Showing papers on "Phase space published in 1980"
TL;DR: In this article, a modified jet evolution equation is proposed which resums large corrections to the usual leading logarithmic approximation when phase-space constraints expose the singular infrared structure of QCD.
234 citations
TL;DR: In this article, a pair of weakly coupled van der Pol oscillators are studied and the bifurcations of phase-locked periodic motions which occur as the coupling coefficients are varied are investigated.
Abstract: We study a pair of weakly coupled van der Pol oscillators and investigate the bifurcations of phase-locked periodic motions which occur as the coupling coefficients are varied. Perturbation methods are used and their relation to the topological structure of solutions in the four dimensional phase space is discussed. While the problem is formulated for general linear coupling, the case of detuning plus diffusive coupling via displacement and velocity is discussed in more detail. It is shown that up to four phase-locked periodic motions can exist in this case.
218 citations
TL;DR: In this article, a generalized definition of local and normal modes and the associated identification of normal modes as a (1:1) resonance between local zeroth order oscillators are presented.
Abstract: Normal and local mode behavior in molecular systems and the transition between them is explored using nonlinear mechanics techniques. Significant insight into this behavior and into the structure of phase space results from a generalized definition of local and normal modes and the associated identification of normal modes as a (1:1) resonance between local zeroth order oscillators. In addition to qualitative insight, this approach yields a simple formula [Eq. (28)] for the level of excitation at which local modes become possible. Applications to H2O and to the overtone spectroscopy of the dihalomethanes, benzene, and TMS are provided.
172 citations
TL;DR: In this paper, the tree-like evolution of a QCD jet in a planar gauge is studied, with particular emphasis on the soft parton region of the phase space.
137 citations
TL;DR: In this paper, the dynamics of collisions are described by classical trajectories as in the widely used quasiclassical method, but the initial and final internal states are represented in phase space in a quantum statistical way, using the Wigner distribution function.
Abstract: A new approach is presented in which classical mechanics is combined with quantum statistics to describe molecular collisions. In this approach, the dynamics of collisions is described by classical trajectories as in the widely used quasiclassical method. However, initial and final internal states are represented in phase space in a quantum statistical way, using the Wigner distribution function. Results of calculations performed on a collinear He–H2 collision indicate that this new method is more accurate than the quasiclassical method, especially when the initial vibrational energy is low. Moreover, the new method is capable of describing classically forbidden processes that cannot be accounted for by the quasiclassical method.
119 citations
TL;DR: In this paper, the authors proposed new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules, and found that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment.
Abstract: This paper proposes new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules. Several hallmarks of stochastic and nonstochastic behavior are identified, some of which are readily available from spectral data. We find that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment. This unsettling feature arises as a quantum analog of the necessity, in classical mechanics, of specifying the a priori known integrals of the motion before ergodic or stochastic behavior can be defined. In quantum mechanics, it is not possible to have flow or measure local properties (analog of trajectories and phase space measure) without some uncertainty in the integrals of the motion (most often the energy). This paper addresses the problems this creates for the definition of stochastic flow. Sev...
117 citations
TL;DR: In this article, the contraction of the description of Brownian motion from phase space to position space is discussed by means of non-Markovian Langevin equations in position space.
Abstract: The contraction of the description of Brownian motion from phase space to position space is discussed by means of non-Markovian Langevin equations in position space. A Fokker-Planck equation valid for any time is derived for the harmonic oscillator, and the overdamped, critical, and infradamped cases are discussed. For anharmonic potentials systematic corrections to the Smoluchowski equation are derived. Such corrections can be interpreted in this context as an expansion in powers of the correlation time of the “colored” stochastic noise appearing in the Langevin equation. The breakdown of the Fokker-Planck approximation is also discussed.
109 citations
TL;DR: In this article, a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states is derived for the simplest semisimple Lie group SU (2) and its classical consequences are investigated.
Abstract: Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two‐dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.
108 citations
TL;DR: In this article, the authors analyzed the Weyl and Wick symbols of path integrals in a phase space and found that these symbols are always discontinuous, which is a result of the assumption that the paths are always continuous.
Abstract: Feynman path integrals in a phase space are analyzed in detail. The analysis is based on the theory of operator symbols, in contrast with the traditional approach based on the direct use of canonical commutation relations. Particular attention is paid to the Weyl and Wick symbols, which are the most important in applications. The set of paths on which the integral is concentrated is studied. It is found that these paths are always discontinuous. This discontinuity is responsible for errors in certain papers on path integrals in a phase space. The most important properties of the Weyl and Wick symbols are reviewed.
88 citations
TL;DR: In this paper, a general introduction to and some results from studies of a procedure called variational transition-state theory are presented, which is a fundamental assumption of this theory is that the net rate of forward reaction at equilibrium equals the equilibrium flux in the product direction through the transition state.
Abstract: A general introduction to and some results from studies of a procedure called variational transition-state theory are presented. A fundamental assumption of this theory is that the net rate of forward reaction at equilibrium equals the equilibrium flux in the product direction through the transition state where the transition state is a surface in phase space dividing reactants from products. Classical generalized-transition-state-theory calculations for nine collinear systems are compared to classical trajectory calculations. This new technique should provide useful insight into the successes and failures of the conventional theory and useful quantitative estimates of possible errors on the predictions of conventional transition-state theory. This should also contribute to a more accurate theory now available for the practical calculations of chemical reaction rates and thermochemical and structural interpretations of rate processes. (BLM)
66 citations
TL;DR: In this paper, the time evolution of wavefunctions Φ( t ) composed of superpositions of energy eigenstates are compared when the wave functions are initiated in (1) the non-integrable and (2) the integrable regimes in the phase space of the Henon - Heiles system.
TL;DR: In this paper, a method for directly evaluating the boundary of reactivity bands is proposed, which is iterative, convergent and at each iteration step provides improved upper and lower bounds to the reaction probability.
Abstract: Instead of finding regions of reactivity in the asymptotic reactants (products) phase space, involving a two dimensional search, one may directly evaluate the boundary of reactivity bands. Here we provide a practical method, for the regime in which transition state theory is not exact, for directly evaluating such boundaries. The method is iterative, convergent and at each iteration step provides improved upper and lower bounds to the reaction probability. A numerical application to the hydrogen exchange reaction giving product distributions and reaction probabilities over a wide range of energies is provided. We find that the existence of two bounded trajectories that are not periodic is crucial to understanding the dynamics of the system.
TL;DR: In this paper, an analytic expression is obtained for an invariant attracting curveCα(a) in phase space, which becomes the central object of study, and regions of persistence and escape are described for characteristic values ofa.
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01 Jan 1980TL;DR: In this paper, three types of groups and their application to composite particle theory are described, i.e., the symmetric group associated with exchange, orbital partitions and the supermultiplet scheme.
Abstract: In the theory of nuclear structure and reactions, one often splits the full system into composite systems and studies the dynamics of these composite systems. In this report, I shall describe some methods of group theory which we have developed for dealing with these systems. We shall describe three types of groups and their application to composite particle theory. The symmetric group will be associated with exchange, orbital partitions and the supermultiplet scheme. The general linear group and its representations will be applied to exchange decompositions. The inhomogeneous symplectic transformations of classical phase space and their representations will be used to describe the kinematics and dynamics of composite particles.
TL;DR: In this article, the authors derived nonlinear identities for each standard "diffraction catastrophe" wave function and interpreted them in terms of projections of Wigner functions from phase space on to coordinate space.
Abstract: For each standard 'diffraction catastrophe' wavefunction psi , describing the interference pattern near a stable caustic, the authors derive two nonlinear identities. These relate the intensity mod psi mod 2 to an integral over the wavefunction psi corresponding to the same catastrophe, or to a less singular one. The identities are interpreted in terms of projections of Wigner functions from phase space on to coordinate space.
TL;DR: In this article, the authors present the first laboratory of measurements of divergence rates (or characteristic exponents), using a system of coupled tunnel diode relaxation oscillators, which is reliably associated with broadband spectra and both methods are used to characterize the motion as a function of the coupling strength and natural frequency ratio of the two oscillators.
Abstract: The exponential divergence of nearby phase space trajectories is a hallmark of nonperiodic (chaotic) behavior in dynamical systems. We present the first laboratory of measurements of divergence rates (or characteristic exponents), using a system of coupled tunnel diode relaxation oscillators. This property of sensitive dependence on initial conditions is reliably associated with broadband spectra, and both methods are used to characterize the motion as a function of the coupling strength and natural frequency ratio of the two oscillators. A simple piecewise linear model correctly predicts the major periodic and non-periodic regions of the parameter space, thus confirming that the chaotic behavior involves only a few degrees of freedom.
TL;DR: In this paper, the Schrodinger equation for a Hamiltonian H that is a general second order polynomial in the canonical coordinates qi and momenta pi is discussed, and the solution is implemented fully for n=1,2 and its structu...
Abstract: The solution of the Schrodinger equation for a Hamiltonian H that is a general second order polynomial in the canonical coordinates qi and momenta pi is discussed. Examples of such Hamiltonians abound in various fields of physics and, e.g., the problem of small vibrations has been solved a long time ago. In the general case we first use linear canonical transformations [the group Sp (2n, R)] to reduce H to a simplified representative for HR. Next we imbed each HR into a complete set of commuting second order integrals of motion, making use of a recently obtained classification of maximal Abelian subalgebras of the algebra sp (2n, R). This imbedding is then used to separate variables in the representative Schrodinger equations and to obtain complete sets of their eigenfunctions. Finally the solutions of the representative equations can be transformed back into those of the original one making use of representations of the canonical transformations. The program is implemented fully for n=1,2 and its structu...
TL;DR: In this article, a phase space is introduced and the equations of motion of a harmonic crystal, which need not be a primitive one, are explicitly solved by several methods using the classical Kubo-Martin-Schwinger (KMS) condition.
TL;DR: In this paper, it was shown that if Wigner's function is a delta function, it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In the classical limit the W space becomes the phase space parametrised by the canonical variables.
Abstract: According to Weyl one may associate a function with a dynamical operator; these functions depend on the parameters p and q and can be displayed in a p, q manifold, the W space. In the classical limit the W space becomes the phase space parametrised by the canonical variables. The function associated in this manner with the density operator is Wigner's function. It turns out that if Wigner's function is a delta function it cannot represent the density operator of a physically realisable state unless the argument of the delta-function is linear in the parameters a and q. In all other cases Wigner's function associated with a physically realisable state has a finite width, proportional to h 2 3 . Consequently straightness (linear combination of p and q) has a fundamental significance in the W space. Since this property is preserved under linear inhomogeneous transformations the W space will have a geometry generated by these transformations, the affine geometry of Euler, Moebius and Blaschke. In the present note we show how this comes about, how it simplifies the semiclassical approximations of Wigner's function, and makes one understand how in the classical limit this geometry is lost, allowing to be replaced by the geometry of canonical transformations.
TL;DR: In this paper, the structure of fully dispersed waves in dusty gases is investigated using the Navier-Stokes equations for the gas phase and for the particle phase, which is solved analytically in phase space by expanding the variables of state in power series.
Abstract: The structure of one-dimensional fully dispersed waves in dusty gases is investigated using the Navier-Stokes equations for the gas phase and for the particle phase. The resulting system of six ordinary nonlinear differential equations is reduced to a system of four autonomous nonlinear differential equations which is solved analytically in phase space by expanding the variables of state in power series.
TL;DR: In this paper, it was shown that almost all solutions to the linearized variational equations derived from bounded, integrable hamiltonian systems exhibit an average linear growth with time, becoming unbounded at t → ∞.
TL;DR: In this article, a covariant extension of classical field theory for extended phase space V8 has been proposed to incorporate Born's reciprocity, which demands equal status for the variables q and p. The present formulation is covariant under the extended Poincare group acting in V8.
Abstract: Classical field theory is developed in the arena of extended phase space V8, the space of position, time, momentum, and energy. This enables one to incorporate Born’s reciprocity which demands equal status for the variables q and p. The present formulation is covariant under the extended Poincare group P8 acting in V8. Variational methods for classical field theory are generalized. Besides the usual concept of the total 4‐momentum, one encounters the notion of average position and time of the field distributions. The total charge emerges from a dynamical viewpoint. The Dirac and Duffin–Kemmer algebras are generalized in this setting. The corresponding wave equations would lead to a dynamical theory of the elementary particles. The symplectic structure is not considered because of the difficulties to represent spinors.
TL;DR: In this paper, it is argued that even in non-relativistic quantum mechanics the coordinate and momentum variables cannot be interpreted directly as observables, and that only by defining these by suitable test bodies can one obtain verifiable predictions.
Abstract: It is being argued that even in non-relativistic quantum mechanics the coordinate and momentum variables cannot be interpreted directly as observables. Only by defining these by suitable test bodies can one obtain verifiable predictions. When these arguments are implemented on a Wigner distribution the author shows that positive phase space probabilities always ensue, and hence this function can be used as a quantum mechanical phase space function.
TL;DR: The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied in this article by transforming the time-dependant Hartree Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the HartreeFock energy surface in the phase space.
Abstract: The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For two-body interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the ''librations'' and ''rotations'' in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.
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TL;DR: In this article, the Schrodinger equation is replaced by the Wigner distribution function, which is a useful mathematical tool in spite of its poor physical properties, and is applied to the breaking of highly nonlinear waves in a cold plasma (usually treated by the Lagrangian method).
Abstract: The Schrodinger equation describes the motion of a particle in a statistical sense. It consequently possesses the two main properties of the Vlasov equation (dynamic and statistic) and can replace this last equation provided we take sophisticated initial conditions. The scheme must be considered as a new attempt to discretize intelligently the amount of information contained in the phase space distribution and to stop, without destroying it, the flow of information which usually goes to high wavenumbers in velocity space. The method is applied to the breaking of highly nonlinear waves in a cold plasma (usually treated by the Lagrangian method) and to double beam instability. It is shown that such an Eulerian scheme works quite well with a much smaller number of discretized functions than are required in the regular Fourier—Fourier or Fourier-Hermite methods. The central point is the introduction of the phase space Wigner distribution function which is a useful mathematical tool in spite of its poor physical properties.
01 Jan 1980
TL;DR: In this paper, the authors define the Gibbs states as the probability measures which are compatible with a given collection of conditional probabilities, defined in terms of the potential energy for the system being considered.
Abstract: Equilibrium behaviour in classical Statistical mechanics can be described in terms of probability measures (called Gibbs ensembles) on an appropriate phase space. In their simplest form the Gibbs ensembles correspond to systems lying in a bounded region of space. However, in order to study phenomena such as phase transitions it is necessary to consider infinite volume limits of these ensembles. The work of Dobrushin (1968 a), (1968 b), (1969), Lanford and Ruelle (1969) and Ruelle (1970) shows that these infinite volume limits, called Gibbs states, can be defined as the probability measures which are compatible with a given collection of conditional probabilities. These conditional probabiIities are defined in terms of the potential energy for the system being considered.
TL;DR: In this paper, the authors derived projective changes of variables under which the classical orbits of any such system are put in one-to-one correspondence with SO (3, 1) and/or SO (4) invariant sets of curves on similarly invariant surfaces.
Abstract: By adding to the force between an electric and a magnetic point charge a central force arising from a specially chosen potential, one can construct a system known to have the same SO (3,1) and/or SO (4) dynamical symmetry algebra as the Kepler system. We derive projective changes of variables under which the classical orbits of any such system are put in one‐to‐one correspondence with SO (3,1)‐ and/or SO (4)‐invariant sets of curves on similarly invariant surfaces. This extends results hitherto established only for the Kepler system. This is surprising in that there is a sense in which the phase space of such a magnetic system is a truncation of the Kepler phase space and so one might have expected such global properties not to generalize. Our transformations apparently do not permit transcription of the corresponding Schrodinger equation into a manifestly SO (3,1)‐ and/or SO (4)‐symmetric form, in contrast to the pure Kepler case. Such magnetic systems play roles in the theory of quantum fields in Taub–NUT space‐times, and in the theory of quantum‐mechanical fluctuations about extended magnetic monopoles in supersymmetric gauge theories. In passing, we use the properties of the magnetic systems to formulate a very short and direct proof that the classical orbits of the Kepler system are conic sections.
TL;DR: In this article, a nonlinear diffusion approximation for a previously derived master equation describing an inhomogeneous Boltzmann gas in a lumped phase space is proposed, which differs from the usual Langevin equations in three essential properties: the drift and random force are nonlinear, the random noise obeys a generalized fluctuation-dissipation theorem, and there is no reference to equilibrium.
Abstract: A nonlinear diffusion approximation for a previously derived master equation describing an inhomogeneous Boltzmann gas in a lumped phase space is proposed. A fluctuating kinetic equation is obtained which differs from the usual Langevin equations in three essential properties: the drift and random force are nonlinear, the random noise obeys a generalized fluctuation-dissipation theorem, and there is no reference to equilibrium. Relations with other approaches to hydrodynamic fluctuations are discussed. © 1980 The American Physical Society.
TL;DR: In this paper, the strange attractor of the Lorenz model is well approximated by suitably chosen two-dimensional invariant manifolds through the three stationary points of the flow in phase space.
Abstract: The strange attractor of the Lorenz model is found to be well approximated by suitably chosen two-dimensional invariant manifolds through the three stationary points of the flow in phase space. The stationary probability density, defined by the two-dimensional flow on the invariant manifolds, is determined in the vicinity of the origin of the phase space in terms of two parameters and compared with the numerically determined stationary distribution on the Lorenz attractor.
TL;DR: In this article, a repeated ring kinetic theory describing the reversible chemical reaction A+B?C+D is obtained, where the solute is diluted such that the kinetic theory describes isolated solute pairs interacting with the solvent.
Abstract: A repeated ring kinetic theory describing the reversible chemical reaction A+B?C+D is obtained. We derive a kinetic equation for the equilibrium time correlation function of the solute pair phase space density. The solute–solvent and solvent–solvent interactions are described by a hard sphere potential. The solute–solute interactions consists of both hard and soft potential terms. This choice permits the use of a pseudo‐Liouville operator formalism to obtain the kinetic theory. This greatly simplifies the derivation by comparison with a Liouville operator formalism for continuous potentials. The solute is dilute such that the kinetic theory describes isolated solute pairs interacting with the solvent. The static memory function of the kinetic equation is treated exactly. It describes solute propagation induced by direct interaction, Enskog collisions with the solvent and by the potential of mean force operating between the solute pair. The dynamic memory function is obtained at the repeated ring level of ...