A path integral approach to inclusive processes

The single-particle inclusive differential cross-section for a reaction is written as the imaginary part of a correlation function in a forward scattering amplitude for in a modified effective theory. In this modified theory the interaction Hamiltonian equals in the original theory up to a certain time. Then there is a sign change and becomes nonlocal. This is worked out in detail for scalar field models and for QED plus the abelian gluon model. A suitable path integral for direct calculations of inclusive cross sections is presented. Received: 8 March 2000 / Published online: 6 July 2000     

Sound waves in a liquid with polydisperse vapor–gas bubbles

A mathematical model is presented for the propagation of plane, spherical, and cylindrical sound waves in a liquid containing polydisperse vapor–gas bubbles with allowance for phase transitions. A system of integro-differential equations is constructed to describe perturbed motion of a two-phase mixture, and a dispersion relation is derived. An expression for equilibrium sound velocity is obtained for a gas–liquid or vapor–liquid mixture. The theoretical results agree well with the known experimental data. The dispersion curves obtained for the phase velocity and the attenuation coefficient in a mixture of water with vapor–gas bubbles are compared for various values of vapor concentration in the bubbles and various bubble distributions in size. The evolution of pressure pulses of plane and cylindrical waves is demonstrated for different values of the initial vapor concentration in bubbles. The calculated frequency dependence of the phase sound velocity in a mixture of water with vapor bubbles is compared with experimental data.     

Generalized Noether theorems in canonical formalism for field theories and their applications

A generalization of Noether's first theorem in phase space for an invariant system with a singular Lagrangian in field theories is derived and a generalization of Noether's second theorem in phase space for a noninvariant system in field theories is deduced. A counterexample is given to show that Dirac's conjecture fails. Some preliminary applications of the generalized Noether second theorem to the gauge field theories are discussed. It is pointed out that for certain systems with a noninvariant Lagrangian in canonical variables for field theories there is also a Dirac constraint. Along the trajectory of motion for a gauge-invariant system some supplementary relations of canonical variables and Lagrange multipliers connected with secondary first-class constraints are obtained.     

Quantum Nonlocality of Photon Pairs in Interference in a Mach-Zehnder Interferometer

Nonlocal character of entangled photon pairs generated in spontaneous parametric downconversion is demonstrated. One photon from a pair propagates through a Mach-Zehnder interferometer and is detected in coincidence-count measurement with its twin. Width of the coincidence-count interference pattern (measured for various values of path difference in the Mach-Zehnder interferometer) depends on spectral width of the twin as a result of entanglement of photons in a pair. The experimental setup is analyzed for a Gaussian spectral filter and a Fabry-Perot resonator. It is shown that nonlocal interference is much stronger for cw pumping in comparison with femtosecond pulsed pumping for values of parameters commonly used in spontaneous parametric downconversion experiments.     

Cross sections for the photodisintegration of a deuteron and neutron capture by a proton calculated for the case of nonlocal potential

The processes of deuteron photodisintegration by a gamma-quantum and the radiative capture of a neutron by a proton with the emission of a gamma-quantum are considered. Interaction between nucleons is described by a nonlocal potential of the Yamaguchi type but allowing for repulsion due to the nucleon cores. In contrast to other potentials, the Schroedinger equation is solved exactly for the proposed potential. The potential is more exactly defined in comparison with the previously obtained values for the parameters; information on this potential is important in solving certain fundamental problems in nuclear theory. The effective cross sections for photodisintegration of a deuteron and for radiative capture are computed. Calculations show that the proposed potential makes it possible to describe the photodisintegration processes quite accurately for intermediate (up to 20 MeV) energies. The cross section computed for radiative capture is in somewhat better agreement with experiment than is the same cross section determined for other potentials.     

A collapsing bubble in a fluid: a statistical mechanical approach

The first non-equilibrium statistical mechanical theory is presented for the mechanical and thermal behaviour of the collapse of a microsropically small bubble in a liquid. First the number density and temperature space-time profiles for the special case of weakly interacting particles, the perfect gas model, are obtained. This is then generalized to a model in which the motion of the molecules is characterized by a single finite diffusion constant. The results for the collapse of a small bubble in a typical fluid are compared with those recently obtained through computer simulation. The agreement with the simulation is remarkably good for the perfect gas model; very high temperatures, sufficient for sonoluminescence, appear in a simple and natural way. An unexpected conclusion is that the perfect gas model agrees better with computer simulation than the model characterized by a single bulk diffusion constant. This may be because the collapse of the bubble is controlled by the leading shell of the fluid where the fluid density is low.     

Plane wave propagation in a uniaxial bianisotropic medium with an application to a polarization transformer

Uniaxial bianisotropic medium is a generalization of the well-studied bi-isotropic and chiral media. It is obtained, for example, when microscopic helices with parallel axes are positioned in a host dielectric in random locations. Plane wave propagation in such a medium is studied and a simple solution for the dispersion equation and for the eigenwaves are found. As a numerical example, polarization properties of a transverse wave propagating in a uniaxial bianisotropic medium is considered. The results give a simple possibility to construct a polarization transformer with a transversely uniaxial chiral medium for changing the polarization of a propagating plane wave.     

Tunneling in a “breathing” double well: Adiabatic and antiadiabatic limits and tunneling suppression

Tunneling in a piecewise harmonic potential coupled to a harmonic oscillator is considered by means of the path integral technique. The reduced propagator for the tunneling particle is calculated explicitly and the tunneling splitting is found in semiclassical approximation. The result holds for arbitrary values of the parameters of the system. From this the adiabatic and antiadiabatic approximations are obtained as particular cases and compared with the results obtained differently. The limit of a strong interaction is also considered. It is found that for strong interaction or equivalently for the harmonic frequency tending to zero the preexponential factor in the tunneling splitting tends to zero which results in a suppression of tunneling. Implications of this result for tunneling in a more general potential are discussed.     

An analytic study of the friedel criterion in disordered magnetic states in transition metals in the presence ofS-D hybridisation,using the cluster bethe lattice approach

In this paper we present a new theory for including both short range order and long range order simultaneously in the well known cluster Bethe lattice method for binary alloys. We have used this theory for obtaining the Friedel criterion for the appearance of magnetic moments in disordered states using the single band Hubbard model. This is followed by a study of this criterion in a two-bands-d hybridised Hubbard model, which is considered as a simulation of real transition metals. A new technique for solving this problem in the Bethe lattice network is presented, which yields an analytic solution for the critical correlation strength in the presence of hybridisation and short range order. It is found that in all cases hybridisation tends to diminish the tendency for magnetisation, which is in accord with physical expectations.     

A semi-discrete central scheme for magnetohydrodynamics on orthogonal–curvilinear grids

This work describes a novel scheme for the equations of magnetohydrodynamics on orthogonal–curvilinear grids within a finite-volume framework. The scheme is based on a combination of central-upwind techniques for hyperbolic conservation laws and projection–evolution methods originally developed for Hamilton–Jacobi equations. The scheme is derived in semi-discrete form, and a full-fledged version is obtained by applying any stable and accurate solver for integration in time. The divergence-free condition of the magnetic field is a built-in property of the scheme by virtue of a constrained-transport ansatz for the induction equation. From the general formulation second-order accurate schemes for cylindrical grids and spherical grids are introduced in some more detail pointing out their potential importance in many applications. Special emphasis in this context is put to a treatment of the geometric axis implying severe complications because of the presence of coordinate singularities and associated grid degeneracy. An attempt to tackle these problems is presented. Numerical experiments illustrate the overall robustness and performance of the scheme for a small suite of tests.     

Revisiting the moving force problem

The problem of the vibration of a beam subject to a travelling force is considered. The purpose of the study is to develop simple tools for finding the maximum deflection of a beam for any given velocity of the travelling force. It is shown that, for given boundary conditions, there exists a unique response-velocity dependence function. A technique to determine this function is suggested, which is based on the assumption that the maximum beam response can be adequately approximated by means of the first beam mode. To illustrate this, the maximum response function is calculated analytically for a simply supported (SS) beam and constructed numerically for a clamped-clamped beam. The effect of the higher modes on the maximum response is investigated, and the relative error of the one-mode approximation for a SS beam is constructed. The estimates obtained substantiate the assumption about adequacy of the one-mode approximation in a wide range of velocities; in particular, the relative error in the neighborhood of the velocity that results in the largest response is less than one percent.     

Potential surface waves at the boundary of a metal with a magnetoactive plasma at finite pressure

The propagation of surface-type potential waves along the interfacial boundary of a plasma with an ideally conducting metal in an external magnetic field perpendicular to the boundary is examined. It is shown that a necessary condition for the existence of these waves in the system is a finite gas kinetic pressure. Dispersion relations for these waves and expressions for the penetration depth of the wave fields into the plasma are obtained, and they are studied numerically for various plasma parameters. The frequency region for propagation of these waves is found. It is also shown that in a nonzero external magnetic field a system of this kind has a range of frequencies in which the wave is a generalized surface wave. Zh. Tekh. Fiz. 69, 30–33 (November 1999)     

Optical theorem and Aharonov-Bohm scattering

A rigorous derivation of the optical theorem (OT) from the conservation of probability flux (CPF) is presented for scattering on an arbitrary spherically symmetric potential inN-spatial dimensions (ND). The constructed expression for the OT is found to yield the corresponding well-known results for two- and three-dimensional cases in a rather natural way. The Aharonov-Bohm (AB) effect is considered as a scattering event of an electron by a magnetic field confined in an infinitely long shielded solenoid and a similar derivation is attempted for an appropriate optical theorem. Our current understanding of the scattering theory is found to be inadequate for the purpose. The reason for this is discussed in some detail.     

Functional Fokker-Planck treatment of electromagnetic field propagation in a thermal medium

The quantum statistics of continuous space time dependent electromagnetic fields is analyzed by means of functionals. The case of a field propagating in a thermal reservoir serves as a simple example to illustrate the succeeding steps: a masterequation is derived for the density operator which is a functional of the field operators. By means of the coherent state representation for continuous fields the masterequation is transformed into a functional differential equation in the function space, spanned by the coherent state amplitudes. This equation is of the Fokker-Planck type and determines a Gaussian process for a continuum of variables or a field. It is solved by determining the characteristics in function space of the associated equation of motion for the characteristic functional and subsequent functional integration. The solution is used to calculate some correlation functions and the spectral function of the field.     

On classical and quantum relativistic dynamics

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.     

A numerical method for acoustic normal modes for shear flows

The normal modes and their propagation numbers for acoustic propagation in wave guides with flow are the eigenvectors and eigenvalues of a boundary value problem for a non-standard Sturm-Liouville problem. It is non-standard because it depends non-linearly on the eigenvalue parameter. (In the classical problem for ducts with no flow, the problem depends linearly on the eigenvalue parameter.) In this paper a method is presented for the fast numerical solution of this problem. It is a generalization of a method that was developed for the classical problem. A finite difference method is employed that combines well known numerical techniques and a generalization of the Sturm sequence method to solve the resulting algebraic eigenvalue problem. Then a modified Richardson extrapolation method is used that dramatically increases the accuracy of the computed eigenvalues. The method is then applied to two problems. They correspond to acoustic propagation in the ocean in the presence of a current, and to acoustic propagation in shear layers over flat plates.     

Time adaptive variational integrators: A space-time geodesic approach

The goal of this paper is to show that the space-time geodesic approach of classical mechanics can be used to generate a time adaptive variational integration scheme. The only assumption we make is that the Lagrangian for the system is in a separable form. The geometric structure which is preserved in the integration scheme is made explicit and the algorithm is illustrated with simulation for a compact case, a non-compact case, a chaotic system which arises as a perturbation of an integrable system and the figure eight solution for a three body problem.     

Singular scaling functions in clustering phenomena

We study clustering in a stochastic system of particles sliding down a fluctuating surface in one and two dimensions. In steady state, the density-density correlation function is a scaling function of separation and system size. This scaling function is singular for small argument — it exhibits a cusp singularity for particles with mutual exclusion, and a divergence for noninteracting particles. The steady state is characterized by giant fluctuations which do not damp down in the thermodynamic limit. The autocorrelation function is a singular scaling function of time and system size. The scaling properties are surprisingly similar to those for particles moving in a quenched disordered environment that results if the surface is frozen.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

The response of a resonator under a turbulent boundary layer to a high amplitude non-harmonic sound field

A method for finding the acoustic response of a Helmholtz resonator in the wall of a pipe containing fully developed turbulent pipe flow is described, for the case of an intense sound field that can be non-harmonic. The equation of motion for the fluid in the orifice is solved numerically in the time domain, and the orifice resistance is made to vary, during the oscillation cycle, in a way that is governed by the grazing flow parameters and the instantaneous mean flow velocity in the orifice. The mass end correction of the orifice is also allowed to be a function of the grazing flow. Predictions of the cavity pressure variation are presented, for a specified pipe pressure variation, and are compared to measurements; generally good agreement between the two is noted.