Global Conservation Laws Research Articles

Normal scaling in globally conserved interface-controlled coarsening of fractal clusters.

We find that globally conserved interface-controlled coarsening of diffusion-limited aggregates exhibits dynamic scale invariance (DSI) and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The general sharp-interface limit of this equation is introduced and reduced to volume preserving motion by mean curvature. A simple example of globally conserved interface-controlled coarsening system: the sublimation/deposition dynamics of a solid and its vapor in a small closed vessel, is presented in detail. The results of the numerical simulations show that the scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal dynamic scaling. A decrease of the cluster radius with time, predicted by DSI, is observed. The difference between global and local conservation is discussed.

Area-preserving dynamics of a long slender finger by curvature: a test case for globally conserved phase ordering.

A long and slender finger can serve as a simple "test bed" for different phase-ordering models. In this work, the globally conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics is reducible to area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of the Mullins finger (a well-known analytic solution) whose width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in very good agreement with the numerical solution.

Analytical solution of combined electroosmotic/pressure driven flows in two-dimensional straight channels: finite Debye layer effects.

Analytical results for the velocity distribution, mass flow rate, pressure gradient, wall shear stress, and vorticity in mixed electroosmotic/pressure driven flows are presented for two-dimensional straight channel geometry. We particularly analyze the electric double-layer (EDL) region near the walls and define three new concepts based on the electroosmotic potential distribution. These are the effective EDL thickness, the EDL displacement thickness, and the EDL vorticity thickness. We show that imposing Helmholtz-Smoluchowski velocity at the edge of the EDL as the velocity matching condition between the EDL and the bulk flow region is incomplete under spatial bulk flow variations across the finite EDL. However, the Helmholtz-Smoluchowski velocity can be used as the appropriate slip velocity on the wall. We discuss the limitations of this approach in satisfying the global conservation laws.

The Continuous Galerkin Method Is Locally Conservative

We examine the conservation law structure of the continuous Galerkin method. We employ the scalar, advection–diffusion equation as a model problem for this purpose, but our results are quite general and apply to time-dependent, nonlinear systems as well. In addition to global conservation laws, we establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements. These results are somewhat surprising and contradict the widely held opinion that the continuous Galerkin method is not locally conservative.

Lepton number violation in supersymmetric grand unified theories

We argue that the nature of the global conservation laws in Supersymmetric Grand Unified Theories is determined by the basic vacuum configuration in the model rather than its Lagrangian. It is shown that the suppression of baryon number violation in a general ( R-parity violating) superpotential can naturally appear in some extended SU( N) SUSY GUTs which, among other degenerate symmetry-breaking vacua, have a missing VEV vacuum configuration giving a solution to the doublet-triplet splitting problem. We construct SU(7) and SU(8) GUTs where the effective lepton number violating couplings immediately evolve, while the baryon number non-conserving ones are safely projected out as the GUT symmetry breaks down to that of the MSSM. However at the next stage, when SUSY breaks, the radiative corrections shift the missing VEV components to some nonzero values of order M SUSY, thereby inducing the ordinary Higgs doublet mass, on the one hand, and tiny baryon number violation, on the other. So, a missing VEV solution to the gauge hierarchy problem leads at the same time to a similar hierarchy of baryon vs lepton number violation.

Convective heat conduction and diffusion in one-dimensional hydrodynamics

The impurity concentration in localized structures is described on the basis of analytic solutions of model equations for convective diffusion in the one-dimensional hydrodynamic approximation without pressure. The simplicity of the derivation of the analytic results depends on the ratio of the kinetic coefficients of the liquid (the Prandtl numbers). For the same kinetic coefficients, any time-dependent problem can be reduced to problems for the conventional heat conduction equation. For integer Prandtl numbers the problem of time-dependent convective diffusion in the flow field of a uniformly moving shock wave likewise reduces to problems for the heat conduction equation. Relations are established between problems whose Prandtl numbers differ by an integer. Various representations of the Green’s functions for the equations of convective diffusion are analyzed. For integer Prandtl numbers they can be expressed in terms of error functions. The asymptotic character of the solutions depends strongly on the satisfaction of global conservation laws. For global conservation of the impurity mass, coalescence of shock waves corresponds to merging of impurity solitons, i.e., clustering.

Canonical Symmetries in the Functional Formalism

Based on the phase-space generating functionalof the Green function, the canonical Ward identities(CWI) under local, nonlocal, and global transformationsin phase space for a system with a regular and singular Lagrangian have been derived. Therelation of global canonical symmetries to conservationlaws at the quantum level is presented. The advantage ofthis formulation is that one does not need to carry out the integration over canonicalmomenta in a phase-space path (functional) integral asin the traditional treatment in configuration space. Ingeneral, the connection between global canonicalsymmetries and conservation laws in classical theories isno longer preserved in quantum theories. Applications ofour formulation to the non-Abelian Chern-Simons (CS)theory are given, and new forms for CS gauge-ghost field proper vertices and the quantal conservedangular momentum of this system are obtained; thisangular momentum differs from the classical one in thatone needs to take into account the contribution of angular momenta of ghost fields.

Distributional curvature of time dependent cosmic strings

Colombeau's theory of generalized functions is used to calculate the contributions, at the rotation axis, to the distributional curvature for a time-dependent radiating cosmic string, and hence the mass per unit length of the string source. This mass per unit length is compared with the mass at null infinity, giving evidence for a global energy conservation law.

On global conservation laws at null infinity

In this paper the standard values for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity whether to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to N ther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, the conserved quantities are explicitly related to the sources of gravity through Einstein's equations. The relations depend, however, on a rule for mapping spacetimes onto backgrounds.

Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

Starting from the phase-space generating functional of the Green function for a system with singular higher order Lagrangian, the generalized canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, and the quantal conservation laws are obtained for such a system. We give a preliminary application to higher derivative Yang-Mills theory; a generalized quantal BRS conserved quantity is found.

Global well-posedness and conservation laws for the water wave interaction equation

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem foris locally well posed in the largest space where the three conservationscan be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

Nonequilibrium phase ordering with a global conservation law.

In all dimensions, infinite-range Kawasaki spin exchange in a quenched Ising model leads to an asymptotic length-scale $L \sim (\rho t)^{1/2} \sim t^{1/3}$ at $T=0$ because the kinetic coefficient is renormalized by the broken-bond density, $\rho \sim L^{-1}$. For $T>0$, activated kinetics recovers the standard asymptotic growth-law, $L \sim t^{1/2}$. However, at all temperatures, infinite-range energy-transport is allowed by the spin-exchange dynamics. A better implementation of global conservation, the microcanonical Creutz algorithm, is well behaved and exhibits the standard non-conserved growth law, $L \sim t^{1/2}$, at all temperatures.

Conservation laws for systems of extended bodies in the first post-Newtonian approximation.

The general form of the global conservation laws for $N$-body systems in the first post-Newtonian approximation of general relativity is considered. Our approach applies to the motion of an isolated system of $N$ arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies and uses a framework recently introduced by Damour, Soffel and Xu (DSX). We succeed in showing that seven of the first integrals of the system (total mass-energy, total dipole mass moment and total linear momentum) can be broken up into a sum of contributions which can be entirely expressed in terms of the basic quantities entering the DSX framework: namely, the relativistic individual multipole moments of the bodies, the relativistic tidal moments experienced by each body, and the positions and orientations with respect to the global coordinate system of the local reference frames attached to each body. On the other hand, the total angular momentum of the system does not seem to be expressible in such a form due to the unavoidable presence of irreducible nonlinear gravitational effects.

Utilitarian Models of the Solar Nebula

Models of the primitive solar nebula based on a combination of theory, observations of T Tauri stars, and global conservation laws are presented. The models describe the motions of nebular gas, mixing of interstellar material during the formation of the nebula, and evolution of thermal structure in terms of several characteristic parameters. The parameters describe key aspects of the protosolar cloud (its rotation rate and collapse rate) and the nebula (its mass relative to the Sun, decay time, and density distribution). For most applications, the models are heuristic rather than predictive. Their purpose is to provide a realistic context for the interpretation of solar system data, and to distinguish those nebula characteristics that can be specified with confidence, independently of the assumptions of particular models, from those that are poorly constrained. It is demonstrated that nebular gas typically experienced large radial excursions during the evolution of the nebula and that both inward and outward mean radial velocities on the order of meters per second occurred in the terrestrial planet region, with inward velocities predominant for most of the evolution. However, the time history of disk size, surface density, and radial velocities are sensitive to the total angular momentum of the protosolar cloud, which cannot be constrained by purely theoretical considerations. It is shown that a certain amount of "formational" mixing of interstellar material was an inevitable consequence of nebular mass and angular momentum transport during protostellar collapse, regardless of the specific transport mechanisms involved. Even if the protosolar cloud was initially homogeneous, this mixing was important because it had the effect of mingling presolar material that had experienced different degrees of thermal processing during collapse and passage through the accretion shock. Nebular thermal structure is less sensitive to poorly constrained parameters than is dynamical history. A simple criterion is derived for the condition that silicate grains are evaporated at midplane, and it is argued that this condition was probably fulfilled early in nebular history. Cooling of a hot nebula due to coagulation of dust and consequent local reduction of optical depth is examined, and it is shown how such a process leads naturally to an enrichment of rock-forming elements in the gas phase.

Integrability and attractors

We discuss global and local integrability and also attractors from a physicist's standpoint. A local Lie method is used to form an analytic and geometric picture of integrability via conservation laws. The starting point is Lie's proof that two-dimensional flows, even dissipative ones, have a conservation law and so are integrable. Through conservation laws, integrable higher-dimensional flows reduce to the two-dimensional case. We consider a class of models defined by the Euler-Lagrange equations for a set of nonintegrable velocities (nonholonomic coordinates). The destruction of the global conservation laws of this conservative system by the inclusion of linear damping and constant external driving leads to self-confinement and attractors in phase space. In particular, the Lorenz model belongs to this class and can be seen as a damped, driven symmetric top.

Reduced energy conservation law for magnetized plasma

A global energy conservation law for a magnetized plasma is studied within the context of a quasiparticle description. A reduced energy conservation law is derived for low-frequency, as compared to the gyromagnetic frequency, plasma motions with regard to both non-uniform mean flows and fluctuations in the plasma. The mean value of plasma energy is calculated and sufficient stability conditions for non-equilibrium plasmas are derived.

A note on the unimodular theory of gravitation

Since the covariant divergence law for the energy tensor in general relativity does not lead to a global conservation law one has the freedom to modify it. Rather than also assuming this law in the unimodular theory, invoking only the Bianchi identity, one ends up with a theory without the cosmological constant.

Critical dynamics, spinodal decomposition, and conservation laws.

A simple acceleration algorithm for Ising systems with conserved magnetization (model B) is presented. The dynamical critical exponent and the domain growth exponent in spinodal decomposition are found to be equal to those observed for model A systems with no conservation law. Our results demonstrate that systems with global conservation laws are in the same dynamical universality class as systems with no conservation laws.

Deterministic and stochastic spin diffusion in classical Heisenberg magnets

This computer simulation study provides further evidence that spin diffusion in the one-dimensional classical Heisenberg model at T=∞ is anomalous: 〈Sj(t)⋅Sj〉 ∼t−α1 withα1 ≳1/2. However, the exponential instability of the numerically integrated phase-space trajectories transforms the deterministic transport of spin fluctuations into a computationally generated stochastic process in which the global conservation laws are still satisfied to high precision. This may cause a crossover in 〈Sj(t)⋅Sj〉 from anomalous spin diffusion (α1 ≳ 1/2) to normal spin diffusion (α1 = 1/2) at some characteristic time lag that depends on the precision of the numerical integration.

Comment on "Critical dynamics and global conservation laws"

Comment on "Critical dynamics and global conservation laws"