Showing papers on "Phase space published in 1976"

Wigner phase space method: Analysis for semiclassical applications

Abstract: We investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found not to reduce to classical dynamics in most circumstances, even as h→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct h‐expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.

Fokker-Planck equations for simple non-Markovian systems

Abstract: Exact generalized Fokker–Planck equations are derived from the linear Mori–Kubo generalized Langevin equation for the case of Gaussian but non‐Markovian noise. Fokker–Planck equations which generate the momentum and phase space probability distribution functions (pdf’s) for free Brownian particles and the phase space pdf for Brownian oscillators are presented. Also given is the generalized diffusion equation for the free Brownian particle pdf in the zero inertia limit. The generalized Fokker–Planck equations are similar in structure to the corresponding phenomenological equations. They, however, involve time‐dependent friction and frequency functions rather than phenomenological constants. Explicit results for the frequency and friction functions are given for the Debye solid model. These functions enter as simple multiplicative factors rather than as retarded kernels. Further the phase space Fokker–Planck equations contain an extra diffusive term, a mixed phase space second partial derivative, not occurr...

Parity operator and quantization of δ-functions

Abstract: In the Weyl quantization scheme, the δ-function at the origin of phase space corresponds to the parity operator. The quantization of a functionf(υ) on phase space is the operator ef(υ/2)W(υ)dυM, whereM is the parity andW(υ) the Weyl operator.

Quantum mechanics in phase space I. Unicity of the Wigner distribution function

Abstract: The joint distribution function in phase space is related to the density matrix by an integral transformation which depends on the rule of correspondence used. All the requirements which can be restrictive for the kernel function defining the transformation are studied. It is shown that the conditions of Galilei invariance, unitarity, reality and normalization lead to the Wigner kernel function in a unique way. Galilei invariance, the requirement that the free particle behaves classically, and the conditions to obtain the correct mixed distributions also lead to the same result.

Kinetic theory of a relativistic beam

Abstract: A Fokker–Planck equation is derived to study the evolution of a stable, low‐current beam propagating in a gas‐plasma medium. Small‐angle scattering of the beam particles by the medium causes diffusion in the phase space projected transverse to the direction of propagation. The projected components of dynamical friction vanish. As a result, there is a continued input of energy into the transverse particle motions, which is taken up in expansion against the pinch field. A quasi‐static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker–Planck equation with scale radius increasing in accord with Nordsieck’s formula. An H theorem is proved and the Bennett distribution is shown to minimize both H and −dH/dt; hence, it is the time‐dependent asymptotic state. The predicted current profile and radius are shown to be in fair agreement with experiment.

Dynamical quantization of the Kepler manifold

Abstract: The dynamical quantization of the ’’Kepler manifold’’ in any number of degrees of freedom is constructed. The Kepler manifold is the phase space of the regularized Kepler motion and is shown to be an SO(n,2) ‐homogeneous symplectic manifold, corresponding to an extremely singular orbit in the co‐adjoint representation; the quantization is obtained by ’’approximating’’ this orbit by more regular ones, which are equivalent to homogeneous bounded domains. The most relevant result is that the usual quantum‐mechanical ’’hydrogen atom’’ model is recovered in the particular representation introduced by Fock in 1935 [SO(n) ‐homogeneous integral equation in momentum space].

Stochastic wave‐kinetic theory in the Liouville approximation

Abstract: The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic, random media is investigated on the basis of a stochastic, collisionless, Liouville‐type equation governing the temporal evolution of a phase‐space Wigner distribution density function. Within the framework of the first‐order smoothing approximation, a general diffusion–convolution‐type kinetic or transport equation is derived for the mean phase‐space distribution function containing generalized (nonloral, with memory) diffusion, friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long‐time, Markovian, diffusion approximation, a general set of Fokker–Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.

Semiclassical eigenvalues for non-separable bound systems from classical trajectories

Abstract: A new form of the semiclassical quantum conditions in non-separable systems is proposed. In two dimensions (2D) it has the form (ħ = 1) where CΣ is the path of a classical trajectory closed in phase space, Nx and Ny are the number of circuits in the x and y ‘senses’ on the invariant toroid and Jx and Jy are the ‘good’ action variables on the toroid; these action variables, Jx and Jy , must have the values 2π(n 1 + ½) and 2π(n 2 + ½) respectively where n 1 and n 2 are the integer quantum numbers. Closed classical trajectories occur only for the exceptional toroids with rational frequency ratios. In the general case we imply that the trajectory has closed on itself to some arbitrary accuracy. Results for the 2D potentials studied are in agreement with previously published work. It is shown how the method may be extended to 3D systems.

Correspondence rules and path integrals

Abstract: Using the Weyl correspondence, Mizrahi has derived an expression for the transition amplitude as a path integral in phase space. It is shown that the same result follows if any correspondence rule is used.

Quantum two‐particle scattering in fuzzy phase space

Abstract: The concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy‐phase‐space representation spaces L2(Γs), 0

Anomalous transport due to long-lived fluctuations in plasma Part 1. A general formalism for two-time fluctuations

Abstract: A general formalism for describing two-time fluctuations in magnetized plasma is presented. Two-time expectations of one-body operators (phase functions) are written in terms of the phase space density autocorrelation function where δ N is the fluctuation in the singular Klimontovich microdensity. It is shown that is the first member of a set of two-time quantities which collectively obeys the linearized BBGKY cumulant hierarchy in the ( X i , t ) variables, with initial conditions successively smaller in the plasma parameter . We study in detail the case of fluctuations in thermal equilibrium, although the general formalism holds also for the non-equilibrium case. To lowest order in e P , Γ obeys the linearized Vlasov equation. From this are recovered all of Rostoker's results for fluctuations excited by Cherenkov emission and absorbed by Landau damping, as well as a constructive proof of the test particle superposition principle. To first order, Γ obeys (in the Markovian approximation) the linearized Balescu-Guernsey-Lenard equation. For frequencies and wavenumbers in the hydrodynamic regime, the velocity moments of Γ obey linearized fluid equations with classical transport coefficients (i.e. essentially those computed by Braginskii in the 3-D case). It has been found that the classical theory is in disagreement with certain computer and laboratory experiments performed in strong magnetic fields. This defect is attributed to the absence in the classical theory of contributions to the collision operator, hence transport coefficients, of fluctuations long-lived on the Vlasov scale. Analogous difficulties arise in the theory of hydrodynamics in neutral fluids. To improve the plasma theory, a renormalization of the two-time hierarchy is proposed which sums selected terms from all orders in e P and thus treats the hydrodynamic fluctuations self-consistently. The resulting theory retains appropriate fluid conservation laws, thereby avoiding erroneous results encountered in certain diffusing orbit theories, when the fluid viscosity is indiscriminantly replaced by the test particle diffusion coefficient. In order to explain the results of the computer simulations, the theory is applied in part 2 to the problem of anomalous hydrodynamic contributions to the transport coefficients.

Kinetic theory of turbulent compressible flows and comparison with classical theory

Abstract: A complete set of turbulent correlations is given in terms of expansion coefficients in a double series of Hermite polynomials of a two‐particle correlation in phase space. Only two of these coefficients, corresponding to Reynolds’ stress and turbulent heat flux, are shown to appear in gasdynamic equations of turbulence. This is accomplished by incorporating the phase‐space correlation in the collision integral of Boltzmann’s equation, and by deriving generalized Navier–Stokes and Fourier transport relations. In contrast to the classical formalism, the simple expression remains invariant in form whether the flow is compressible or not. Nevertheless, a bilinear transformation of fluctuating quantities shows that the two formalisms are identical with regard to terms of double correlations. The kinetic theory justifies nonexistence of the higher order correlations characteristic of compressible turbulence equations of the classical regime.

Canonical Transformations and Phase Space Path Integrals

Abstract: A previous discussion of canonical transformations and path integrals is extended to the phase space path integral method. Within this approach a broader class of canonical transformations can be introduced than within the Lagrangian approach, including coordinate transformations and essentially all infinitesimal tranformations.

The classical limit for quantum dynamical semigroups

Abstract: We describe a class of single-particle quantum-mechanical dynamical semigroups which, in the classical limit, give rise to Markov semigroups on phase space.

Approximately Relativistic Hamiltonians for Interacting Particles

Abstract: In the relativistic canonical formalism of Bakamjian and Thomas describing direct particle interactions the generators are defined in terms of the total momentum, the center-of-mass position, and a complete set of additional intrinsic canonical variables. In the interaction region of phase space the transformation linking these variables to individual particle coordinates and momenta is not determined by basic principles. In this paper canonical transformations to single-particle variables valid to order c/sup -2/ and the corresponding approximate Hamiltonians are constructed for a two-particle system; approximate many-body Hamiltonians are then constructed from the two-body ones, maintaining the Lie algebra of the Poincare group to the same order. If, and only if, the nonrelativistic limit of the potential is velocity independent (except for a possible spin-orbit interaction) it is possible to require, to order c/sup -2/, transformation properties of the position operators corresponding to the classical world-line conditions. This requirement implies restrictions on admissible canonical transformations to single-particle variables. The cluster separability condition is then automatically satisfied. In the classical limit the class of approximately relativistic Hamiltonians for spinless particles is identical with that obtained by Woodcock and Havas from expansion of an exact Poincare-invariant Fokker-type variational principle automatically satisfying the world-line conditions.more » Conversely, direct quantization of their classical Hamiltonians is shown to lead to the approximate quantum-mechanical ones resulting from the Bakamjian-Thomas theory. The relation of these results to various approximately relativistic Hamiltonians built up by several authors starting from the nonrelativistic theory is discussed, as well as their implications for phenomenological nucleon-nucleon potentials. (AIP)« less

Dispersion of ensembles of non-interacting particles

Abstract: The dynamics of an ensemble of particles emanating from a common point with a distribution of velocities is modeled as a continuum of particles described by a phase space distribution function. A general solution for the distribution function and the associated spatial density function is obtained for a general dynamical system. The special cases of linear dynamical systems and slow dispersion from a circular orbit are treated in detail. A transcendental equation is derived, the roots of which determine the time since initial dispersion from knowledge of the spatial density function at later times.

Localizability of relativistic particles in fuzzy phase space

Abstract: The concept of fuzzy phase space Gamma s, recently introduced in non-relativistic quantum mechanics, is extended to the relativistic case The L2( Gamma s) representation of the wave packet is used as a basis for discussing localizability in Gamma s for non-zero mass particles as well as for the photon

Two new proofs of the test particle superposition principle of plasma kinetic theory

Abstract: The test particle superposition principle of plasma kinetic theory is discussed in relation to the recent theory of two‐time fluctuations in plasma given by Williams and Oberman. Both a new deductive and a new inductive proof of the principle are presented; the deductive approach appears here for the first time in the literature. The fundamental observation is that two‐time expectations of one‐body operators are determined completely in terms of the (x,v) phase space density autocorrelation, which to lowest order in the discreteness parameter obeys the linearized Vlasov equation with singular initial condition. For the deductive proof, this equation is solved formally using time‐ordered operators, and the solution is then re‐arranged into the superposition principle. The inductive proof is simpler than Rostoker’s although similar in some ways; it differs in that first‐order equations for pair correlation functions need not be invoked. It is pointed out that the superposition principle is also applicable to the short‐time theory of neutral fluids.

Aperiodic Trajectories and Stationary Points in a Three-Component Spectral Model of Atmospheric Flow.

Abstract: Aperiodic solutions to spectrally truncated models based on the vorticity equation are considered for the case of a zonal flow interacting nonlinearly with two other components both having the same zonal wavenumber. It is shown that all such aperiodic trajectories proceed asymptotically to either a stationary point in the phase space of coefficients or to a periodic solution with steady amplitudes. It is also shown that the set of such solutions is of measure zero on surfaces of constant energy in phase space. Thus if the initial coefficients for a nonlinear, three-component flow are selected at random, then the resulting flow will in all probability be periodic.

The N -body problem of celestial mechanics

Abstract: A selective survey of then-body problem of celestial mechanics is given where the emphasis is on the asymptotic behavior of all solutions ast→∞, the possible configurations the particles can assume in phase space and in physical space, and collision and non-collision singularities.

The uniform phase space representation of product state distributions of elementary chemical reactions

Abstract: The uniform phase space (UPS) representation of translational or internal degrees of freedom of the products of a chemical reaction is presented. Procedures are developed to deal with cases in which the final state distribution is discrete or approximately continuous; extensions cover multidimensional distributions. Explicit realistic models are used to demonstrate these situations.In the UPS representation, the statistical or phase space distribution over product states is constant, yielding equal volumes of phase space for equal intervals along the coordinate in the UPS representation. As a consequence, an actual distribution over product states, cast into the UPS representation, becomes a direct measure of its deviance from a statistical distribution. The information content (in the information theoretic sense) is invariant under transformation to this representation, and its value can be readily approximated by a simple histogrammic procedure. Calculations for realistic model problems illustrate the s...

Invariances of approximately relativistic Hamiltonians and the center-of-mass theorem

Abstract: In an earlier paper we considered a class of Lagrangians for directly interacting particles, arising from a slow-motion approximation in various special- and general-relativistic field theories. It was shown that if the Lagrangian is invariant under time and space translations this implies invariance under an additional three-parameter set of infinitesimal transformations, which leads directly to the center-of-mass theorem. This result is rederived here in a Hamiltonian formalism, in which these infinitesimal transformations are shown to be generators of a Lie symmetry group in phase space. Then we consider the problem of the most general form possible of a canonical post-Newtonian theory that is a realization of the Lie algebra of the Poincar\'e group to order ${c}^{\ensuremath{-}2}$ and that arises from a theory of the usual Newtonian type with two-body interactions. It is found that in such a theory the world-line condition is satisfied to order ${c}^{\ensuremath{-}2}$. This canonical theory encompasses all the approximately relativistic interactions, found recently by Woodcock and Havas, which follow from a Fokker-type special-relativistic variational principle for particles with direct two-body interactions. The relation of our work to various other approaches to approximately relativistic theories of interacting particles is discussed.