Showing papers on "Phase space published in 1972"

Theory of Phase Space Density Granulation in Plasma

Abstract: A novel state of turbulent plasma characterized by small scale phase‐space granulations called “clumps” is proposed. Clumps are produced when regions of different phase space density are mixed by the fluctuating electric field. They move along ballistic orbits and drive the turbulent field in a manner similar to that in which thermal fluctuations are driven by particle discreteness. In the coherent wave limit the clumps become the familiar trapped particle eddies of a Bernstein‐Green‐Kruskal mode. The turbulent state can exist in the absence of linear instability although it is more likely to occur in a linearly unstable plasma. The spectrum contains a ballistic portion as well as resonances at the wave (collective) frequencies. The discreteness of the clumps produces collision‐like process. For example, the average distribution function satisfies a Fokker‐Planck equation instead of a quasilinear diffusion equation.

Analytic continuation of classical mechanics for classically forbidden collision processes.

Abstract: Classically forbidden processes are those that cannot take place via ordinary classical dynamics. Within the framework of classical S‐matrix theory, however, classical mechanics can be analytically continued and classical‐limit approximations obtained for these classically forbidden, or weak transition amplitudes (i.e., S‐matrix elements). The most powerful and general way of analytically continuing classical mechanics for a complex dynamical system is to integrate the equations of motion themselves through the classically inaccessible regions of phase space. Success in calculating these analytically continued trajectories is reported in this work; with certain special features of these complex‐valued trajectories recognized and taken account of, it is seen that they are essentially as easy to deal with numerically as ordinary (i.e., real) classical trajectories. Application to the linear A+BC collision (vibrational excitation) gives excellent results; transition probabilities as small as 10−11 (the smallest ones available for comparison) have been obtained, agreement with the exact quantum mechanical values being within a few percent.

Experimental Observation of Nonlinear Wave-Particle Interactions in a Weak Cold Beam-Plasma System

Abstract: Nonlinear evolution of a beam-plasma instability has been investigated by direct measurements on the energy distribution of beam electrons and the field intensity of an externally driven electrostatic wave. The initial exponential growth of the excited single wave is halted when the wave traps the electrons and rotates them in phase space. At the phase of spatial bunching of the beam, its harmonics are observed to have maximum values. The results agree with those predicted by a single wave model.

On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior

Abstract: A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillator system H=12(p12+p22+q12+q22)+q12q2−13q23. Using procedures developed by Greene [J. Math. Phys. 9, 760 (1968)], empirical evidence has been obtained indicating that this system has a dense or near dense set of unstable periodic orbits throughout its stochastic (unstable) regions of phase space. The extent to which such stochastic regions exhibit C‐system behavior, i.e., ergodicity and mixing, is discussed. Finally, the above Hamiltonian system is shown to be intimately related to the Fermi‐Pasta‐Ulam system as well as to the Toda lattice.

Plasma distribution function near the loss cone of a mirror machine

Abstract: The particle distribution in phase space near the loss cone of a stable mirror machine is determined by an asymptotic analysis based on the smallness of λ, the ratio of transit time between mirrors to mean collision time. Away from the loss cone, the solution is obtained by averaging over a bounce motion. Near the loss Cone, i.e., within a distance scaling as λ1/2, the governing equations are converted into a Wiener-Hopf equation along a boundary line in phase space whose solution yields the desired distribution. It is found that the ratio of plasma density in the mirror throat to central density scales as λ3/4 while the ratio of external density to central density scales as λ. For typical reactor conditions, these ratios correspond to densities of 1011 cm−3 and 108, respectively. Since the distance over which this drop occurs scales as λ1/2 times the local magnetic scale length, it is necessary to re-examine the reflection of unstable plasma waves in the neighbourhood of the mirrors.

On the WKB Approximation in Time‐Dependent Scattering Theory Including Rearrangement Processes

Abstract: The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrodinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner‐Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.

The velocity distribution function for a polymer chain

Abstract: The Gibbs distribution for a polymer molecule contains implicitly the correlation functions for velocity and position of the constituent monomers. The purely spatial part, ignoring potentials, gives the random flight distribution; the velocity correlation function is calculated, exploiting the markovian structure of the problem in phase space. It is shown that the problem can be reduced to an eigenfunction problem and hence solved. The form of the correlation function is quite accurately given by ( nu n1- nu n2)2=(2kT/m)(1-exp(-c mod n1-n2 mod )) where nu n1, nu n2 are the velocities of the n1th and n2th monomers and c equivalent to 1.

Optimization of the Phase Acceptance of the Triumf Cyclotron

Abstract: The horizontal and vertical beam behaviour in the TRIUMF cyclotron has been calculated numerically up to 20 MeV. Effects limiting the cyclotron phase acceptance for an extracted beam with good emittance and energy resolution are discussed, as wel1 as ways of overcoming these effects. Two effects which are critical because of their strong phase dependence are vertical electric focusing at the dee gaps and coupling between the radial and longitudinal motions. With only the RF fundamental present it is shown that the second of these effects can be reduced considerably by the use of local bumps in either the average magnetic field or its second harmonic component, or both. The addition of a third harmonic RF component suitably phase shifted from the fundamental results in the phase dependence of both effects being considerably reduced over a wide phase range. For an initial beam emittance of 0.5π in.‐mrad and an extracted energy resolution of ±600 keV the net phase acceptance is expected to be ∼30 deg with fundamental RF only and ∼60 deg with 24% third harmonic RF added leading 8 deg in phase.

A pictorial study of an invariant torus in phase space of four dimensions

Abstract: An investigation was conducted with the aid of a computer graphics device at Goddard Space Flight Center to study the behavior of the invariant manifolds of a particular fourth-order equation, as a parameter in the equation is varied over the interval from 0 to 1. The equation consists of two coupled Van der Pol equations. For a small parameter value, the manifold is an asymptotically stable torus, where the flow on the torus is simply a rotation. As the value of the parameter is increased, the only thing that changes is the nature of the flow on the torus, which itself persists throughout the parameter variation. It is shown that ultimately the four periodic cycles which appear play a more significant part in the phase profile of the system than does the torus itself.

Energy Distributions of Reaction Product by Statistical Phase Space Theory

Abstract: The phase space theory of chemical reaction in which all accessible channels are uniformly weighted was shown to be inappropriate for describing exothermic reactions. Modification to the theory was introduced by a concept of partial statisticity in which only that portion of the phase space available to each vibrational level was treated statistically. The product rotational and translational distributions of the reaction, Cl + HI, and its isotopic variant, Cl + DI, were computed by this modified formulation and compared with that obtained from a straight application of the phase space theory. The implication of the profound differences found from these two formulations was discussed in light of experimental data.

Contrastreaming instability in a one-dimensional finite-length system.

Abstract: Two independent and oppositely directed electron beams with equal densities and speeds are considered to interpenetrate one another in a finite‐length system with a stationary neutralizing ion background. The nonlinear development of the resulting electrostatic waves is followed both in time and space by numerical simulation. The results include the time‐dependent behavior of the particle trajectories governed by the equation W = 12mv2−| qφ0(t) | coskmaxz. In the cold beam case a hysteresis‐like loop in phase space is observed to develop in the transition from the laminar to a turbulent state. The vortices formed in velocity space are found to be absorbed at the right‐hand boundary of the finite‐length system and coalesce into one another for all warm beam cases. The total electric field energy saturates at approximately 16% of the total system energy. This result is three times greater than for a one‐dimensional system with periodic boundary conditions.

A Different Proof of the Maxwell-Boltzmann Distribution

Abstract: The Maxwell-Boltzmann Distribution (MBD) for a classical ideal gas is obtained as the mean over phase space of the number of momentum components less than some value p, without making any assumptions beyond those implicit in the microcanonical ensemble. The variance of the fluctuations of actual distributions of the momentum around the average given by the MBD is shown to go to zero as N−1, where N is the number of particles. This proof of the MBD has the advantage over the usual type of proof in that it does not employ a subdivision of phase space into cells, a trick which gives rise to certain difficulties discussed in the introduction. Moreover, it is more complete in that the problem of the fluctuations is treated at an elementary level.

Longitudinal phase-space analysis of the reaction π+p → π+pπ+π− at 11.7 GeV/c and comparison of the reactions π±p → π±π+π−p at the same energy

Abstract: The reaction π+p → π+pπ+π− at 11.7 GeV/c is analysed using longitudinal phase space (L.P.S.) plots with a sample of about 6000 events. The results are compared with those obtained by Kittel, Ratti and Van Hove in the analysis of the reaction π−p → π−pπ−π+ at 11 and 16 GeV/c. The four dominant sectors 1) π+p → (3π)p, 2) π+p → π+(pπ+π−), 3) π+p → → (π+π+)(pπ−), 4) π+p → (π+π−)(pπ+), are examined in some detail. In particular, sectors 1) and 2) are very similar to the corresponding ones observed for π−p at the two momenta. The sectors 3) and 4) are compared in detail with the equivalent configurations observed in π−p at 11 GeV/c,i.e. 3′) π−p → (π−π−)(pπ+), 4′) π−p → (π+π−)(pπ−).

Adiabatic Switching-Off of Interactions and Foundations of the Boltzmann Equation

Abstract: The Boltzmann equation for a monatomic gas is derived with the aid of the adiabatic switching-off of interactions, starting with the Heisenberg equation of motion for a number operator in phase space introduced by Ono. It is shown that neither Kirkwood's time-averaging procedure nor the random phase approximation is necessary by virtue of the switching-off of interactions and a new method of time-differentiation. The effectiveness of the present formalism is demonstrated in an analysis of higher order interactions, and the Boltzmann equation is corrected so as to include multiple scattering effects.

Model amplitudes for e+e− annihilation into pions at high energies

Abstract: The high-energy limit of e+e− annihilation into hadrons is studied within the framework of Feynman graphs, by supposing that the virtual photon couples to a hadronic system via a pair of elementary hadrons We study the role played by the topology of the graphs in determining the distribution of the final-state particles in the multidimensional phase space, usingφ3 theory as a tool In these models there emerge two competing phenomena analogous to « pionization » and « fragmentation » in two-body hadronic production processes The effect of including spin and internal-symmetry requirements is studied for the special example of the peripheral production of three pions off a fermion loop Specific predictions of this model are discussed

Invariant and angular variables for five particle processes.

Abstract: Angles and rapidity variables which are simply related to the invariants of a five-particle process are derived and the connecting equations established. Classification of the different forms of the phase space integral, obtained by successive use of these equations, is given. The symmetric treatment clarifies the meaning of several (commonly used or here introduced) sets of variables.

Certain general questions of fast-electron transport II. Kinetic equations and conditions governing the accelerated motion of fast electrons in dielectrics in an electric field

Abstract: A general kinetic equation for the differential density of fast particles moving in a medium in an external field is derived on the basis of the continuity equation in phase space. An equation is written for the differential flux in the case of fixed target particles. This equation is used to derive equations for fast electrons; account is taken of the coupling of energy-loss and scattering events in an electric field for various particular problems analogous to those studied in the theory of electron transport in the absence of a field. The kinetic equations are used to analyze the conditions governing accelerated motion of electrons in a dielectric in an external electric field in the continuous-deceleration approximation. Account is taken of fluctuations in the energy loss and of multiple scattering. There are two energy ranges of particles moving in a dielectric in which accelerated motion can occur; in the case of an electron beam with a continuous energy spectrum, this acceleration would be accompanied by monochromatization of the beam.

Irreversibility in Many-Body Systems

Abstract: Introduction. Part of the time(viz. after Gibbs) during which the question of irreversibility has been on the agenda of Theoretical Physics, Liouville’s theorem on the motion of volume elements in Phase Space has been the basis from which this discussion takes off: The points, —each of which may represent a separate mechanical system - move around like an incompressible flow in the 2s-dimensional Cartesian space spanned by the coordinates (q_) and the momenta (p).