Fluctuation-dissipation relation for nonlinear Langevin equations

It is shown that the fluctuation-dissipation theorem is satisfied by the solutions of a general set of nonlinear Langevin equations with a quadratic free-energy functional (constant susceptibility) and field-dependent kinetic coefficients, provided the kinetic coefficients satisfy the Onsager reciprocal relations for the irreversible terms and the antisymmetry relations for the reversible terms. The analysis employs a perturbation expansion of the nonlinear terms, and a functional integral calculation of the correlation and response functions, and it is shown that the fluctuation-dissipation relation is satisfied at each order in the expansion.

Thermo and photoluminescence properties of Eu3+ activated hexagonal, monoclinic and cubic gadolinium oxide nanorods

Different phases of Eu3+ activated gadolinium oxide (Gd (OH)(3), GdOOH and Gd2O3) nanorods have been prepared by the hydrothermal method with and without cityl trimethyl ammonium bromide (GAB) surfactant. Cubic Gd2O3:Eu (8 mol%) red phosphor has been prepared by the dehydration of corresponding hydroxide Gd(OH)(3):Eu after calcinations at 350 and 600 degrees C for 3 h, respectively. When Eu3+ ions were introduced into Gd(OH)(3), lattice sites which replace the original Gd3+ ions, a strong red emission centered at 613 nm has been observed upon UV illumination, due to the intrinsic Eu3+ transition between D-5(0) and F-7 configurations. Thermoluminescence glow curves of Gd (OH)(3): Eu and Gd2O3:Eu phosphors have been recorded by irradiating with gamma source ((CO)-C-60) in the dose range 10-60 Gy at a heating rate of 6.7 degrees C sec(-1). Well resolved glow peaks in the range 42-45, 67-76,95-103 and 102-125 degrees C were observed. When gamma-irradiation dose increased to 40 Gy, the glow peaks were reduced and with increase in gamma-dose (50 and 60 Gy) results the shift in first two glow peak temperatures at about 20 degrees C and a new shouldered peak at 86 degrees C was observed. It is observed that there is a shift in glow peak temperatures and variation in intensity, which is mainly attributed to different phases of gadolinium oxide. The trapping parameters namely activation energy (E), order of kinetics (b) and frequency factor were calculated using peak shape and the results are discussed. (C) 2010 Elsevier B.V. All rights reserved.

The Adomian method applied to some extraordinary differential equations

Several ''extraordinary'' differential equations are considered for their solutions via the decomposition method of Adomian. Verifications are made with the solutions obtained by other methods.

Direct Cartesian-Space Solutions of Generalized Bloch Equations in the Rotating Frame

Analytical solutions of the generalized Bloch equations for an arbitrary set of initial values of the x, y, and z magnetization components are given in the rotating frame. The solutions involve the decoupling of the three coupled differential equations such that a third-order differential equation in each magnetization variable is obtained. In contrast to the previously reported solutions given by Torrey, the present attempt paves the way for more direct physical insight into the behavior of each magnetization component. Special cases have been discussed that highlight the utility of the general solutions. Representative trajectories of magnetization components are given, illustrating their behavior with respect to the values of off-resonance and initial conditions. (C) 1995 Academic Press, Inc.

Similarity solution of unsteady boundary layer equations with a magnetic field

The unsteady laminar incompressible boundary layer flow of an electrically conducting fluid in the stagnation region of two-dimensional and axisymmetric bodies with an applied magnetic field has been studied. The boundary layer equations which are parabolic partial differential equations with three independent variables have been reduced to a system of ordinary differential equations by using suitable transformations and then solved numerically using a shooting method. Here, we have obtained new solutions which are solutions of both the boundary layer and Navier-Stokes equations.

The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

Analytical solutions to problems in finite elasticity are most often derived using the semi-inverse approach along with the spatial form of the equations of motion involving the Cauchy stress tensor. This procedure is somewhat indirect since the spatial equations involve derivatives with respect to spatial coordinates while the unknown functions are in terms of material coordinates, thus necessitating the use of the chain rule. In this classroom note, we derive compact expressions for the components of the divergence, with respect to orthogonal material coordinates, of the first Piola-Kirchhoff stress tensor. The spatial coordinate system is also assumed to be an orthogonal curvilinear one, although, not necessarily of the same type as the material coordinate system. We show by means of some example applications how analytical solutions can be derived more directly using the derived results.

Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations

We study small perturbations of three linear Delay Differential Equations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reduction as a first step. We demonstrate that the method of multiple scales, on simply discarding the infinitely many exponentially decaying components of the complementary solutions obtained at each stage of the approximation, can bypass the explicit center manifold calculation. Analytical approximations obtained for the DDEs studied closely match numerical solutions.

An extended Cahn-Hilliard model for interfaces with cubic anisotropy

For studying systems with a cubic anisotropy in interfacial energy sigma, we extend the Cahn-Hilliard model by including in it a fourth-rank term, namely, gamma (ijlm) [partial derivative (2) c/(partial derivativex(i) partial derivativex(j))] [partial derivative (2) c/(partial derivativex(l) partial derivativex(m))]. This term leads to an additional linear term in the evolution equation for the composition parameter field. It also leads to an orientation-dependent effective fourth-rank coefficient gamma ([hkl]) in the governing equation for the one-dimensional composition profile across a planar interface. The main effect of a non-negative gamma ([hkl]) is to increase both sigma and interfacial width w, each of which, upon suitable scaling, is related to gamma ([hkl]) through a universal scaling function. In this model, sigma is a differentiable function of interface orientation (n) over cap, and does not exhibit cusps; therefore, the equilibrium particle shapes (Wulff shapes) do not contain planar facets. However, the anisotropy in the interfacial energy can be large enough to give rise to corners in the Wulff shapes in two dimensions. In particles of finite sizes, the corners become rounded, and their shapes tend towards the Wulff shape with increasing particle size.

Pseudodynamic systems approach based on a quadratic approximation of update equations for diffuse optical tomography

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associated with diffuse optical tomography, adoption of the quadratic update scheme combined with the pseudotime integration appears not only to yield higher convergence, but also a muted sensitivity to the regularization parameters, which include the pseudotime step size for integration. These observations are validated through reconstructions with both numerically generated and experimentally acquired data. (C) 2011 Optical Society of America

A Multidimensional Kinetic Upwind Method for Euler Equations

The study of directional derivative lead to the development of a rotationally invariant kinetic upwind method (KUMARI)3 which avoids dimension by dimension splitting. The method is upwind and rotationally invariant and hence truly multidimensional or multidirectional upwind scheme. The extension of KUMARI to second order is as well presented.

Quasichemical equations for oxygen and sulphur in liquid binary alloys

An equation has been derived for predicting the activity coefficient of oxygen or sulphur in dilute solution in binary alloys, based on the quasichemical approach, where the metal atoms and the oxygen atoms are assigned different bond numbers. This equation is an advance on Alcock and Richardson's earlier treatment where all the three types of atoms were assigned the same coordination number. However, the activity coefficients predicted by this new equation appear to be very similar to those obtained through Alcock and Richardson's equation for a number of alloy systems, when the coordination number of oxygen in the new model is the same as the average coordination number used in the earlier equation. A second equation based on the formation of “molecular species” of the type XnO and YnO in solution is also derived, where X and Y atoms attached to oxygen are assumed not to make any other bonds. This equation does not fit experimental data in all the systems considered for a fixed value of n. Howover, if the strong oxygen-metal bonds are assumed to distort the electronic configuation around the metal atoms bonded to oxygen and thus reduce the strength of the bonds formed by these atoms with neighbouring metal atoms by approximately a factor of two, the resulting equation is found to predict the activity coefficients of oxygen that are in good agreement with experimental data in a number of binary alloys.

Gibbs free energies of formation of ZnAl2O4 and ZnCr2O4

The standard free energies of formation of zinc aluminate and chromite were determined by measuring the oxygen potential over a solid CuZn alloy, containing 10 at.−% Zn, in equilibrium with ZnO, ZnAl2O4+Al2O3(χ) and ZnCr2O4+Cr2O3, in the temperature range 700–900°C. The oxygen potential was monitored by means of a solid oxide galvanic cell in which a Y2O3 ThO2 pellet was sandwiched between a CaOZrO2 crucible and tube. The temperature dependence of the free energies of formation of the interoxidic compounds can be represented by the equations, The heat of formation of the spinels calculated from the measurements by the “Second Law method” is found to be in good agreement with calorimetrically determined values. Using an empirical correlation for the entropy of formation of cubic spinel phases from oxides with rock-salt and corundum structures and the measured high temperature cation distribution in ZnAl2O4, the entropy of transformation of ZnO from wurtzite to rock-salt structure is evaluated.