Showing papers on "Equations of motion published in 2009"

On the Extra Mode and Inconsistency of Horava Gravity

Abstract: We address the consistency of Horava's proposal for a theory of quantum gravity from the low-energy perspective. We uncover the additional scalar degree of freedom arising from the explicit breaking of the general covariance and study its properties. The analysis is performed both in the original formulation of the theory and in the Stueckelberg picture. A peculiarity of the new mode is that it satisfies an equation of motion that is of first order in time derivatives. At linear level the mode is manifest only around spatially inhomogeneous and time-dependent backgrounds. We find two serious problems associated with this mode. First, the mode develops very fast exponential instabilities at short distances. Second, it becomes strongly coupled at an extremely low cutoff scale. We also discuss the "projectable" version of Horava's proposal and argue that this version can be understood as a certain limit of the ghost condensate model. The theory is still problematic since the additional field generically forms caustics and, again, has a very low strong coupling scale. We clarify some subtleties that arise in the application of the Stueckelberg formalism to Horava's model due to its non-relativistic nature.

The Euler equations as a differential inclusion

Abstract: We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in R n with n 2. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

On the extra mode and inconsistency of Hořava gravity

Abstract: We address the consistency of Hořava's proposal for a theory of quantum gravity from the low-energy perspective. We uncover the additional scalar degree of freedom arising from the explicit breaking of the general covariance and study its properties. The analysis is performed both in the original formulation of the theory and in the Stuckelberg picture. A peculiarity of the new mode is that it satisfies an equation of motion that is of first order in time derivatives. At linear level the mode is manifest only around spatially inhomogeneous and time-dependent backgrounds. We find two serious problems associated with this mode. First, the mode develops very fast exponential instabilities at short distances. Second, it becomes strongly coupled at an extremely low cutoff scale. We also discuss the ``projectable'' version of Hořava's proposal and argue that this version can be understood as a certain limit of the ghost condensate model. The theory is still problematic since the additional field generically forms caustics and, again, has a very low strong coupling scale. We clarify some subtleties that arise in the application of the Stuckelberg formalism to Hořava's model due to its non-relativistic nature.

Light-Front Holography: A First Approximation to QCD

Abstract: Starting from the Hamiltonian equation of motion in QCD, we identify an invariant light-front coordinate {zeta} which allows the separation of the dynamics of quark and gluon binding from the kinematics of constituent spin and internal orbital angular momentum. The result is a single variable light-front Schroedinger equation for QCD which determines the eigenspectrum and the light-front wavefunctions of hadrons for general spin and orbital angular momentum. This light-front wave equation is equivalent to the equations of motion which describe the propagation of spin-J modes on anti-de Sitter (AdS) space. This allows us to establish formally a gauge/gravity correspondence between an effective gravity theory defined on AdS5 and light front QCD.

Oblique Whistler-Mode Waves in the Inhomogeneous Magnetospheric Plasma: Resonant Interactions with Energetic Charged Particles

Abstract: A consistent account is given of the theory of resonant interactions between energetic charged particles and a whistler-mode wave propagating obliquely to the non-uniform geomagnetic field in the inhomogeneous magnetospheric plasma. The basic equations for the wave field and charged particle dynamics are presented, with the emphasis being placed on the parameters governing the problem. A Hamiltonian approach is consistently used in the analysis of the particle equations of motion which are discussed in detail and solved analytically in various cases. Two applications of the theory are considered. First, we calculate the growth (or damping) rate for a whistler-mode wave propagating obliquely to geomagnetic field in the magnetosphere. Secondly, we estimate the proton precipitation into the upper atmosphere induced by a VLF transmitter signal.

Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations

Abstract: We discuss the use of a Langevin equation with a colored (correlated) noise to perform constant-temperature molecular dynamics. Since the equations of motion are linear in nature, it is easy to predict the response of a Hamiltonian system to such a thermostat and to tune at will the relaxation time of modes of different frequency. This allows one to optimize the time needed for equilibration and to generate independent configurations. We show how this frequency-dependent response can be exploited to control the temperature of Car-Parrinello-like dynamics without affecting the adiabatic separation of the electronic degrees of freedom from the vibrations of the ions.

A fast new public code for computing photon orbits in a kerr spacetime

Abstract: Relativistic radiative transfer problems require the calculation of photon trajectories in curved spacetime. We present a novel technique for rapid and accurate calculation of null geodesics in the Kerr metric. The equations of motion from the Hamilton-Jacobi equation are reduced directly to Carlson's elliptic integrals, simplifying algebraic manipulations and allowing all coordinates to be computed semianalytically for the first time. We discuss the method, its implementation in a freely available FORTRAN code, and its application to toy problems from the literature.

Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets

Abstract: We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal $L^2$ basis and preserve their type under propagation in Schrodinger equations with quadratic potentials. We build a fully explicit, time-reversible time-stepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semiclassical limit. It reduces to the Strang splitting of the Schrodinger equation in the limit of the full basis set, and it advances positions and momenta by the Stormer-Verlet method for the classical equations of motion. The algorithm allows for the treatment of multiparticle problems by thinning out the basis according to a hyperbolic cross approximation and of high-dimensional problems by Hartree-type approximations in a moving coordinate frame.

Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution

Abstract: Basing on the First-order Shear Deformation Theory (FSDT), this paper focuses on the dynamic behaviour of moderately thick functionally graded parabolic panels and shells of revolution. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some symmetric and asymmetric material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. The governing equations of motion are expressed as functions of five kinematic parameters. For the discretization of the system equations the Generalized Differential Quadrature (GDQ) method has been used. Numerical results concerning four types of parabolic shell structures illustrate the influence of the parameters of the power-law distribution on the mechanical behaviour of shell structures considered.

Delocalization of wave packets in disordered nonlinear chains.

Abstract: We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schrodinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy the Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single-site, single-mode, and general finite-size excitations and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.

Peridynamics as an Upscaling of Molecular Dynamics

Abstract: Peridynamics is a formulation of continuum mechanics based on integral equations. It is a nonlocal model, accounting for the effects of long-range forces. Correspondingly, classical molecular dynamics is also a nonlocal model. Peridynamics and molecular dynamics have similar discrete computational structures, as peridynamics computes the force on a particle by summing the forces from surrounding particles, similarly to molecular dynamics. We demonstrate that the peridynamics model can be cast as an upscaling of molecular dynamics. Specifically, we address the extent to which the solutions of molecular dynamics simulations can be recovered by peridynamics. An analytical comparison of equations of motion and dispersion relations for molecular dynamics and peridynamics is presented along with supporting computational results.

A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation

Abstract: A numerical algorithm for fully dynamical lubrication problems based on the Elrod― Adams formulation of the Reynolds equation with mass-conserving boundary conditions is described. A simple but effective relaxation scheme is used to update the solution maintaining the complementarity conditions on the variables that represent the pressure and fluid fraction. The equations of motion are discretized in time using Newmark's scheme, and the dynamical variables are updated within the same relaxation process just mentioned. The good behavior of the proposed algorithm is illustrated in two examples: an oscillatory squeeze flow (for which the exact solution is available) and a dynamically loaded journal bearing. This article is accompanied by the ready-to-compile source code with the implementation of the proposed algorithm.

Self-Consistent Cosmological Simulations of DGP Braneworld Gravity

Abstract: We perform cosmological N-body simulations of the Dvali-Gabadadze-Porrati (DGP) braneworld model, by solving the full nonlinear equations of motion for the scalar degree of freedom in this model, the brane-bending mode. While coupling universally to matter, the brane-bending mode has self-interactions that become important as soon as the density field becomes nonlinear. These self-interactions lead to a suppression of the field in high-density environments, and restore gravity to general relativity. The code uses a multigrid relaxation scheme to solve the nonlinear field equation in the quasistatic approximation. We perform simulations of a flat self-accelerating DGP model without cosmological constant. However, the type of nonlinear interactions of the brane-bending mode, which are the focus of this study, are generic to a wide class of braneworld cosmologies. The results of the DGP simulations are compared with standard gravity simulations assuming the same expansion history, and with DGP simulations using the linearized equation for the brane-bending mode. This allows us to isolate the effects of the nonlinear self-couplings of the field which are noticeable already on quasilinear scales. We present results on the matter power spectrum and the halo mass function, and discuss the behavior of the brane-bending mode within cosmological structure formation. We find that, independently of cosmic microwave background constraints, the self-accelerating DGP model is strongly constrained by current weak lensing and cluster abundance measurements.

Small-scale effect on the free in-plane vibration of nanoplates by nonlocal continuum model

Abstract: In the present paper, small-scale effects on the free in-plane vibration (FIV) of nanoplates are investigated employing nonlocal continuum mechanics. Equations of motion of the nonlocal plate model for the aforementioned study are derived and presented. Explicit relations for natural frequencies are obtained through direct separation of variables. It has been shown that nonlocal effects are quite significant in in-plane vibration studies and need to be included in the continuum model of nanoplates such as in graphene sheets.

Fermion localization and resonances on two-field thick branes

Abstract: We consider (4, 1)-dimensional branes constructed with two scalar fields $\ensuremath{\phi}$ and $\ensuremath{\chi}$ coupled to a Dirac spinor field by means of a general Yukawa coupling. The equation of motion for the coefficients of the chiral decomposition of the spinor in curved spacetime leads to a Schr\"odinger-like equation whose solutions allow to obtain the masses of the fermionic modes. The simplest Yukawa coupling $\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\chi}\ensuremath{\Psi}$ is considered for the Bloch brane model and fermion localization is studied. We found resonances for both chiralities and related their appearance to branes with internal structure.

Thermal noise and stochastic strings in AdS/CFT

Abstract: We clarify the structure of thermal noise in AdS/CFT by studying the dynamics of an equilibrated heavy quark string. Using the Kruskal extension of the correspondence to generate the dynamics of the field theory on the Keldysh contour, we show that the motion of the string is described by the classical equations of motion with a stochastic boundary condition on the stretched horizon. The form of the stochastic boundary condition is consistent with the dissipation on this surface and is found by integrating out the fluctuations inside of the stretched horizon. Solving the equations of motion for the fluctuating string we determine the full frequency dependence of the random force on the boundary quark and show that it is consistent with the frequency dependent dissipation. We show further that the stochastic motion reproduces the bulk to bulk two point functions of the Kruskal formalism. These turn out to be related to the usual retarded bulk to bulk propagator by KMS relations. Finally we analyze the stochastic equations and give a bulk picture of the random boundary force as a flip-flopping trailing string solution. The basic formalism can be applied to the fluctuations of gravitons, dilatons, and other fields.

Relativistic Viscous Fluid Dynamics and Non-Equilibrium Entropy

Abstract: Fluid dynamics corresponds to the dynamics of a substance in the long wavelength limit. Writing down all terms in a gradient (long wavelength) expansion up to second order for a relativistic system at vanishing charge density, one obtains the most general (causal) equations of motion for a fluid in the presence of shear and bulk viscosity, as well as the structure of the non-equilibrium entropy current. Requiring positivity of the divergence of the non-equilibrium entropy current relates some of its coefficients to those entering the equations of motion. I comment on possible applications of these results for conformal and non-conformal fluids.

On the stability of elliptical vortices in accretion discs

Abstract: Context. The existence of large-scale and long-lived 2D vortices in accretion discs has been debated for more than a decade. They appear spontaneously in several 2D disc simulations and they are known to accelerate planetesimal formation through a dust trapping process. In some cases, these vortices may even lead to an efficient way to transport angular momentum in protoplanetary discs when MHD instabilities are inoperative. However, the issue of the stability of these structures to the imposition of 3D disturbances is still not fully understood, and it casts doubts on their long term survival Aims. We present new results on the 3D stability of elliptical vortices embedded in accretion discs, based on a linear analysis and several non-linear simulations. Methods. We introduce a simple steady 2D vortex model which is a non-linear solution of the equations of motion, and we show that its core is made of elliptical streamlines. We then derive the linearised equations governing the 3D perturbations in the core of this vortex, and we show that they can be reduced to a Floquet problem. We solve this problem numerically in the astrophysical regime, including a simplified model to take into account vertical stratification effects. We present several analytical limits for which the mechanism responsible for instability can be explained. Finally, we compare the results of the linear analysis to some high resolution numerical simulations obtained with spectral and finite difference methods. A discussion is provided, emphasising the astrophysical consequences of our findings for the dynamics of vortices. Results. We show that most anticyclonic vortices are unstable due to a resonance between the turnover time and the local epicyclic oscillation period. A small linearly stable domain is found for vortex cores with an aspect-ratio of around 5. However, our simulations show that it is only the vortex core that is stable, with the instability still appearing on the vortex boundary. In addition, we find numerically that results obtained under the assumption of incompressibility are not affected by the introduction of a moderate compressibility. Finally, we show that a strong vertical stratification does not create any additional stable domain of aspect ratio, but it significantly reduces growth rates for relatively weak (and therefore elongated) vortices. Conclusions. Elliptical vortices are always unstable, whatever the horizontal or vertical aspect-ratio is. The instability can however be weak and is often found at small scales, making it difficult to detect in low-order finite-difference simulations.

Dissipative Solitary Waves in Granular Crystals

Abstract: We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a common dissipation exponent with a material-dependent prefactor.