Conservation laws, uncertainty relations, and quantum limits of measurements

The uncertainty relation between the noise operator and the conserved quantity leads to a bound on the accuracy of general measurements. The bound extends the assertion by Wigner, Araki, and Yanase that conservation laws limit the accuracy of "repeatable," or "nondisturbing," measurements to general measurements, and improves the one previously obtained by Yanase for spin measurements. The bound represents an obstacle to making a small quantum computer.     

Dynamic Scaling in Miscible Viscous Fingering

We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Woodings model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.     

Continuous Correspondence of Conservation Laws of the Semi-discrete AKNS System

In this paper we investigate the semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchy, and specifically their Lax pairs and infinitely many conservation laws, as well as the corresponding continuum limits. The infinitely many conserved densities derived from the Ablowitz-Ladik spectral problem are trivial, in the sense that all of them are shown to reduce to the first conserved density of the AKNS hierarchy in the continuum limit. We derive new and nontrivial infinitely many conservation laws for the sdAKNS hierarchy, and also the explicit combinatorial relations between the known conservation laws and our new ones. By performing a uniform continuum limit, the new conservation laws of the sdAKNS system are then matched with their counterparts of the continuous AKNS system.     

求解二维双曲型方程的一种基本守恒差分格式

构造了一种求解二维双曲型方程的基本守恒型差分格式,并证明了该格式的数值解是全变差有界的,在光滑区域具有二阶精度,按L¹范数及L^∞范数稳定,且其几乎处处有界收敛的极限解是微分方程的物理解。     

Two-Body Correlation Transport Theory for Heavy Ion Collisions Ⅳ.Conservation Laws

The conservation laws under physical hierarchy - truncation in Two-Body Correlation Dynamics (TBCD) are discussed. The numerical results show that the different physical hierarchy truncations in Two-Body Correlation Transport Theory (TBCTT) are all compatible with conservation laws.     

重离子碰撞两体关联输运理Ⅳ.守恒定律

讨论了在物理截断下两体关联动力学中守恒定律保持与破坏的一般特点．数值计算表明：两体关联输运理论在不同的等级截断下均较好地保持了有关的守恒定律。     

Spin,torsion, forces

A theory of gravitation with torsion that is derived from a potential is developed. An explicit material action is presented that gives rise to the correct conservation laws and equations of motion. It is shown that the Noether identities yield the same conservation laws as the Bianchi identities, and the Papapetrou method is used to develop the propagation equations and the force law. The equations are put in the nonrelativistic limit and the 3-vector formulation is displayed.     

Relativistic thermodynamics of fluids. I

In 1953, Stueckelberg and Wanders derived the basic laws of relativistic linear nonequilibrium thermodynamics for chemically reacting fluids from the relativistic local conservation laws for energy-momentum and the local laws of production of substances and of non-negative entropy production by the requirement that the corresponding currents (assumed to depend linearly on the first derivatives of the state variables) should not be independent. Generalizing their method, we determine the most general allowed form of the energy-momentum tensor T^αβ and of the corresponding rate of entropy production under the same restriction on the currents. The problem of expressing this rate in terms of thermodynamic forces and fluxes is discussed in detail; it is shown that the number of independent forces is not uniquely determined by the theory, and several possibilities are explored. A number of possible new cross effects are found, all of which persist in the Newtonian (low-velocity) limit. The treatment of chemical reactions is incorporated into the formalism in a consistent manner, resulting in a derivation of the law for rate of production, and in relating this law to transport processes differently than suggested previously. The Newtonian limit is discussed in detail to establish the physical interpretation of the various terms of T^αβ. In this limit, the interpretation hinges on that of the velocity field characterizing the fluid. If it is identified with the average matter velocity following from a consideration of the number densities, the usual local conservation laws of Newtonian nonequilibrium thermodynamics are obtained, including that of mass. However, a slightly different identification allows conversion of mass into energy even in this limit, and thus a macroscopic treatment of nuclear or elementary particle reactions. The relation of our results to previous work is discussed in some detail.     

Thermodyamic Bounds on Drude Weights in Terms of Almost-conserved Quantities

We consider one-dimensional translationally invariant quantum spin (or fermionic) lattices and prove a Mazur-type inequality bounding the time-averaged thermodynamic limit of a finite-temperature expectation of a spatio-temporal autocorrelation function of a local observable in terms of quasi-local conservation laws with open boundary conditions. Namely, the commutator between the Hamiltonian and the conservation law of a finite chain may result in boundary terms only. No reference to techniques used in Suzuki’s proof of Mazur bound is made (which strictly applies only to finite-size systems with exact conservation laws), but Lieb-Robinson bounds and exponential clustering theorems of quasi-local C* quantum spin algebras are invoked instead. Our result has an important application in the transport theory of quantum spin chains, in particular it provides rigorous non-trivial examples of positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ spin 1/2 chain (Prosen, in Phys Rev Lett 106:217206, 2011).     

Some conservation issues for the dynamical cores of NWP and climate models

The rationale for designing atmospheric numerical model dynamical cores with certain conservation properties is reviewed. The conceptual difficulties associated with the multiscale nature of realistic atmospheric flow, and its lack of time-reversibility, are highlighted. A distinction is made between robust invariants, which are conserved or nearly conserved in the adiabatic and frictionless limit, and non-robust invariants, which are not conserved in the limit even though they are conserved by exactly adiabatic frictionless flow. For non-robust invariants, a further distinction is made between processes that directly transfer some quantity from large to small scales, and processes involving a cascade through a continuous range of scales; such cascades may either be explicitly parameterized, or handled implicitly by the dynamical core numerics, accepting the implied non-conservation. An attempt is made to estimate the relative importance of different conservation laws. It is argued that satisfactory model performance requires spurious sources of a conservable quantity to be much smaller than any true physical sources; for several conservable quantities the magnitudes of the physical sources are estimated in order to provide benchmarks against which any spurious sources may be measured.     

A general theory of conservation laws,their violation,and spontaneous phenomena

We formulate a general theory of conservation laws and other invariants for a physical system through equivalence relations. The conservation laws are classified according to the type of equivalence relation, with group equivalence, homotopical equivalence, and other types of equivalence relations giving respective kinds of conservation laws. The stability properties in the topological (and differentiable) sense are discussed using continuous deformations with respect to control parameters. The conservation laws due to the Abelian symmetries are shown to be stable through application of well-known theorems.     

Conservation laws in acoustics

The method earlier proposed by the author for obtaining a complete set of conservation laws is applied to a number of simple acoustic problems. Some of the conservation laws presented are derived for the first time. Special attention is given to the physical interpretation of the results.     

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.     

Potentials of divergent equations and new additional conservation laws for two-dimensional steady gas flows

An algorithm is constructed that compares a new divergent equation (the additional conservation law) to each of two divergent equations. Both the starting divergent equations themselves and their potentials participate in the additional conservation law, and both the first and second participate symmetrically. A characteristic feature of such additional conservation laws is that not only are the functions of gas-dynamic parameters and their derivatives taken along streamlines but so are the integrals, i.e., the functionals, and their derivatives participate in them. All these facts reveal the physical sense of the topical conservation laws constructed. A comparison with the asymmetric conservation laws constructed previously by the author (Doklady Physics, 2002) is performed. As an example, the relation that connects four additional laws comparable by dimensionality is constructed as an example. This is the conservation law of the momentum and its three analogs. Two laws are asymmetric (from Doklady Physics, 2002), while two others are constructed in this study.     

Onsager Relations and Eulerian Hydrodynamic Limit for Systems with Several Conservation Laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.     

Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

Additive Invariant Functionals for Dynamical Systems

We consider the problem of defining completely a class of additive conservation laws for the generalized Liouville equation whose characteristics are given by an arbitrary system of first-order ordinary differential equations. We first show that if the conservation law, a time-invariant functional, is additive on functions having disjoint compact support in phase space, then it is represented by an integral over phase space of a kernel which is a function of the solution to the Liouville equation. Then we use the fact that in classical mechanics phase space is usually a direct product of physical space and velocity space (Newtonian systems). We prove that for such systems the aforementioned representation of the invariant functionals will hold for conservation laws which are additive only in physical space; i.e., additivity in physical space automatically implies additivity in the whole phase space. We extend the results to include non-degenerate Hamiltonian systems, and, more generally, to include both conservative and dissipative dynamical systems. Some applications of the results are discussed.     

High-Speed Scattering of Charged and Uncharged Particles in General Relativity

After a brief consideration of the high-speed scattering of two point charges we thoroughly discuss high-speed scattering for a charged particle by a fixed mass and of two uncharged particles of comparable masses. We use perturbation technique over Minkowski spacetime in the de Donder gauge and solve the field equations and the resulting equations of motion (which take the reaction of the particles' quasistatic self-field into account) by iteration. The obtained energy-momentum conservation laws allow the computation of second-order corrections for the scattering angle and the cross section. The asymptotic structure of the far-field indicates synchrotron radiation (electromagnetic and gravitational, respectively) which causes an energy loss whose reaction on the motion is briefly considered in the low-velocity limit including bound motion. (For neutral particles this is a third-order effect).     

Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids

The development of a compact fourth-order finite volume method for solutions of the Navier–Stokes equations on staggered grids is presented. A special attention is given to the conservation laws on momentum control volumes. A higher-order divergence-free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes. Three forms of the nonlinear correction for staggered grids are proposed and studied. The accuracy of each approximation is assessed comparatively in Fourier space. The importance of higher-order approximations of pressure is discussed and numerically demonstrated. Fourth-order accuracy of the complete scheme is illustrated by the doubly-periodic shear layer and the instability of plane-channel flow. The efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations. The proposed scheme is highly accurate and efficient. At the same level of accuracy, the fourth-order scheme can be ten times faster than the second-order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.