Lagrange method and boundary conditions for the intensity correlation functions in turbulent media

An exact functional integral representation for the two-point intensity correlation function was previously obtained by the author for a collimated beam wave by solving the moment equation. The variable functions of integration involved therein can be effectively limited to a set of functions determined so that the entire phase term of the integrand becomes stationary against arbitrary variation of the variable functions, exactly according to the Lagrange variational principle in dynamics. The result is free from any expansion and is presented with a set of unperturbed equations of closed form. When making a formal expansion, it leads to the zeroth- and first-order expressions similar to those obtained by an improved two-scale method. With exactly the same procedure, the three-point intensity correlation and the two-frequency intensity correlation were also obtained.The Lagrange method leads to the 'equation of motion' subjected to boundary conditions to continue the phase term from the incident beam wave. The boundary conditions were previously found based on a physical reasoning, while the same conditions are found here purely based an the variational principle. A focused beam wave is assumed for the incident wave, including both spherical and plane waves as special cases.     

Lagrange method and boundary conditions for the intensity correlation functions in turbulent media*

Abstract

An exact functional integral representation for the two-point intensity correlation function was previously obtained by the author for a collimated beam wave by solving the moment equation. The variable functions of integration involved therein can be effectively limited to a set of functions determined so that the entire phase term of the integrand becomes stationary against arbitrary variation of the variable functions, exactly according to the Lagrange variational principle in dynamics. The result is free from any expansion and is presented with a set of unperturbed equations of closed form. When making a formal expansion, it leads to the zeroth- and first-order expressions similar to those obtained by an improved two-scale method. With exactly the same procedure, the three-point intensity correlation and the two-frequency intensity correlation were also obtained.The Lagrange method leads to the ‘equation of motion’ subjected to boundary conditions to continue the phase term from the incident beam wave. The boundary conditions were previously found based on a physical reasoning, while the same conditions are found here purely based an the variational principle. A focused beam wave is assumed for the incident wave, including both spherical and plane waves as special cases.     

Thermodynamic basis for a variational model for crystal growth

Variational models provide an alternative approach to standard sharp interface models for calculating the motion of phase boundaries during solidification. We present a correspondence between objective functions used in variational simulations and specific thermodynamic functions. We demonstrate that variational models with the proposed identification of variables are consistent with nonequilibrium thermodynamics. Variational models are derived for solidification of a pure material and then generalized to obtain a model for solidification of a binary alloy. Conservation laws for internal energy and chemical species and the law of local entropy production are expressed in integral form and used to develop variational principles in which a "free energy," which includes an interfacial contribution, is shown to be a decreasing function of time. This free energy takes on its minimum value over any short time interval, subject to the laws of conservation of internal energy and chemical species. A variational simulation based on this model is described, and shown for small time intervals to provide the Gibbs-Thomson boundary condition at the solid-liquid interface.     

Funktionalmittelwerte in der statistischen Theorie von quantenmechanischen und klassischen Systemen vieler Teilchen

The grand-canonical partition function of an interacting many-particle-system is represented as a functional integral with Gaussian random variables. The representation can be regarded as a Gaussian average over the partition function of free particles in an external fluctuating potential. The latter partition function is studied by means of diagrammatical techniques. The set of diagrams of a particularly simple structure is summed up by introducing the full scattering amplitude for the scattering in the external potential. The thermodynamicalGibbs' potential proves to be stationary with respect to the true particle density. It is shown that a variational procedure leads directly to an approximation which may be regarded as the renormalized form of the well-known Random-Phase-Approximation (RPA). The main feature of the approximation is thatGibbs' potential is stationary with respect to the two-particle-density correlation function. The classical limit of the renormalized RPA yields the results of the Debye-Hückel theory. In case of an hard-core potential the approximation applies only to the long-range part of the potential. The results are similar to recent developments in the theory of the Ising model and of real gases.     

Symmetry violating trial wave functions

A variational procedure using trial wave functions projected on to symmetry eigenstates is investigated. In the individual particle model it leads to generalized self-consistent single-particle equations being dependent on the symmetry eigenvalue. A perturbation expansion appropriate for the case of strong symmetry violations is described. Model independent expressions for the moment of inertia and the decoupling parameter are derived by applying the method to rotational symmetry.     

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.     

Gauge invariance,charge conservation,and variational principles

We present new results on the correspondence between symmetries, conservation laws and variational principles for field equations in general non-abelian gauge theories. Our main result states that second order field equations possessing translational and gauge symmetries and the corresponding conservation laws are always derivable from a variational principle. We also show by the way of examples that the above result fails in general for third order field equations.     

Variational principles for collective motion: Relation between invariance principle of the Schrödinger equation and the trace variational principle

The invariance principle of the Schrödinger equation provides a basis for theories of collective motion with the help of the time-dependent variational principle. It is formulated here with maximum generality, requiring only the motion of intrinsic state in the collective space. Special cases arise when the trial vector is generalized coherent state and when it is a uniform superposition of collective eigenstates. The latter example yields variational principles uncovered previously only within the framework of the equations of motion method.     

First quantized electrodynamics

The parametrized Dirac wave equation represents position and time as operators, and can be formulated for many particles. It thus provides, unlike field-theoretic Quantum Electrodynamics (QED), an elementary and unrestricted representation of electrons entangled in space or time. The parametrized formalism leads directly and without further conjecture to the Bethe–Salpeter equation for bound states. The formalism also yields the Uehling shift of the hydrogenic spectrum, the anomalous magnetic moment of the electron to leading order in the fine structure constant, the Lamb shift and the axial anomaly of QED.     

Distribution functions for the Brownian motion of particles in a periodic potential driven by an external force

The stationary distribution functions for the Brownian motion of particles driven by an external force are calculated by expanding the velocity part into Hermite functions and the space part into a Fourier series. Insertion into the Fokker-Planck equation leads to a matrix continued fraction for the lowest two coefficients of the Hermite functions. Higher order terms are found by reverse iteration. Results are shown for a cosine potential. The good convergence allows the calculation in the full range of damping constants. For small friction the distribution function is in good agreement with previous results and the maxima are given by the solutions without noise.     

Variational model for one-dimensional quantum magnets

A new variational technique for investigation of the ground state and correlation functions in 1D quantum magnets is proposed. A spin Hamiltonian is reduced to a fermionic representation by the Jordan–Wigner transformation. The ground state is described by a new non-local trial wave function, and the total energy is calculated in an analytic form as a function of two variational parameters. This approach is demonstrated with an example of the XXZ-chain of spin-1/2 under a staggered magnetic field. Generalizations and applications of the variational technique for low-dimensional magnetic systems are discussed.     

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.      

Variational DVR Calculations

In the discrete variable representation (DVR) method the potential energy matrix has a particular simple form. It is diagonal with values of the interaction potential at the discrete points. However, this simple form is obtained by making approximations in the calculation of the matrix elements of . As a consequence the results cannot be considered as variational estimates. We will show how to recover the variational character of the method using the discrete variable representation eigenvectors as trial functions and performing a variational calculation in a restricted Hilbert space.     

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrödinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy,momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.     

Polarization of spin-1 particles in a uniform magnetic field

The exact solution of the Corben–Schwinger equations is obtained for spin-1 particles without an anomalous magnetic moment in a uniform magnetic field. The exact Hamiltonian in the Foldy–Wouthuysen representation is derived. The conservation of projections of the polarization operator onto four directions is proved. The approximate conservation of projections of this operator onto the horizontal axes of the cylindrical coordinate system is established. For spin-1 particles with the anomalous magnetic moment, the Hamiltonian in the Foldy–Wouthuysen representation is deduced within first order terms in the Planck constant. Dynamics of spin-1 particles with the anomalous magnetic moment and their spins in the strong uniform magnetic field are calculated.     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.     

Coulomb interactions in a computer simulation of a system periodic in two directions

The integral representation of the gamma function and the Poisson summation formula are used to calculate the interaction energy of charged particles in a 3-dimensional system periodic in two directions. A parallelogram shape simulation box is considered. Calculations are carried out for interactions described by any inverse power, and analytical continuation of the energy function leads to the final expression for the Coulomb interaction energy. Summation over the simulation box replica along one or the other side of the box base is replaced by summation in reciprocal space. Therefore there are two equivalent formulas for the potential energy that offer the possibility of avoiding slowly convergent series. The energy expressions are identical to those obtained from the Lekner method. The special case is considered where the functions defining the energy are infinite, i.e. when two charges lie on a line parallel to the simulation box side that was chosen to convert real space summation into reciprocal space.     

A variational principle for the scattered wave

A Schwinger-type variational principle is presented for the scattered field in the case of scalar wave scattering with an arbitrary field incident on an object of arbitrary shape with homogeneous Dirichlet boundary conditions. The result is variationally invariant at field points ranging from the surface of the scatterer to the farfield and is an important extension of the usual Schwinger variational principle for the scattering amplitude, which is a farfield quantity. Also, a generic procedure, physically motivated by the general principles of boundary conditions and shadowing, is presented for constructing simple trial functions to approximate the fields. The variational principle and the trial function design are tested for the special case of a spherical scatterer and accurate answers are found over the entire frequency range.     

Variational approaches to the problems of plasma stability and of nonlinear plasma dynamics

A variational method is developed in order to investigate the nonlinear dynamics and stability of plasma using hydrodynamic plasma models, namely, the one-fluid, Hall, and electron MHD models. The key idea of the method is to adequately take into account variational symmetries and the associated conservation laws inherent in these hydrodynamic models. This approach is applied to derive variational criteria for the stability of a steadily moving plasma and to propose a variational method of the adiabatic separation of fast and slow motions, which makes it possible to simplify (reduce) the basic hydrodynamic models.