Magnetic relaxation in superconductors with varying creep behavior

The relaxation of an electromagnetic field inside low-and high-temperature superconductors in the case when the magnetic flux occupies the cross section incompletely is studied theoretically in the self-similar approximation. On the macroscopic level, creep is described by phenomenological equations for exponential and power I-V characteristics. A relation between the nonlinearity of the I-V curves and the magnetic relaxation process is found. The range of low electric fields is shown to be of considerable importance for the relaxation in high-temperature superconductors. In this range, the equations for exponential and power I-V curves may differ substantially from those at high fields. Complete analytical solutions are contrasted with a numerical solution to the problem.     

Dynamical process of complex systems and fractional differential equations

Behavior of dynamical process of complex systems is investigated. Specifically we analyse two types of ideal complex systems. For analysing the ideal complex systems, we define the response functions describing the internal states to an external force. The internal states are obtained as a relaxation process showing a “power law” distribution, such as scale free behaviors observed in actual measurements. By introducing a hybrid system, the logarithmic time, and double logarithmic time, we show how the “slow relaxation” (SR) process and “super slow relaxation” (SSR) process occur. Regarding the irregular variations of the internal states as an activation process, we calculate the response function to the external force. The behaviors are classified into “power”, “exponential”, and “stretched exponential” type. Finally we construct a fractional differential equation (FDE) describing the time evolution of these complex systems. In our theory, the exponent of the FDE or that of the power law distribution is expressed in terms of the parameters characterizing the structure of the system.     

NMR relaxation rate in metallic carbon nanotubes

NMR relaxation rate, T₁⁻¹, of the metallic carbon nanotube is discussed based on Tomonaga–Luttinger-liquid theory. It is found that the Coulomb interaction leads to increase of (T₁T)⁻¹ by a power law with decreasing temperature, T. The dependence on temperature of (T₁T)⁻¹ in the multi-wall nanotube (MWNT) is shown to be strongly suppressed by existence of the metallic shells in the MWNTs.     

Hierarchical description for a non-Markoff relaxation process

A phenomenal model is proposed in this paper to describe a non-Markoff relaxation process. The main feature of the model is the fractal time concept introduced to take account of the coupling of a relaxation mode with a heat bath. The theoretical investigation indicates that a weakly non-Markoff relaxation process can be related to a set of hierarchically distributed Markoff relaxation processes and the fractal exponential law exp [?(t/τ)^1?σ] is the character of a non-Markoff relaxation process. Calculations about the temperature and time scale dependence of the exponent σ have been made, and comparison with recent experiments is also given.     

Notes on the absorption of sound in high magnetic fields

We study the absorption of sound by a free electron-impurity system in a quantizing magnetic field and derive a formula for the absorbed power on the basis of linear response theory. In the approximation of constant relaxation time and in the limit of high temperatures (1/τ?kT) this formula is essentially equivalent to the formulas used by Gurevichet al. and by Svirskii. Without these restricting assumptions we obtain general conditions for the appearance of quantum oscillations. We discuss the influence of spin flip scattering on the shape of the 0⁺-absorption line. Furthermore, we show that the usual kinetic approach leads to the incorrect results given by Quinn.     

Physical mechanisms responsible for the relaxation time distribution in disordered dielectrics

The relaxation time distribution function F(τ) is calculated in the framework of the random-field theory. The function F(τ) is expressed through the distribution function f(E) of a random electric field E with due regard for the derived dependence of the relaxation time τ on the electric field. The distribution function F(τ) is calculated in terms of the statistical theory within the random-field approximation. The nonlinear random-field contributions and spatial effects of correlations between randomly distributed electric dipoles are taken into account. The calculations are performed for a mixed ferroelectric glassy phase in which the short-range and long-range polar orders coexist. It is demonstrated that the inclusion of nonlinear contributions of the random field leads to an asymmetric relaxation time distribution function F(τ), whereas allowance made only for the linear random-field contributions results in a symmetric function F(τ). A comparison of the calculated functions F(τ) with empirical functions derived from the Cole-Cole (CC), Davidson-Cole (DC), Kohlrausch-Williams-Watts (KWW), and Havriliak-Negami (HN) laws for the dielectric response shows that these laws correspond to disordered systems in which the long-range and short-range orders coexist. Different forms of the function F(τ) are determined by either linear (the CC law) or nonlinear (the DC, KWW, and HN laws) contributions of the random field.     

Bulk model of rapid thermal oxidation of silicon

A model of rapid thermal oxidation of silicon in dry oxygen based on the reaction of volume oxidation is constructed. It is assumed that the coefficient of oxygen diffusion for silicon dioxide decreases because of internal compressive stress, which is at a maximum near the SiO₂-Si interface; as the distance from the interface increases, this stress decreases according to the time-dependent exponential law because of viscoelastic relaxation from the value of the diffusion coefficient for strained oxide to that for fused quartz. The characteristic relaxation time of the coefficient of oxygen diffusion in silicon dioxide correlates with the relaxation time of internal stress in silicon dioxide films on silicon and with the relaxation time of the refraction index. Because the refraction index is related to the density of silicon dioxide, we arrive at the conclusion that the relaxation of the diffusion coefficient is related not only to the relaxation of internal mechanical stress, but also to the relaxation of the density of silicon dioxide.     

Anisotropic Dark Energy Bianchi Type-II Cosmological Models in Lyra Geometry

The spatially homogeneous and totally anisotropic Bianchi type-II cosmological model has been discussed in general relativity in the presence of a hypothetical anisotropic dark energy fluid with constant deceleration parameter within the frame work of Lyra’s manifold with uniform and time varying displacement field vector. With the help of special law of variation for Hubble’s parameter proposed by Bermann (Nuovo Cimento 74B:182, 1983) a dark energy cosmological model is obtained in this theory. We use the power law relation between average Hubble parameter H and average scale factor R to find the solution. The assumption of constant deceleration parameter leads to two models of universe, i.e. power law model and exponential model. Some physical and kinematical properties of the model are also discussed.     

The one-dimensional kinetic Ising model: A series expansion study

Exact power series expansions (through eight terms) in the time are derived for relaxation in the one-dimensional Ising model with nearest-neighbor interactions for a general rate parameter where the activation energy is a variable fraction of the energy required to break nearest-neighbor bonds. It is found that the qualitative nature of the relaxation is very dependent on this parameter, varying from nearly simple exponential decay (as with Glauber dynamics) for an intermediate value of this parameter, to an initial rate of change that is either much slower or faster than a simple exponential at the extremes of the range of variation of the parameter. The rate equations for the limit of rapid internal diffusion (internal equilibration) are integrated for several special values of the rate parameter. In general the internal equilibration approximation is not a good representation of the relaxation except when the relaxation is similar to Glauber dynamics.     

Investigation of glassy behavior in lead barium ytterbium tantalate relaxors

The glassy behavior of lead barium ytterbium tantalate relaxors has been investigated. The frequency dependence of the temperature, T′_m (the temperature corresponding to the maximum value dielectric constant) is analyzed using Vogel-Fulcher relation and power law. The fitting parameters reveal that the power law explains the relaxation mechanism much better than the Vogel-Fulcher relation. The temperature dependence of dielectric constant at temperatures much higher and lower than T′_m is analyzed by two exponential functions, which gives an idea about the production of polar clusters at high temperature and the distribution of freezing temperatures at lower temperature.     

Anomalously slow relaxation of a nonwetting liquid in the disordered confinement of a nanoporous medium

The time evolution of the water–disordered nanoporous medium Libersorb 23 (L23) system has been studied after complete filling at elevated pressure followed by full release of overpressure. It is established that relaxation of the L23 rapidly flows out during the overpressure relief time, following the variation in pressure. At a temperature below that of the dispersion transition (T < T _d = 284 K), e.g., at T = 277 K, the degree of filling θ decreases from 1 to 0.8 within 10 s. The degree of filling varies with time according to the power law θ ~ t ^–α with the exponent α < 0.1 over a period of t ~ 10⁵ s. This process corresponds to slow relaxation of a metastable state of a nonwetting liquid in a porous medium. At times t > 10⁵ s, the metastable state exhibits decay, manifested as the transition to a power dependence of θ(t) with a larger exponent. The relaxation of the metastable state of nonwetting liquid in a disordered porous medium is described in the mean field approximation as a continuous sequence of metastable states with a barrier decreasing upon a decrease in the degree of filling. Using this approach, it is possible to qualitatively explain the observed relaxation process and crossover transition to the stage described by θ(t) with a larger exponent.     

<Emphasis Type="Italic">β</Emphasis>-distribution for Reynolds stress and turbulent heat flux in relaxation turbulent boundary layer of compression ramp

A challenge in the study of turbulent boundary layers (TBLs) is to understand the non-equilibrium relaxation process after sep-aration and reattachment due to shock-wave/boundary-layer interaction. The classical boundary layer theory cannot deal with the strong adverse pressure gradient, and hence, the computational modeling of this process remains inaccurate. Here, we report the direct numerical simulation results of the relaxation TBL behind a compression ramp, which reveal the presence of intense large-scale eddies, with significantly enhanced Reynolds stress and turbulent heat flux. A crucial finding is that the wall-normal profiles of the excess Reynolds stress and turbulent heat flux obey a β-distribution, which is a product of two power laws with respect to the wall-normal distances from the wall and from the boundary layer edge. In addition, the streamwise decays of the excess Reynolds stress and turbulent heat flux also exhibit power laws with respect to the streamwise distance from the corner of the compression ramp. These results suggest that the relaxation TBL obeys the dilation symmetry, which is a specific form of self-organization in this complex non-equilibrium flow. The β-distribution yields important hints for the development of a turbulence model.     

Absorption of intense regular and noise waves in relaxing media

An integro-differential equation is written down that contains terms responsible for nonlinear absorption, visco-heat-conducting dissipation, and relaxation processes in a medium. A general integral expression is obtained for calculating energy losses of the wave with arbitrary characteristics—intensity, profile (frequency spectrum), and kernel describing the internal dynamics of the medium. It is shown that for weak waves, the general integral leads to well-known results of a linear approximation. Profiles of stationary solutions are constructed both for an exponential relaxation kernel and for other types of kernels. Energy losses at the front of week shock waves are calculated. General integral formulas are obtained for energy losses of intense noise, which are determined by the form of the kernel, the structure of the noise correlation function, and the mean square of the derivative of realization of a random process.     

Parallel cluster labeling for large-scale Monte Carlo simulations

We present an optimized version of a cluster labeling algorithm previously introduced by the authors. This algorithm is well suited for large-scale Monte Carlo simulations of spin models using cluster dynamics on parallel computers with large numbers of processors. The algorithm divides physical space into rectangular cells which are assigned to processors and combines a serial local labeling procedure with a relaxation process across nearest-neighbor processors. By controlling overhead and reducing inter-processor communication this method attains good computational speed-up and efficiency. Large systems of up to 65536² spins have been simulated at updating speeds of 11 nanosecs/site (90.7 × 10⁶ spin updates/sec) using state-of-the-art supercomputers. In the second part of the article we use the cluster algorithm to study the relaxation of magnetization and energy on large Ising models using Swendsen-Wang dynamics. We found evidence that exponential and power law factors are present in the relaxation process as has been proposed by Hackl et al. The variation of the power-law exponent λ_M taken at face value indicates that the value of Z_M falls in the interval 0.31–0.49 for the time interval analysed and appears to vanish asymptotically.     

Unified index to quantifying heterogeneity of complex networks

Although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function, a unified definition of the heterogeneity of a network with any degree distribution is absent. In this paper, we define a heterogeneity index 0≤H<1 to quantify the degree heterogeneity of any given network. We analytically show the existence of an upper bound of H=0.5 for exponential networks, thus explain why exponential networks are homogeneous. On the other hand, we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0.5 if and only if its degree exponent is between 2 and 2.5. We further show that for any power law network with a degree exponent greater than 2.5, there always exists an exponential network such that both networks have the same heterogeneity index. This may help to explain why 2.5 is a critical degree exponent for some dynamic behaviors on power law networks.     

Compton scattering of an X-ray photon by an open-shell atom

A nonrelativistic quantum theory for the nonresonant Compton scattering of an X-ray photon by a free many-electron atom with an open shell in the ground state has been constructed in the single-configuration Hartree-Fock approximation outside the impulse approximation widely used in the literature. The transition to an atom with closed shells reproduces the results obtained previously in [6, 7]. The results of a test calculation for atoms with open (Ti, Fe) and closed (Zn) 3d core shells are presented. The effects of the radial relaxation of one-electron states in the field of core vacancies have been taken into account. The results of the calculation agree well with the experimental results [15, 16]. It has been established that the results of the impulse approximation in the investigated X-ray photon energy ranges disagree with those of our theory not only quantitatively but also qualitatively. In particular, the impulse approximation near the elastic (Thomson and Rayleigh) scattering line leads to a gross overestimation of the contributions from the deep atomic shells involved in the inelastic photon scattering only virtually to the scattering probability. The presented theory is general in character and its applicability to a particular element of the Mendeleev table with an open core shell or to a many-electron atomic ion is limited only by the requirement that the nonrelativistic Hartree-Fock approximation be properly used in describing the scattering-state wave functions.     

Optical control of the coherent dynamics of a bismuth lattice at liquid-helium temperature at low and high excitation levels

The fully symmetric A_1g phonons of bismuth have been investigated at liquid-helium temperature by the coherent control method for various crystal excitation levels. It has been found that large-amplitude coherent phonons exhibit the “rigidity” of the phase, which is absent at a small amplitude. The impossibility of changing the phase of phonon oscillations appears at the excitation levels at which their amplitude relaxation law changes from exponential to power. The modification of the phase properties and relaxation law of the excitations of the crystal lattice can be understood in terms of the concept of the condensation of phonons, which occurs with an increase in the crystal excitation level.     

Dielectric relaxation of the cowle-cowle type and self-similar relaxation processes

In order to describe relaxation processes not obeying an exponential law a model of a self-similar relaxation process is proposed which is described by an equation containing fractional differentiation operators. It turns out that the complex susceptibility corresponding to such a model system has a form which agrees with the known empirical Cowle—Cowle expression. Kazan State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 6–11, April, 1997.     

A study of molecular motion and Raman processes in K3H(SO4)2 and KHSO4 single crystals by observation of H and K spin-lattice relaxation

The spin-lattice relaxation rates for ¹H and ³⁹K nuclei in K₃H(SO₄)₂ and KHSO₄ single crystals, which are potential candidate materials for use in fuel cells, were determined as a function of temperature. The spin-lattice relaxation recovery of ¹H can be represented for both crystals with a single exponential function, but cannot be represented by the Bloembergen-Purcell-Pound (BPP) function, so is not related to HSO₄ motion. The recovery traces of ³⁹K, which predominantly undergoes quadrupole relaxation, can be represented by a linear combination of two exponential functions. The temperature dependences of the relaxation rates for ³⁹K can be described with a simple power law T₁⁻¹=αT². The spin-lattice relaxation rates for the ³⁹K nucleus in K₃H(SO₄)₂ and KHSO₄ crystals are in accordance with a Raman process dominated by a phonon mechanism.     

Bianchi Type-II Dark Energy Model in f(R,T) Gravity

The spatially homogeneous and totally anisotropic Bianchi Type-II space-time dark energy model with EoS parameter is considered in the presence of a perfect fluid source in the framework of f(R,T) gravity proposed by Harko et al. (Phys. Rev. D, 84:024020, 2011). With the help of special law of variation for Hubble’s parameter proposed by Berman (Nuovo Cimento B, 74:182, 1983) a dark energy cosmological model is obtained in this theory. We consider f(R,T) model and investigate the modification R+f(T) in Bianchi type-II cosmology with an appropriate choice of a function f(T)=λT. We use the power law relation between average Hubble parameter H and average scale factor R to find the solution. The assumption of constant deceleration parameter leads to two models of universe, i.e. power law model and exponential model. Some physical and kinematical properties of the model are also discussed.