Showing papers on "Conservation law published in 2009"

Emergence of the persistent spin helix in semiconductor quantum wells

Abstract: According to Noether's theorem, for every symmetry in nature there is a corresponding conservation law. For example, invariance with respect to spatial translation corresponds to conservation of momentum. In another well-known example, invariance with respect to rotation of the electron's spin, or SU(2) symmetry, leads to conservation of spin polarization. For electrons in a solid, this symmetry is ordinarily broken by spin-orbit coupling, allowing spin angular momentum to flow to orbital angular momentum. However, it has recently been predicted that SU(2) can be achieved in a two-dimensional electron gas, despite the presence of spin-orbit coupling. The corresponding conserved quantities include the amplitude and phase of a helical spin density wave termed the 'persistent spin helix'. SU(2) is realized, in principle, when the strengths of two dominant spin-orbit interactions, the Rashba (strength parameterized by alpha) and linear Dresselhaus (beta(1)) interactions, are equal. This symmetry is predicted to be robust against all forms of spin-independent scattering, including electron-electron interactions, but is broken by the cubic Dresselhaus term (beta(3)) and spin-dependent scattering. When these terms are negligible, the distance over which spin information can propagate is predicted to diverge as alpha approaches beta(1). Here we report experimental observation of the emergence of the persistent spin helix in GaAs quantum wells by independently tuning alpha and beta(1). Using transient spin-grating spectroscopy, we find a spin-lifetime enhancement of two orders of magnitude near the symmetry point. Excellent quantitative agreement with theory across a wide range of sample parameters allows us to obtain an absolute measure of all relevant spin-orbit terms, identifying beta(3) as the main SU(2)-violating term in our samples. The tunable suppression of spin relaxation demonstrated in this work is well suited for application to spintronics.

Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains - eScholarship

Abstract: Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains P.-O. Persson a J. Bonet band .J. Peraire c,* Department of Mathematics, University of California, Be'rkeley, Berkeley, CA 94720-3840, USA b School of Engineering, Swansea University, Swansea SA2 8PP, UK C Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-K utta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries. Key words: Discontinuous Galerkin, Deformable domains, Navier-Stokes, Arbitrary Lagrangian-Eulerian, Geometric Conservation 1. Introduction There is a growing interest in high-order methods for fluid problems, largely because of their ability to produce highly accurate solutions with minimum numerical dispersion. The Discontinuous Calerkin (DC) method produces stable discretizations of the convective operator for any order discretization, Moreover, it can be used with unstructured meshes of simplices, which appears to be a requirement for real-world complex geometries. In this paper, we present a high order DC formulation for computing high order solutions to problems with variable geometries. Time varying geometries appear in a number of practical applications such us rotor-stator flows, flapping flight or fluid-structure interactions, In such cases, it is necessary to properly account for the time variation of the solution domain if accurate solutions are to be obtained. For the Navier-Stokes equations, there has been a considerable effort in the development of Arbitrary Lagrangian Eulerian (ALE) methods to deal with * Corresponding author. Tel.: +1-617-25.3-1981: Fax.: +1-617-258-,514:3. Ematl address: peraire~mit .edu (J. Peraire). Preprint SUbJIlitted to Computer l\Jethods in Applied l\Jechanics and Engineering January 2009

Linearized model Fokker-Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests

Abstract: A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a model collision operator satisfying these properties is described. This operator is based on the exact linearized test-particle collision operator, with approximations to the field-particle terms that preserve conservation laws and an H-theorem. It includes energy diffusion, pitch-angle scattering, and finite Larmor radius effects corresponding to classical real-space diffusion. The numerical implementation in the continuum gyrokinetic code GS2 Kotschenreuther et al., Comput. Phys. Comm. 88, 128 1995 is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation. © 2009 American Institute of Physics. DOI: 10.1063/1.3155085

Quasivelocities and symmetries in non-holonomic systems

Abstract: This article is concerned with the theory of quasivelocities for non-holonomic systems. The equations of non-holonomic mechanics are derived using the Lagrange-d'Alembert principle written in an arbitrary configuration-dependent frame. The article also shows how quasivelocities may be used in the formulation of non-holonomic systems with symmetry. In particular, the use of quasivelocities in the analysis of symmetry that leads to unusual momentum conservation laws is investigated, as is the applications of these conservation laws and discrete symmetries to the qualitative analysis of non-holonomic dynamics. The relationship between asymptotic dynamics and discrete symmetries of the system is also elucidated.

Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem☆

Abstract: A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nother’s identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nother’s theorem leading to conservation laws for fractional Euler–Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler–Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws.

Tensor products and correlation estimates with applications to nonlinear Schrödinger equations

Abstract: We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schrodinger equations and derive a direct proof of Nakanishi's H 1 scattering result for every L 2 -supercritical nonlinearity. We also prove scattering below the energy space for a certain class of L 2 -supercritical equations. In this paper we obtain new 1 a priori estimates for solutions of the nonlinear Schrodinger equation in one and two dimension. We also provide a systematic way to obtain the known interaction a priori estimates for dimensions higher than three. These estimates are monotonicity formulae that take advantage of the conservation of the momentum of the equation. Due to the pioneering work (19), estimates of this type are referred to as Morawetz estimates in the literature. We then apply these estimates to study the global behavior of solutions to the nonlinear Schrodinger equation. To be more precise we want to study the global-in-time behavior of solutions to the following initial value problem

An Engquist-Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

Abstract: We consider scalar conservation laws with the spatially varying flux $H(x)f(u)+(1-H(x))g(u)$, where $H(x)$ is the Heaviside function and $f$ and $g$ are smooth nonlinear functions. Adimurthi, Mishra, and Veerappa Gowda [J. Hyperbolic Differ. Equ., 2 (2005), pp. 783-837] pointed out that such a conservation law admits many $L^1$ contraction semigroups, one for each so-called connection $(A,B)$. Here we define entropy solutions of type $(A,B)$ involving Kruzkov-type entropy inequalities that can be adapted to any fixed connection $(A,B)$. It is proved that these entropy inequalities imply the $L^1$ contraction property for $L^\infty$ solutions, in contrast to the “piecewise smooth” setting of Adimurthi, Mishra, and Veerappa Gowda. For a fixed connection, these entropy inequalities include a single adapted entropy of the type used by Audusse and Perthame [Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253-265]. We prove convergence of a new difference scheme that approximates entropy solutions of type $(A,B)$ for any connection $(A,B)$ if a few parameters are varied. The scheme relies on a modification of the standard Engquist-Osher flux, is simple as no $2\times2$ Riemann solver is involved, and is designed such that the steady-state solution connecting $A$ to $B$ is preserved. In contrast to most analyses of similar problems, our convergence proof is not based on the singular mapping or compensated compactness methods, but on standard spatial variation estimates away from the flux discontinuity. Some numerical examples are presented.

Time propagation of the Kadanoff-Baym equations for inhomogeneous systems.

Abstract: We have developed a time-propagation scheme for the Kadanoff-Baym equations for general inhomogeneous systems. These equations describe the time evolution of the nonequilibrium Green function for interacting many-body systems in the presence of time-dependent external fields. The external fields are treated nonperturbatively whereas the many-body interactions are incorporated perturbatively using Phi-derivable self-energy approximations that guarantee the satisfaction of the macroscopic conservation laws of the system. These approximations are discussed in detail for the time-dependent Hartree-Fock, the second Born, and the GW approximation.

Dynamics of emitting electrons in strong laser fields

Abstract: A new derivation of the motion of a radiating electron is given, leading to a formulation that differs from the Lorentz–Abraham–Dirac equation and its published modifications. It satisfies the proper conservation laws. Particularly, it conserves the generalized momentum, eliminating the symmetry-breaking runaway solution. The equation allows a consistent calculation of the electron current, the radiation effect on the electron momentum, and the radiation itself, for a single electron or plasma electrons in strong electromagnetic fields. The equation is then applied to a simulation of a strong laser pulse interaction with a plasma target. Some analytical solutions are also provided.

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

Abstract: In this paper, analytically investigated is a higher-order dispersive nonlinear Schrodinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

Fractional Electromagnetic Equations Using Fractional Forms

Abstract: The generalized physics laws involving fractional derivatives give new models and conceptions that can be used in complex systems having memory effects. Using the fractional differential forms, the classical electromagnetic equations involving the fractional derivatives have been worked out. The fractional conservation law for the electric charge and the wave equations were derived by using this method. In addition, the fractional vector and scalar potentials and the fractional Poynting theorem have been derived.

Maxwell's macroscopic equations, the energy-momentum postulates, and the Lorentz law of force

Abstract: We argue that the classical theory of electromagnetism is based on Maxwell's macroscopic equations, an energy postulate, a momentum postulate, and a generalized form of the Lorentz law of force. These seven postulates constitute the foundation of a complete and consistent theory, thus eliminating the need for actual (i.e., physical) models of polarization $\mathbf{P}$ and magnetization $\mathbf{M}$, these being the distinguishing features of Maxwell's macroscopic equations. In the proposed formulation, $\mathbf{P}(\mathbf{r},t)$ and $\mathbf{M}(\mathbf{r},t)$ are arbitrary functions of space and time, their physical properties being embedded in the seven postulates of the theory. The postulates are self-consistent, comply with the requirements of the special theory of relativity, and satisfy the laws of conservation of energy, linear momentum, and angular momentum. One advantage of the proposed formulation is that it sidesteps the long-standing Abraham-Minkowski controversy surrounding the electromagnetic momentum inside a material medium by simply ``assigning'' the Abraham momentum density $\mathbf{E}(\mathbf{r},t)\ifmmode\times\else\texttimes\fi{}\mathbf{H}(\mathbf{r},t)∕{c}^{2}$ to the electromagnetic field. This well-defined momentum is thus taken to be universal as it does not depend on whether the field is propagating or evanescent, and whether or not the host medium is homogeneous, transparent, isotropic, dispersive, magnetic, linear, etc. In other words, the local and instantaneous momentum density is uniquely and unambiguously specified at each and every point of the material system in terms of the $\mathbf{E}$ and $\mathbf{H}$ fields residing at that point. Any variation with time of the total electromagnetic momentum of a closed system results in a force exerted on the material media within the system in accordance with the generalized Lorentz law.

Dynamics of Emitting Electrons in Strong Electromagnetic Fields

Abstract: We derive a modified non-perturbative Lorentz-Abraham-Dirac equation. It satisfies the proper conservation laws, particularly, it conserves the generalized momentum, the latter property eliminates the symmetry-breaking runaway solution. The equation allows a consistent calculation of the electron current, the radiation effect on the electron momentum, and the radiation itself, for a single electron or plasma electrons in strong electromagnetic fields. The equation is applied to a simulation of a strong laser pulse interaction with a plasma target. Some analytical solutions are also provided.