Global Conservation Laws Research Articles

Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

AbstractIn this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

BRST analysis of general mechanical systems

We study the groups of local BRST cohomology associated to the general systems of ordinary differential equations, not necessarily Lagrangian or Hamiltonian. Starting with the involutive normal form of the equations, we explicitly compute certain cohomology groups having clear physical meaning. These include the groups of global symmetries, conservation laws and Lagrange structures. It is shown that the space of integrable Lagrange structures is naturally isomorphic to the space of weak Poisson brackets. The last fact allows one to establish a direct link between the path-integral quantization of general not necessarily variational dynamics by means of Lagrange structures and the deformation quantization of weak Poisson brackets.

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

AbstractThe local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system

In this paper, local energy and momentum conservation laws are proposed for the coupled nonlinear Schrödinger system. The two local conservation laws are more essential than global conservation laws since they are independent of the boundary conditions. Based on the rule that numerical algorithms should conserve the intrinsic properties of the original problems as much as possible, we propose local energy-preserving and momentum-preserving algorithms for the problem. The proposed algorithms conserve the local energy and momentum conservation laws in any local time–space region, respectively. With periodic boundary conditions, we prove the proposed algorithms admit the charge, global energy and global momentum conservation laws. Numerical experiments are conducted to show the performance of the proposed methods. Numerical results verify the theoretical analysis.

Can the Abraham Light Momentum and Energy in a Medium Constitute a Lorentz Four-Vector?

By analyzing the Einstein-box thought experiment with the principle of relativity, it is shown that Abraham’s light momentum and energy in a medium cannot constitute a Lorentz four-vector, and they consequentially break global momentum and energy conservation laws. In contrast, Minkowski’s momentum and energy always constitute a Lorentz four-vector no matter whether in a medium or in vacuum, and the Minkowski’s momentum is the unique correct light momentum. A momentum-associated photon mass in a medium is exposed, which explains why only the Abraham’s momentum is derived in the traditional “center-of-mass-energy” approach. The EM boundary-condition matching approach, combined with Einstein light-quantum hypothesis, is proposed to analyze this thought experiment, and it is found for the first time that only from Maxwell equations without resort to the relativity, the correctness of light momentum definitions cannot be identified. Optical pulling effect is studied as well.

Numerical dispersion analysis of a multi-symplectic scheme for the three dimensional Maxwell’s equations

In this paper, we study a multi-symplectic scheme for three dimensional Maxwell’s equations in a simple medium. This is a system of PDEs with multi-symplectic structures. We prove that this multi-symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence. Furthermore, we extend the discussion to several dispersion properties of the multi-symplectic scheme including the numerical dispersion relation, the numerical group velocity, the effect of large time steps and the CFL condition.

Electromagnetic energy, momentum, and angular momentum in an inhomogeneous linear dielectric

In a previous work, Optics Communications 284 (2011) 2460–2465, we considered a dielectric medium with an anti-reflection coating and a spatially uniform index of refraction illuminated at normal incidence by a quasimonochromatic field. Using the continuity equations for the electromagnetic energy density and the Gordon momentum density, we constructed a traceless, symmetric energy–momentum tensor for the closed system. In this work, we relax the condition of a uniform index of refraction and consider a dielectric medium with a spatially varying index of refraction that is independent of time, which essentially represents a mechanically rigid dielectric medium due to external constraints. Using continuity equations for energy density and for Gordon momentum density, we construct a symmetric energy–momentum matrix, whose four-divergence is equal to a generalized Helmholtz force density four-vector. Assuming that the energy–momentum matrix has tensor transformation properties under a symmetry group of space–time coordinate transformations, we derive the global conservation laws for the total energy, momentum, and angular momentum.

Explicit Multisymplectic Fourier Pseudospectral Scheme for the Klein—Gordon—Zakharov Equations

Applying the Fourier pseudospectral method to space derivatives and the symplectic Euler rule to time derivatives in the multisymplectic form of the Klein-Gordon-Zakharov equations, we derive an explicit multisymplectic scheme. The semi-discrete energy and momentum conservation laws are given. Some numerical experiments are carried out to show the accuracy of the numerical solutions. The performance of the scheme in preserving the global energy and momentum conservation laws are also checked.

Statistical mechanics of quasi-geostrophic flows on a rotating sphere

Statistical mechanics provides an elegant explanation for the appearance of coherentstructures in two-dimensional inviscid turbulence: while the fine-grained vorticity field,described by the Euler equation, becomes more and more filamentary through time, itsdynamical evolution is constrained by some global conservation laws (energy, Casimirinvariants). As a consequence, the coarse-grained vorticity field can be predicted throughstandard statistical mechanics arguments (relying on the Hamiltonian structure ofthe two-dimensional Euler flow), for any given set of the integral constraints.It has been suggested that the theory applies equally well to geophysical turbulence;specifically in the case of the quasi-geostrophic equations, with potential vorticity playingthe role of the advected quantity. In this study, we demonstrate analytically thatthe Miller–Robert–Sommeria theory leads to non-trivial statistical equilibria forquasi-geostrophic flows on a rotating sphere, with or without bottom topography.We first consider flows without bottom topography and with an infinite Rossbydeformation radius, with and without conservation of angular momentum. When theconservation of angular momentum is taken into account, we report a second-order phasetransition associated with spontaneous symmetry breaking. In a second step,we treat the general case of a flow with an arbitrary bottom topography and afinite Rossby deformation radius. Previous studies were restricted to flows in aplanar domain with fixed or periodic boundary conditions with a beta-effect.In these different cases, we are able to classify the statistical equilibria for the large-scaleflow through their macroscopic features. We build the phase diagrams of the system anddiscuss the relations of the various statistical ensembles.

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov–Kuznetsov (ZK) equation and the Kadomtsev–Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.

Persistence Results for Chemical Reaction Networks with Time-Dependent Kinetics and No Global Conservation Laws

New checkable criteria for persistence of chemical reaction networks are proposed, which extend and complement existing ones. The new results allow the consideration of reaction rates which are time-varying, thus incorporating the effects of external signals, and also relax the assumption of existence of global conservation laws, thus allowing for inflows (production) and outflows (degradation). For time-invariant networks, parameter-dependent conditions for persistence of certain classes of networks are provided. As an illustration, two networks arising in the systems biology literature are analyzed, namely a hypoxia network and an apoptosis network.

Symplectic and multisymplectic numerical methods for Maxwell’s equations

In this paper, we compare the behaviour of one symplectic and three multisymplectic methods for Maxwell’s equations in a simple medium. This is a system of PDEs with symplectic and multisymplectic structures. We give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments. We also show that these numerical methods preserve the divergence. Furthermore, we extend the discussion on dispersion for (multi)symplectic methods applied to PDEs with one spatial dimension, to include anisotropy when applying (multi)symplectic methods to Maxwell’s equations in two spatial dimensions. Lastly, we demonstrate how varying the Courant–Friedrichs–Lewy (CFL) number can cause the (multi)symplectic methods in our comparison to behave differently, which can be explained by the study of backward error analysis for PDEs.

Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations

The exact global energy conservation laws for the full and truncated versions of the nonlinear electromagnetic gyrokinetic equations in general magnetic geometry are presented. In each version, the relation between polarization and magnetization effects in the gyrokinetic Poisson and Ampère equations and the quadratic ponderomotive gyrocenter Hamiltonian is emphasized.

Conditions for phase equilibrium in supernovae, protoneutron, and neutron stars

We investigate 1st order phase transitions in a general way, if not the single particle numbers of the system but only some particular charges like e.g. baryon number are conserved. In addition to globally conserved charges, we analyze the implications of locally conserved charge fractions, like e.g. local electric charge neutrality or locally fixed proton or lepton fractions. The conditions for phase equilibrium are derived and it is shown, that the properties of a phase transformation do not depend on the locally conserved fractions but only on the number of globally conserved charges. Finally, the general formalism is applied to the liquid-gas phase transition of nuclear matter and the hadron-quark phase transition for typical astrophysical environments like in supernovae, protoneutron, or neutron stars. We demonstrate that the Maxwell construction known from cold-deleptonized neutron star matter with two locally charge neutral phases requires modifications and further assumptions concerning the applicability for hot lepton-rich matter. All relevant combinations of local and global conservation laws are analyzed, and the physical meaningful cases are identified. Several new kinds of mixed phases are presented, as e.g. a locally charge neutral mixed phase in protoneutron stars which will disappear during the cooling and deleptonization of the protoneutron star.

Gas Flow in Micro Channels-Experimental, Computational and Kinetic-Theoretical Investigations

The investigation of fluid flow in a tube or channel commenced from studying experimentally the liquid flow along a circular tube under the pressure difference imposed at both the ends, the results were verified by the exact solution of the Navier-Stokes Eq. under constant pressure gradient assumption (Poiseuille flow). The experimental work has been playing a pioneering role in the investigation of channel flows. In the case of gas for the tube (or channel) flow, according to the global mass conservation law, the pressure gradient is never a constant. The theoretical studies on rarefied gas Poiseuille flow based on constant pressure gradient assumption still compared well with the early experimental work because when pressure difference between the inlet and the outlet is small in comparison with the average pressure, the pressure distribution is approximately a linear one. Recently non-linear pressure distribution was experimentally found in the integrated micro- channel/pressure sensor systems with gas flow in the transitional regime. The mass flow rate of micro channel was exactly measured, challenging the rarefied gas dynamics community to put forward computation or simulation means for the calculation of micro-channel flow, and the flow characteristics in other micro devices. This paper reviews some of these computational efforts. It also presents a strict kinetic solution of the finite length micro-channel flow problem based on the global mass conservation and the exact kinetic theoretical solution of the Poiseuille flow at each section. Calculations and simulations are compared with reviewed experiments to check the agreement between the computational or theoretical results with experimental data. Keywords: Microchannel flow, Poiseuille flow, finite length channel flow, gas flow in channels, lattice Boltzmann method (LBM), information preservation (IP) method, kinetic theoretical solution of microchannel flow, global mass conservation for channel flow

Multiplicity fluctuations and correlations in limited momentum space bins in relativistic gases

Multiplicity fluctuations and correlations are calculated within thermalized relativistic ideal quantum gases. These are shown to be sensitive to the choice of statistical ensemble as well as to the choice of acceptance window in momentum space. It is furthermore shown that global conservation laws introduce nontrivial correlations between disconnected regions in momentum space, even in the absence of any dynamics.

Optimal reference states for maximum accessible entanglement under the local-particle-number superselection rule

Global conservation laws imply superselection rules (SSR) which restrict the operations that are possible on any given state. Imposing the additional constraint of local operations and classical communication (LOCC) forbids the transfer of quantum systems between spatially separated sites. In the case of particle conservation this imposes a SSR for local particle number. That is, the coherences between subspaces of fixed particle number at each site are not accessible and any state is therefore equivalent to its projection onto these subspaces. The accessible entanglement under the SSR is less than (or equal to) that available in the absence of the SSR. An ancilla can be used as a reference system to increase the amount of accessible entanglement. We examine the relationship between local particle number uncertainty and the accessible entanglement and consider the optimal reference states for recovering entanglement from certain systems. In particular we derive the optimal ancilla state for extracting entanglement for a single shared particle and make steps towards the optimum for general systems.

Error analysis of discrete conservation laws for Hamiltonian PDEs under the central box discretizations

The central box scheme has been the most successful of the multisymplectic integrators for Hamiltonian PDEs. In this paper, we investigate conservative properties of the central box scheme for Hamiltonian PDEs and derive the error formulas of discrete local and global conservation laws of energy and momentum. We apply these results to the nonlinear Schrödinger equation and Klein–Gordon equation. Numerical experiments are presented to verify the theoretical predications.

Global conservation laws and femtoscopy of small systems

It is increasingly important to understand, in detail, two-pion correlations measured in $p+p$ and $d+A$ collisions. In particular, one wishes to understand the femtoscopic correlations to compare to similar measurements in heavy-ion collisions. However, in the low-multiplicity final states of these systems, global conservation laws generate significant $N$-body correlations that project onto the two-pion space in nontrivial ways and complicate the femtoscopic analysis. We discuss a formalism to calculate and account for these correlations in collisions dominated by a single particle species (e.g., pions). We also discuss effects on two-particle correlations between nonidentical particles, the understanding of which may be important in the study of femtoscopic space-time asymmetries.

A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzynski (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzynski helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.