Noether symmetry in the higher order gravity theory

Noether symmetry for scalar tensor theory including curvature quadratic term has been explored, with the introduction of an auxiliary variable. Introduction of an auxiliary variable in the action facilitates in transforming the fourth order field equations to the second order field equations. Introduction of Noether symmetry in the action yield the coupling function () and the potential V(). The application of Noether symmetry further turned out to be powerful tool to find the solution of the field equations. A few physically reasonable solutions like power law inflation are presented.     

A mode coupling theory of higher order time correlation functions

Kawasaki's mode coupling theory [Ann. Phys. 61 (1970) 1] is used to compute time correlation functions of the form 〈A_k0(t₀) … A_kn(t_n)〉, where A_k(t) represents some slowly varying quantity. The Gaussian and Bare Vertex approximations are made, thus yielding extremely simple expressions for these higher order correlation functions. These do not contain any bare transport coefficients and suggest relatively simple tests by which the theory could be checked. Examples relating to light scattering in nonequilibrium systems and the hydrodynamics of simple fluids are presented.     

Effective elastic moduli under hydrostatic stress—I. quasi-harmonic theory

Fourth-order finite strain expressions for the effective elastic moduli of a solid under hydrostatic stress are derived from a general expression for effective elastic moduli. Expressions in terms of the strain tensors E and η are given. The expressions are then written in terms of the moduli and their pressure derivatives evaluated at an arbitrary reference state. The temperature dependence of these expressions is derived from the fourth-order quasi-harmonic expression for the lattice vibrational energy. Some general thermodynamic relations are derived between the parameters which specify the thermal effects and the pressure and temperature derivatives of the elastic moduli at the reference state. General relations between isothermal and isentropic elastic moduli and their pressure and temperature derivatives are also given. Much of the development is valid for materials of arbitrary symmetry, but the complete development is given only for materials of cubic symmetry.     

Ginzburg-Landau theory for higher order phase transition

Ginzburg-Landau theory for studying phase transitions of higher order has been derived using coarse graining and lattice formulation within Ehrenfest thermodynamics. Our developed Hamiltonian leads directly to the functional of the system. We studied the evolution of the order parameter using our developed model equations for third and fourth order phase transitions. The periodic nature of the system can be likened to spatially varying periodic soliton/antisoliton lattice of holes in condensate. This is different from what one observes for any conventional solitary wave in the second order phase regime.     

On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals

A randomly inhomogeneous material may have macroscopic properties (elasticity, conductivity) scattered over some uncertainty intervals, despite the idealistic uniqueness assumption of homogenization theory. Based on minimum energy principles and certain statistical isotropy-symmetry hypotheses, our partly third-order bounds on the effective properties of random polycrystals are expected to estimate those scatter ranges. Explicit expressions are given and calculated for the elastic moduli of the random aggregates of some known monoclinic and triclinic crystals, which yield results in agreement with those calculated for higher-symmetry crystals: the moduli are determinable within an accuracy of two or three significant digits in most cases. It is shown, however, that with some real-world exotic crystals the bounds may fall far apart, and further theoretical and experimental studies on them deserve attention.     

A higher order dynamic theory for viscoelastic plates and layered composites

By using a new technique proposed by the first author [1] approximate theories are developed for the dynamic response of viscoelastic plates and layered composites. The originality of the new technique lies in the fact that it permits the approximate theory to satisfy correctly the lateral boundary conditions of a plate, or the interface (continuity) conditions of a layered composite. This, in turn, enables the approximate theory to describe accurately the geometric dispersion of waves propagating in a plate or layered composite. The approximate equations of a single viscoelastic plate are first derived by making use of the new technique. To develop the approximate theory for viscoelastic layered composites made of two alternating layers it is noted that the approximate equations of a single plate already established also hold in each layer of the composite. The theory is completed by adding the continuity conditions to these equations and using a smoothing operation. The equations thus obtained constitute a continuum (homogeneous) model (CM) which simplifies the determination of the dynamic response of viscoelastic layered composites when the number of layers in the composite is large. The proposed approximate theories are open to improvement in the sense that their region of validity in the wave number-frequency plane can be enlarged as much as one wishes by increasing the orders of the theories and continuity conditions.     

Near integrability of kink lattice with higher order interactions

We make use of Manton's analytical method to investigate the force between kinks and anti-kinks at large distances in 1+1 dimensional field theory.The related potential has infinite order corrections of exponential pattern,and the coefficients for each order are determined.These coefficients can also be obtained by solving the equation of the fluctuations around the vacuum.At the lowest order,the kink lattice represents the Toda lattice.With higher order correction terms,the kink lattice can represent one kind of generic Toda lattice.With only two sites,the kink lattice is classically integrable.If the number of sites of the lattice is larger than two,the kink lattice is not integrable but is a near integrable system.We make use of Flaschka's variables to study the Lax pair of the kink lattice.These Flaschka's variables have interesting algebraic relations and non-integrability can be manifested.We also discuss the higher Hamiltonians for the deformed open Toda lattice,which has a similar result to the ordinary deformed Toda.     

Statistical mechanics of elastic moduli

We consider continuous systems of particles in the framework of classical statistical mechanics and derive a general expression for the static elastic moduli tensor in terms of correlation functions. We find sufficient conditions for the vanishing of the shear modulus. Relationships between these conditions and others insuring translational or rotational invariance are discussed.     

Third-order elastic moduli of single-layer graphene

In the framework of the Keating model with allowance made for the anharmonic constant of the central interaction between the nearest neighbors μ, analytical expressions have been obtained for three third-order independent elastic constants c _ijk (μ, ζ) of single-layer graphene, where ζ = (2α − β)/(4α + β) is the Kleinman internal displacement parameter and α and β are the harmonic constants of the central interaction between the nearest neighbors and the noncentral interaction between the next-nearest neighbors, respectively. The dependences of the second-order elastic constants on the pressure p have been determined. It has been shown that the moduli c ₁₁ and c ₂₂ differently respond to the pressure. Therefore, graphene is isotropic in the harmonic approximation, whereas the inclusion of anharmonicity leads to the appearance of the anisotropy.     

Zero-frequency elastic moduli of uniform fluids

A statistical mechanical treatment of equilibrium elasticity of a uniform fluid phase based on density functional theory is presented. Bulk expressions for the stress tensor and the zero-frequency elastic moduli tensor involving the direct correlation function are found.     

Fourth-order contributions to μμ elastic scattering in the gauge theory of weak interactions

The fourth-order contributions of WW, Z? and ZZ intermediate states to the S-matrix elements of μμ elastic scattering have been calculated by dispersion theoretic techniques in the U-gauge. It is shown that the contributions are finite and very small compared to the low-order contributions. It is also shown that the main part of these finite contributions comes from the WW intermediate state.     

Self-consistent theory of second order elastic constants for nonionic anharmonic crystals

Elastic constants of strongly anharmonic nonionic dielectric crystals are studied within the pseudoharmonic approximation by means of two-time Green functions. The relation between the method of homogeneous deformation by Leibfried and Ludwig and that of long waves is investigated. These considerations lead to a generalization of the method of homogeneous deformation for the case of strongly anharmonic crystals. A comparison between the results of the two methods shows that the pseudoharmonic approximation satisfies the exact elastic sum rule. On leave from the Institute of Theoretical Physics University of Wrocław, Wrocław, Poland.