–Landau parameter κ and ρ. The main result of this paper is a set of inequalities obeyed by the critical value of the Ginzburg–Landau parameter—the value of κ for which the surface energy changes sign—and its derivative with respect to ρ. In addition we prove a number of inequalities of the functional dependence of the surface energy on the parameters of the problem and obtain a numerical solution of the Ginzburg–Landau equations. Finally a criterion for the types of colour superconductivity (type I or type II) is established in the weak coupling approximation.     

Effect of Quenching on the Grain Boundary Relaxation in PM2000 ODS Alloy

Grain boundary relaxation in a Fe-based ODS alloy is studied by internal friction measurements. It is found that a grain-boundary peak appears at a lower temperature in the quenched specimens than that in the annealed specimens. The activation energy of the peak is H=2.82±0.11 eV for the former while H = 2.53±0.08 eV for the latter. In addition, a new relaxation peak is observed at the high temperature side of the grain boundary peak in the quenched specimens with an activation energy of 4.41±0.25 eV. The height of the peak increases with increasing quenching temperature. The results suggest that both the shift of the grain-boundary peak and the appearance of the new peak are due to increasing vacancies by quenching that are favourable for the motion of the grain boundaries.     

Influence of Boundary Condition and Diffusion Coefficient on the Accuracy of Diffusion Theory in Steady—State Spatially Resolved Diffuse Relfectance of Biological Tissues

The applicability of diffusion theory for the determination of tissue optical properties from steady-state diffuse reflectance is investigated.Analytical expressions from diffusion theory using the two most commonly assumed boundary conditions at the air-tissue interface and the two definitions of the diffusion coefficient are compared with Monte Carlo simulations.The effects of the choice of the boundary conditions and diffusion coefficients on the accuracy of the findings for the optical parameters are quantified.and criteria for accurate curve-fitting algorithms are developed.It is shown that the error in deriving the optical coefficients is considerably smaller for the solution which uses the extrapolated boundary condition and the diffusion coefficient independence of absorption coeffcient,compared to the other three solutions.