Identical motion in relativistic quantum and classical mechanics

The Klein-Gordon equation for the stationary state of a charged particle in a spherically symmetric scalar field is partitioned into a continuity equation and an equation similar to the Hamilton-Jacobi equation. There exists a class of potentials for which the Hamilton-Jacobi equation is exactly obtained and examples of these potentials are given. The partitionAnsatz is then applied to the Dirac equation, where an exact partition into a continuity equation and a Hamilton-Jacobi equation is obtained.     

Numerical solution of the spinor Bethe-Salpeter equation and the goldstein problem

The spinor Bethe-Salpeter equation describing bound states of a fermion-antifermion pair with massless-boson exchange reduces to a single (uncoupled) partial differential equation for special combinations of the fermion-boson couplings. For spinless bound states with positive or negative parity this equation is a generalization to nonvanishing bound-state masses of the equations studied by Kummer and Goldstein, respectively. In the tight-binding limit the Kummer equation has a discrete spectrum, in contrast to the Goldstein equation, while for loose binding only the generalized Goldstein equation has a nonrelativistic limit. For intermediate binding energies the equations are solved numerically. The generalized Kummer equation is shown to possess a discrete spectrum of coupling constants for all bound-state masses. For the generalized Goldstein equation a discrete spectrum of coupling constants is found only if the binding energy is smaller than a critical value.     

Microdynamics and equations of motion for macrovariables

By means of a new time-dependent projection operator an exact generalized Langevin equation for the macrovariables of a system is derived. This equation is in general nonlinear and also valid far from equilibrium. The projection operator picks up the macroscopic part of an observable which is defined in such a way that it's mean value depends only on the macroscopic state given by the mean values of the considered macrovariables. The exact equation can be separated into an evolution equation for the mean values and an equation for the fluctuations. The second equation contains a nonlinear random force and a term which shows up to be the linearization of the mean value equations around the mean path. The connection with previous works is discussed.     

Darboux Transformation and Explicit Solutions for Drinfel＇d-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.     

Dynamical equations of quantum information and Gaussian channel

A subdynamic based kinetic equation (SKE) for quantum information density (QID) is presented and using this is shown that the Liouville equation, Master equation and Fokker–Planck equation for QID all share the same formalism as the density operator. This allows one to directly use QID for studying quantum communication and to construct a quantum Gaussian channel. The channel is described by a quantum Fokker–Planck equation, which permits harmonic oscillator encoded information to transmit quantum signals with quantum parallelism. The quantum dynamical mutual information for this channel is also calculated.     

Fokker-Planck equation for interacting waves: Dispersion law renormalization and wave turbulence

The multiple timescale method for removing secularities is used to generate the Fokker-Planck (“FP”) equation for a system of interacting waves. This FP equation describes diffusion in the phase space of the angle, as well as the action, variables of all underlying modes. The first moment of the FP equation gives a kinetic (or Boltzmann-type) equation governing the averaged actions, and describing the diffusion of action in time. Angle diffusion leads to a renormalization of the dispersion law. Stationary solutions for the average action (or so-called spectral intensity) are derived for equilibrium and for the driven off-equilibrium state corresponding to a cascade of wave energy from low to high frequencies (wave turbulence). The reduced distribution function for these states is derived.The derivation of the FP equation from the Liouville equation, as well as the derivation of the kinetic equation from the FP equation, requires that the distribution of modes be sufficiently dense. In this limit, cumulants that are initially zero increase at a rate that is thermodynamically sm all. A Langevin equation, governing the evolution of a distinguished oscillator, that is applicable even in off-equilibrium conditions, is derived. The concept of winding numbers is extended to the general phase space motion of action-angle variables through the introduction of a multiple-valued probability density.     

Solving Kadomtsev—Petviashvili Equation via a New Decomposition and Darboux Transformation

Recently,a new decomposition of the (2 1)-dimensional Kadomtsev-Petviashvili(KP) equation to a (1 1)-dimensional Broer-Kaup (BK) equation and a (1 1)-dimensional high-order BK equation was presented by Lou and Hu.In our paper,a unified Darboux transformation for both the BK equation and high-order BK equation is derived with the help of a gauge transformation of their spectral problems.As application,new explicit soliton-like solutions with five arbitrary parameters for the BK equation,high-order BK equation and KP equation are obtained.     

Affine Metrics and Algebroid Structures: Application to General Relativity and Unification

Affine metrics and their associated algebroid bundle are developed. These structures are applied to the general relativity and provide a mathematical structure for unification of gravity and electromagnetism. The final result is a field equation on the associated algebroid bundle that is similar to Einstein field equation but contains Einstein field equation and Maxwell equations simultaneously and contains a new equation that may have new results.     

Spline-Galerkin Solution of Dynamic Equations for Particle Comminution and Collection

The population balance equation is solved for particles undergoing a combination of growth, comminution, and collection. The approximation method is to use a weighted Galerkin technique with cubic B-splines and an implicit scheme for solving the system of ordinary differential equations. The cubic splines are defined on a graded mesh. The performance of the method is investigated by solving a model problem with simple but nonsmooth kernels. The weight function is chosen so that singularities in the equation can be easily treated. A self-similar solution for comminuted particles is shown to be a useful representation for the solution of the population balance equation provided that this equation is solved over a sufficiently long time interval. Stationary solutions of the equation are obtained for a model that describes both particle comminution and collection.     

On the relationship between continuous time random walks and the non-equilibrium Ornstein-Zernike equation

An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.     

Hydrodynamics in two dimensions and vortex theory

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR ². We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.Partially supported by Italian CNR     

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.     

The magnetic susceptibilities of NO and OH

A non-isothermal kinetic equation for the distribution function of a sub-system weakly coupled to a bath is derived by modification of the analysis and assumptions of a previous paper [1]. The equation has the form of a generalized non-isothermal Fokker-Planck equation when it is linearized in thermodynamic gradients and only terms through second order in the coupling parameters are retained. Higher order terms in the coupling parameter do not diverge with time. The equation is compared with certain ‘exact’ model results of Ullersma and with the coherence time method. The equation is used to calculate a jump rate for diffusion of a harmonic particle weakly coupled to a lattice and it is found that the jump rate becomes independent of the mass of the particle for a heavy enough particle. The source of the discrepancy of this result with a similar calculation of Prigogine and Bak is indicated. The model of the jump rate is inappropriate for diffusion in a thermal gradient and more appropriate models of the jump are briefly considered. A brief comparison of the derivation of the kinetic equation with the Fano coherence time approximation is made and a difference is noted.     

An anomalous non-self-similar infiltration and fractional diffusion equation

Problems of anomalous infiltration in porous media are considered. As follows from the analysis of experimental data, modification of the infiltration equation is necessary. A fractional diffusion equation with variable order of the time-derivative operator for describing the liquid infiltration in porous media is proposed. The physical meaning of this fractional equation is explained. This equation provides good agreement with existing experimental data for both the subdiffusion and the superdiffusion. The treatment of experimental data for the absorption of water in a fired-clay brick and for water infiltration in cement mortar using this fractional equation of diffusion is presented. Various formulae, which can be useful for applications, have been developed.     

Dirac equation of spin particles and tunneling radiation from a Kinnersly black hole

In curved space-time, the Hamilton–Jacobi equation is a semi-classical particle equation of motion, which plays an important role in the research of black hole physics. In this paper, starting from the Dirac equation of spin 1/2 fermions and the Rarita–Schwinger equation of spin 3/2 fermions, respectively, we derive a Hamilton–Jacobi equation for the non-stationary spherically symmetric gravitational field background. Furthermore, the quantum tunneling of a charged spherically symmetric Kinnersly black hole is investigated by using the Hamilton–Jacobi equation. The result shows that the Hamilton–Jacobi equation is helpful to understand the thermodynamic properties and the radiation characteristics of a black hole.     

Multipotentializations and nonlocal symmetries: Kupershmidt,Kaup-Kupershmidt and Sawada-Kotera equations

In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetries and that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can be recovered using potentializations.     

Painlevé Property and Allowed Transformations of the Variable-Coefficient Zakharov-Kuznetsov Equation

In this letter, we discuss the Painlevé property and allowed transformation for the variablecoefficient Zakharov-Kuznetsov equation which governs nonlinear ion-acoustic wqves in a magnetized plasma. The general solution of the singular manifold equation and stability solution of the VCZK equation are obtained. To prove some information of the integrability of the ZK equation, we prove the constraints that the variable-coefficient functions for the equation to possess the Painlevé property are not equivalent in order that the equations may be transformed into the constant coefficient equation. So we confirm that the ZK equation is not integrable.     

Mass, energy, and the electron

The two-component solutions of the Dirac equation currently in use are not separately a particle equation or an antiparticle equation. We present a unitary transformation that uncouples the four-component, force-free Dirac equation to yield a two-component spinor equation for the force-free motion of a relativistic particle and a corresponding two-component, time-reversed equation for an antiparticle. The particle-antiparticle nature of the two equations is established by applying to the solutions of these two-component equations criteria analogous to those applied for establishing the four-component particle and antiparticle solutions of the four-component Dirac equation. Wave function solutions of our two-component particle equation describe both a right and a left circularly polarized particle. Interesting characteristics of our solutions include spatial distributions that are confined in extent along directions perpendicular to the motion, without the artifice of wave packets, and an intrinsic chirality (handedness) that replaces the usual definition of chirality for particles without mass. Our solutions demonstrate that both the rest mass and the relativistic increase in mass with velocity of the force-free electron are due to an increase in the rate of Zitterbewegung with velocity. We extend this result to a bound electron, in which case the loss of energy due to binding is shown to decrease the rate of Zitterbewegung.     

A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.