Browsing by Author "Zweifel, Paul F."

Now showing 1 - 20 of 47
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    Analysis of the dispersion function for anisotropic longitudinal plasma waves
    Arthur, M. D.; Bowden, Robert L.; Zweifel, Paul F. (AIP Publishing, 1979-10)
    An analysis of the zeros of the dispersion function for longitudinal plasma waves is made. In particular, the plasma equilibriumdistribution function is assumed to have two relative maxima and is not necessarily an even function. The results of this analysis are used to obtain the Wiener–Hopf factorization of the dispersion function. A brief analysis of the coupled nonlinear integral equations for the Wiener–Hopf factors is also presented.
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    The aperiodically dampened loudspeaker system
    Short, George Elliott III (Virginia Tech, 1988-07-15)
    The theory of the aperiodically damped loudspeaker system is derived and investigated in both the time and frequency domains. The advantages of the aperiodic configuration over sealed box loudspeaker systems are discussed. The aperiodically damped loudspeaker system is described by the parameters used for sealed box loudspeaker systems, with some additions. A real system is constructed which takes full advantage of aperiodic configuration, and its performance is compared with a sealed box system of the same dimensions. Complete derivations of the theory for both the sealed box and aperiodically damped systems are included, as well as methods for measuring all required parameters.
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    The Born-Oppenheimer approximation in scattering theory
    Kargol, Armin (Virginia Tech, 1994-05-05)
    We analyze the Schrödinger equation i𝜖 ¬2â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬-4 and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬2). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.
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    Case eigenfunction expansion for a conservative medium
    Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1976-02)
    By using the resolvent integration technique introduced by Larsen and Habetler, the one‐speed, isotropic scattering,neutron transport equation is treated in the infinite and semi‐infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.
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    The Cauchy problem for the Diffusive-Vlasov-Enskog equations
    Lei, Peng (Virginia Tech, 1993-04-05)
    In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed.
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    Collided-flux-expansion method for the transport of muonic deuterium in finite media
    Rondoni, Lamberto; Zweifel, Paul F. (American Physical Society, 1991-07)
    Transport of muonic deuterium atoms in a slab of thickness d filled with a molecular deuterium gas is described by means of the multiple-collision expansion in the framework of a time-dependent theory. The relevant expressions for the emerging flux are derived. Numerically generated results are presented for several different cases, some of which are under experimental investigation. A justification of the approximations made in a previous work is given.
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    Conservative neutron-transport theory
    Bowden, Robert L.; Cameron, W. L.; Zweifel, Paul F. (AIP Publishing, 1977-02)
    A functional analytic development of the Case full_range and half_range expansions for the neutron transport equation for a conservative medium is presented. A technique suggested by Larsen is used to overcome the difficulties presented by the noninvertibility of the transport operator K _1 on its range. The method applied has considerable advantages over other approaches and is applicable to a class of abstract integro-differential equations.
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    Differential algebraic methods for obtaining approximate numerical solutions to the Hamilton-Jacobi equation
    Pusch, Gordon D. (Virginia Tech, 1990)
    I present two differential-algebraic (DA) methods for approximately solving the Hamilton- Jacobi (HJ) equation. I use the “automatic differentiation” property of DA to convert the nonlinear partial-differential HJ equation into a initial-value problem for a DA-valued first-order ordinary differential equation (ODE), the “HJ/DA equation”. The solution of either form of the HJ/DA equation is equivalent to a perturbative expansion of Hamilton’s principle function about some reference trajectory (RT) through the system. The HJ/DA method also extracts the equations of motion for the RT itself. Hamilton’s principle function generates the canonical transformation, or mapping, between the initial and final state of every trajectory through the system. Since the map is represented by a generating function, it must automatically be symplectic, even in the presence of round-off error. The DA-valued ODE produced by either form of HJ/DA is equivalent tc a hierarchically-ordered system of real-valued ODEs without “feedback” terms; therefore the hierarchy may be truncated at any (arbitrarily high) order without loss of self consistency. The HJ/DA equation may be numerically integrated using standard algorithms, if all mathematical operations are done in DA. I show that the norm of the DA-valued part of the solution is bounded by linear growth. The generating function may be used to track either particles or the moments of a particle distribution through the system. In the first method, all information about the perturbative dynamics is contained in the DA-valued generating function. I numerically integrate the HJ/DA equation, with the identity as the initial generating function. A difficulty with this approach is that not all canonical transformations can be represented by the class of generating functions connected to the identity; one finds that with the required initial conditions, the generating function becomes singular near caustics or foci. One may continue integrating through a caustic by using a Legendre transformation to obtain a new (but equivalent) generating function which is singular near the identity, but nonsingular near the caustic. However the Legendre transformation is a numerically costly procedure, so one would not want to do this often. This approach is therefore not practical for systems producing periodic motions, because one must perform a Legendre transformation four times per period. The second method avoids the caustic problem by representing only the nonlinear part of the dynamics by a generating function. The linearized dynamics is treated separately via matrix techniques. Since the nonlinear part of the dynamics may always be represented by a near-identity transformation, no problem occurs when passing through caustics. I successfully verify the HJ/DA method by applying it to three problems which can be solved in closed form. Finally, I demonstrate the method’s utility by using it to optimize the length of a lithium lens for minimum beam divergence via the moment-tracking technique.
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    Diffusion of muonic atoms
    Rusjan, Edmond; Zweifel, Paul F. (American Physical Society, 1988-10)
    Transport of muonic hydrogen and deuterium atoms in gaseous hydrogen and deuterium is studied in the diffusion approximation and by means of the multiple-collision expansion. The diffusion coefficient is derived. Numerical results of the time-dependent problem in slab geometry are presented for a number of initial energies, temperatures, and pressures.
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    A discrete-velocity, stationary Wigner equation
    Arnold, Anton; Lange, Horst; Zweifel, Paul F. (AIP Publishing, 2000-11)
    This paper is concerned with the one-dimensional stationary linear Wigner equation, a kinetic formulation of quantum mechanics. Specifically, we analyze the well-posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete-velocity model. We find that the problem is uniquely solvable if zero is not a discrete velocity. Otherwise one obtains a differential-algebraic equation of index 2 and, hence, the inflow data make the system overdetermined. (C) 2000 American Institute of Physics. [S0022-2488(00)00112-2].
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    Dissipation in Wigner-Poisson systems
    Lange, Horst; Zweifel, Paul F. (AIP Publishing, 1994-04)
    The Wigner-Poisson (WP) system (or quantum Vlasov-Poisson system) is modified to include dissipative terms in the Hamiltonian. By utilizing the equivalence of the WP system to the Schrodinger-Poisson system, global existence and uniqueness are proved and regularity properties are deduced. The proof differs somewhat from that for the nondissipative case treated previously by Brezzi-Markowich and Illner et al.; in particular the Hille-Yosida Theorem is used since the linear evolution is not unitary, and a Liapunov function is introduced to replace the energy, which is not conserved.
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    Erratum: the half-Hartley and half-Hilbert transforms (vol 35, pg 2648, 1994)
    Paverifontana, S. L.; Zweifel, Paul F. (AIP Publishing, 1994-11)
    Inversion formulas are obtained for the restrictions of the Hartley and Hilbert transforms to R+. Regularity results are derived, and an illustrative example presented.
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    Extension of Case formulas to Lp. Application to half and full space problems
    Larsen, E. W.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1975-05)
    The singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Hölder‐continuous functions to the function spaces X p = {f‖μf (μ) ‐ L p }, where the expansions are now to be interpreted in the X p norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with X p sources and incident distributions.
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    Functional analytic treatment of linear transport equations in kinetic theory and neutron transport theory
    Cameron, William Lyle (Virginia Tech, 1978-01-08)
    The temperature-density equation of Kinetic Theory and the conservative neutron transport equation are studied. In both cases a modified version of the Larsen-Habetler resolvent integration technique is applied to obtain full-range and half-range expansions. For the neutron transport equation the method applied is seen to have notational advantages over previous approaches. In the case of the temperature-density equation this development extends previous results by enlarging the class of expandable functions and has the added advantage of rigor and simplicity. As a natural extension of the Kinetic Theory results, an integral equation for the surface density is derived for half-space problems involving the boundary condition of arbitrary accommodation.
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    Global existence in L1 for the square-well kinetic equation
    Liu, Rongsheng (Virginia Tech, 1993-04-04)
    An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in L¹. global in time and for arbitrary initial data. is proved. A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates. Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved.
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    The half-Hartley and the half-Hilbert transform
    Paverifontana, S. L.; Zweifel, Paul F. (AIP Publishing, 1994-05)
    Inversion formulas are obtained for the restrictions of the Hartley and Hilbert transforms to R+. Regularity results are derived, and an illustrative example presented.
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    Inflow boundary conditions in quantum transport theory
    Lange, Horst; Toomire, Bruce V.; Zweifel, Paul F. (Hindawi Publishing Corporation, 1999)
    A linear (given potential) steady-state Wigner equation is considered in conjunction with inflow boundary conditions and relaxation-time terms. A brief review of the use of inflow conditions in the classical case is also discussed. An analytic expansion of solutions is shown and a recursion relation derived for the given problem under certain regularity assumptions on the given inflow data. The uniqueness of the physical current of the solutions is shown and a brief discussion of the lack of charge conservation associated with the relaxation-time is given.
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    L2-Indices for Perturbed Dirac Operators on Odd Dimensional Open Complete Manifolds
    Gajdzinski, Cezary (Virginia Tech, 1994-05-03)
    For perturbations of the Callias and Anghel type the L2-index of the perturbed Dirac operator on a Spin c -manifold is realized as the result of pairing an element in K -homology with an element of compactly supported K -cohomology. This is achieved by putting the problem of calculating the Fredholm index of the perturbed Dirac operator in the framework of KK-theory and using the identification of K-groups with KK-groups. The formula for the Fredholm index is given in terms of topological data of the Spin c-manifold and the structure of the perturbation.
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    A mean-field method for driven diffusive systems based on maximum entropy principle
    Pesheva, Nina Christova (Virginia Polytechnic Institute and State University, 1989)
    Here, we propose a method for generating a hierarchy of mean-field approximations to study the properties of the driven diffusive Ising model at nonequilibrium steady state. In addition, the present study offers a demonstration of the practical application of the information theoretic methods to a simple interacting nonequilibrium system. The application of maximum entropy principle to the system, which is in contact with a heat reservoir, leads to a minimization principle for the generalized Helmholtz free energy. At every level of approximation the latter is expressed in terms of the corresponding mean—field variables. These play the role of variational parameters. The rate equations for the mean-field variables, which incorporate the dynamics of the system, serve as constraints to the minimization procedure. The method is applicable to high temperatures as well to the low temperature phase coexistence regime and also has the potential for dealing with first-order phase transitions. At low temperatures the free energy is nonconvex and we use a Maxwell construction to find the relevant information for the system. To test the method we carry out numerical calculations at the pair level of approximation for the 2-dimensional driven diffusive Ising model on a square lattice with attractive interactions. The results reproduce quite well all the basic properties of the system as reported from Monte Carlo simulations.
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    Multigroup neutron transport. I. Full range
    Bowden, Robert L.; Sancaktar, Selim; Zweifel, Paul F. (AIP Publishing, 1976-01)
    A functional analytic approach to the N_group, isotropic scattering, particle transport problem is presented. A full_range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half_range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one_group case.