Additional Conservation Laws Research Articles

Additional conservation laws for Rosenau–KdV–RLW equation with power law nonlinearity by Lie symmetry

This paper computes the conservation laws of the Rosenau–KdV–RLW equation with power law nonlinearity by the aid of multiplier approach in Lie symmetry analysis. This equation models the dynamics of dispersive shallow water waves along lake shores and beaches. The usual conservation laws are reported earlier that are computed from basic mathematical principles. The conservation laws in this paper are extracted using Lie symmetry analysis. The corresponding conserved quantities are computed from their respective densities.

Dynamics of the Bose-Hubbard chain for weak interactions

We study the Boltzmann transport equation for the Bose-Hubbard chain in the kinetic regime. The time-dependent Wigner function is matrix-valued with odd dimension due to integer spin. For nearest neighbor hopping only, there are infinitely many additional conservation laws and nonthermal stationary states. Adding longer range hopping amplitudes entails exclusively thermal equilibrium states. We provide a derivation of the Boltzmann equation based on the Hubbard hamiltonian, including general interactions beyond on-site, and illustrate the results by numerical simulations. In particular, convergence to thermal equilibrium states with negative temperature is investigated.

The loss function of dense plasmas and sum rules

Mathematical, particularly, asymptotic properties of the RPA and RPA with dynamic local field corrections of the coupled plasma dielectric function are analyzed within the method of moments which satisfies some exact relations. Particularly f-sum rule,higher-order sum rules and other conservation laws. Thehigher-order sum rules take into account the correlations in the system under scrutiny, so if the system dynamic characteristics, e.g., the dielectric function, do not satisfy these rules which are effectively additional conservation laws, it is difficultto expect the corresponding model to be adequate in the strong-coupling domain.Some other drawbacks and advantages of the above models are pointed out.

Hamiltonian spinfoam gravity

This paper presents a Hamiltonian formulation of spinfoam gravity, which leads to a straightforward canonical quantization. To begin with, we derive a continuum action adapted to a simplicial decomposition of space-time. The equations of motion admit a Hamiltonian formulation, allowing us to perform the constraint analysis. We do not find any secondary constraints, but only get restrictions on the Lagrange multipliers enforcing the reality conditions. This comes as a surprise—in the continuum theory, the reality conditions are preserved in time, only if the torsionless condition (a secondary constraint) holds true. Studying an additional conservation law for each spinfoam vertex, we discuss the issue of torsion and argue that spinfoam gravity may still miss an additional constraint. Finally, we canonically quantize and recover the EPRL (Engle–Pereira–Rovelli–Livine) face amplitudes.Communicated by P R L V Moniz

Adding an energy-like conservation law to the leapfrog integrator

The leapfrog integrator is widely used because of its excellent stability in molecular dynamics simulation. This is recognized as being due to the existence of a discrete variational structure of the equations. We introduce a modified leapfrog method which includes an additional energy-like conservation law by embedding a molecular dynamics simulation within a larger dynamical system.

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.

Matrix-valued Boltzmann equation for the Hubbard chain

We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest-neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The H theorem holds. The nearest-neighbor chain is integrable, which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also nonthermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigate the convergence rate in dependence on the initial conditions.

Criterion of hyperbolicity for non-conservative quasilinear systems admitting a partially convex conservation law

A system of conservation laws admitting an additional convex conservation law can be written as a symmetric t-hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non-conservative hyperbolic models. We generalize the Godunov–Friedrichs–Lax approach to this new class of models. Copyright © 2011 John Wiley & Sons, Ltd.

Self-adjointness and conservation laws of a generalized Burgers equation

A (2 + 1)-dimensional generalized Burgers equation is considered. Having written this equation as a system of two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law.

Conservation Laws and Invariant Measures in Surjective Cellular Automata

We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

Absence of the discontinuous transition in the one-dimensional triplet creation model

Although Hinrichsen in his unpublished work theoretically rebutted the possibility of the discontinuous transition in one-dimensional nonequilibrium systems unless there are additional conservation laws, long-range interactions, macroscopic currents, or special boundary conditions, we have recently observed the resurrection of the claim that the triplet creation (TC) model introduced by Dickman and Tomé [Phys. Rev. A 44, 4833 (1991)] would show the discontinuous transition. By extensive simulations, however, we find that the one-dimensional TC does belong to the directed percolation universality class even for larger diffusion constant than the suggested tricritical point in the literature. Furthermore, we find that the phase boundary is well described by the crossover from the mean field to the directed percolation, which supports the claim that the one-dimensional TC does not exhibit a discontinuous transition.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Quantum limits of measurements induced by multiplicative conservation laws: Extension of the Wigner-Araki-Yanase theorem

The Wigner-Araki-Yanase (WAY) theorem shows that additive conservation laws limit the accuracy of measurements. Recently, various quantitative expressions have been found for quantum limits on measurements induced by additive conservation laws, and have been applied to the study of fundamental limits on quantum-information processing. Here, we investigate generalizations of the WAY theorem to multiplicative conservation laws. The WAY theorem is extended to show that an observable not commuting with the modulus of, or equivalently the square of, a multiplicatively conserved quantity cannot be precisely measured. We also obtain a lower bound for the mean-square noise of a measurement in the presence of a multiplicatively conserved quantity. To overcome this noise it is necessary to make large the coefficient of variation (the so-called relative fluctuation), instead of the variance as is the case for additive conservation laws, of the conserved quantity in the apparatus.

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.

Correlations and fluctuations: Generalized factorial moments

A systematic study of the relations between fluctuations of the extensive multiparticle variables and integrals of the inclusive multiparticle densities is presented. The generalized factorial moments are introduced and their physical meaning discussed. The effects of the additive conservation laws are analyzed.

A non-stationary analogue of Chaplygin's equations in one-dimensional gas dynamics

As an extension of the results obtained in [1], two equivalent uniformly divergent systems of equations are constructed in thespeedograph plane, each of which is the analogue of Chaplygin's equation in the hodograph plane. Each of the systems reduces to a linear second-order equation, in one case for the particle function (the Lagrange coordinate) ψ, and in the other for the time t. These systems possess an infinite set of exact solutions. It is shown that a uniformly divergent system of first-order equations correspond to each of these, and, related to them, the simplest non-linear homogeneous second-order equation in the modified events plane (ψ, t) and the conservation law in the events plant ( x, t). Clear relations are obtained between the velocities of the fronts of constant values of the newly constructed dependent variables and the velocity of sound. Examples are given which demonstrate the relation between the exact solutions with the uniformly divergent equations and the conservation laws of one-dimensional non-stationary gas dynamics and, simultaneously, enable one to compare the newly obtained results (the exact solutions, the equations and conservation laws, and the relations for the velocities of the front) with existing results, including those for plane steady flows. The so-called additional conservation laws, to which Godunov drew attention, are considered.

Weakly nonlocal symplectic structures, Whitham method and weakly nonlocal symplectic structures of hydrodynamic type

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.

Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions

In this paper, two existing one-dimensional mathematical models, one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions, are combined into a model of continuous separation of polydisperse mixtures. This model can be written as a first-order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable. This application motivates the extension of the Kurganov–Tadmor central difference scheme to systems with discontinuous flux. The new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws. These additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step, or solved exactly, that is, the discontinuity parameters are kept constant (with respect to time). Numerical examples and an L 1 error study show that the Kurganov–Tadmor scheme with first-order in time discretization produces spurious oscillations, whereas its semi-discrete version, discretized by a second-order Runge–Kutta scheme, produces good results. The scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved. Numerical examples illustrate the application to separation of polydisperse suspensions.

The Inhomogeneous Kinematic Wave Traffic Flow Model as a Resonant Nonlinear System

The kinematic wave traffic flow model for an inhomogeneous road is studied as a resonant nonlinear system, where an additional conservation law is introduced to model time-invariant road inhomogeneities such as changes in grades or number of lanes. This resonant system has two families of waves, one of which is a standing wave originated at the inhomogeneity. The nature of these waves are examined and their time-space structures are studied under Riemann initial conditions and proper entropy conditions. Moreover, the system is solved numerically with Godunov's method, and the solutions are found to be consistent with those of Daganzo (1995) and Lebacque (1996) for the same initial conditions. Finally, the numerical approximation is applied to model traffic flow on a ring road with a bottleneck and the results conform to expectations.

Variational principles for relativistic smoothed particle hydrodynamics

In this paper we show how the equations of motion for the smoothed particle hydrodynamics (SPH) method may be derived from a variational principle for both non-relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the general relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation.