Showing papers on "Phase space published in 1994"

Constant pressure molecular dynamics algorithms

Abstract: Modularly invariant equations of motion are derived that generate the isothermal–isobaric ensemble as their phase space averages. Isotropic volume fluctuations and fully flexible simulation cells as well as a hybrid scheme that naturally combines the two motions are considered. The resulting methods are tested on two problems, a particle in a one‐dimensional periodic potential and a spherical model of C60 in the solid/fluid phase.

On the out-of-equilibrium relaxation of the Sherrington-Kirkpatrick model

Abstract: Starting from a set of assumptions on the long-time limit behaviour of the nonequilibrium relaxation of mean-field models in the thermodynamic limit, we derive analytical results for the long-time relaxation of the Sherrington-Kirkpatrick model, starting from a random configuration. The system never achieves local equilibrium in any fixed sector of phase space, but remains in an asymptotic out-of-equilibrium regime. We clearly state and motivate the assumptions made. For the study of the out-of-equilibrium dynamics of spin-glass models, we propose as a tool, both numerical and analytical, the use of 'triangle relations' which describe the geometry of the configurations at three (long) different times.

Complex wave-field reconstruction using phase-space tomography.

Abstract: A method is proposed for the experimental determination of the amplitude and phase structure of a quasimonochromatic wave field in a plane normal to its propagation direction. The wave field may represent either a scalar electromagnetic (EM) field or the quantum mechanical (QM) wave function of a matter wave. For coherent EM fields or pure QM states, the method uniquely reconstructs the complex wave fields. For partially coherent EM fields or mixed QM states, it reconstructs the two-point correlation function or density matrix. The experiment uses only intensity measurements and refractive optics (lenses), and the data analysis algorithm is noniterative and requires no deconvolution.

Complex wave-field reconstruction by using phase-space tomography

Abstract: A new class of phase-retrieval methods for 2-D fields is introduced. Phase-space tomography can be used for the experimental determination of the amplitude and phase structure of a quasi-monochromatic wave field in a plane normal to its propagation direction. The wave field ψ(r) may represent either a scalar electromagnetic (EM) field or the quantum-mechanical (QM) wave function of a matter wave. The complex wave field may be coherent or partially coherent, in which case the method reconstructs the two-point spatial correlation function, Γ(r, r′) = ⟨ψ(r)ψ*(r′)⟩. (In the QM case, the analogous quantity is the density matrix.) The experiment uses only intensity measurements and refractive optics (lenses), and the data-analysis algorithm is noniterative and requires no deconvolution.

Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms

Abstract: The general theory of quantum angular momentum is used to derive the unique Wigner distribution function for arbitrary angular-momentum states. We give the explicit distribution for atomic angular-momentum Dicke states, coherent states, and squeezed states that correspond to a collection of N two-level atoms. These Wigner functions W(\ensuremath{\theta},cphi) are represented as a pseudoprobability distribution in spherical phase space with spherical coordinates \ensuremath{\theta} and cphi on the surface of a sphere of radius \ensuremath{\Elzxh} \ensuremath{\surd}j(j+1) where j is the total angular-momentum eigenvalue.

The formulation of quantum statistical mechanics based on the Feynman path centroid density. III. Phase space formalism and analysis of centroid molecular dynamics

Abstract: The formulation of quantum statistical mechanics based on the path centroid variable in Feynman path integration is generalized to a phase space perspective, thereby including the momentum as an independent dynamical variable. By virtue of this approach, operator averages and imaginary time correlation functions can be expressed in terms of an averaging over the multidimensional phase space centroid density. The imaginary time centroid‐constrained correlation function matrix for the phase space variables is then found to define the effective thermal width of the phase space centroid variable. These developments also make it possible to rigorously analyze the centroid molecular dynamics method for computing quantum dynamical time correlation functions. As a result, the centroid time correlation function as calculated from centroid molecular dynamics is shown to be a well‐defined approximation to the exact Kubo transformed position correlation function. This analysis thereby clarifies the underlying role of the equilibrium path centroid variable in the quantum dynamical position correlation function and provides a sound theoretical basis for the centroid molecular dynamics method.

Drift and diffusion in phase space

Abstract: The problem of stability of action variables (i.e. of the adiabatic invariants) in perturbations of completely integrable (real analytic) hamiltonian systems with more than two degrees of freedom is considered. Extending the analysis of [A], we work out a general quantitative theory, from the point of view of dimensional analysis, for a priori unstable systems (i.e. systems for which the unperturbed integrable part possesses separatrices), proving, in general, the existence of the so-called Arnold's diffusion and establishing upper bounds on the time needed for the perturbed action variables to drift by an amount of O(1). The above theory can be extended so as to cover cases of a priori stable systems (i.e. systems for which separatrices are generated near the resonances by the perturbation). As an example we consider the «d'Alembert precession problem in Celestial Mechanics» (a planet modelled by a rigid rotational ellipsoid with small «flatness» η, resolving on a given Keplerian orbit of eccentricity e=η c , c>1, around a fixed star and subject only to Newtonian gravitational forces) proving in such a case the existence of Arnold's drift and diffusion; this means that there exist initial data for which for any η¬=0 small enough, the planet changes, in due (η-dependent) time, the inclination of the precession cone by an amount of O(1). The homo/heteroclinic angles (introduced in general and discussed in detail together with homoclinic splittings and scatterings) in the d'Alembert problem are not exponentially small with η (in site of first order predictions based upon Melnikov type integrals)

The formulation of quantum statistical mechanics based on the Feynman path centroid density. 3: Phase space formalism and analysis of centroid molecular dynamics

Abstract: : The formulation of quantum statistical mechanics based on the path centroid variable in Feynman path integration is generalized to a phase space perspective, thereby including the momentum as an independent dynamical variable. By virtue of this approach, operator averages and imaginary time correlation functions can be expressed in terms of an averaging over the multidimensional phase space centroid density. The imaginary time centroid- constrained correlation function matrix for the phase space variables is then found to define the effective thermal width of the phase space centroid variable. These developments also make it possible to rigorously analyze the centroid molecular dynamics method for computing quantum dynamical time correlation functions. As a result, the centroid time correlation function as calculated from centroid molecular dynamics is shown to be a well-defined approximation to the exact Kubo transformed position correlation function. This analysis thereby clarifies the underlying role of the equilibrium path centroid variable in the quantum dynamical position correlation function and provides a sound theoretical basis for the centroid molecular dynamics method. Chemical dynamics, Computer simulation, Electrochemistry

Structure stability of congestion in traffic dynamics

Abstract: In our previous paper, we proposed a dynamical model, whose equation of motion is expressed as a second order differential equation. This model generates traffic congestion spontaneously. In this paper we study the characteristic properties of the traffic congestion in our model, especially the organization process and the stability of the structure of congestion. It turns out that these phenomena are well described by plotting motions of vehicles in the phase space of velocity and headway. The most remarkable feature is the universality of “the hysterisis loop” in this phase space, which is observed in the final stage of the congestion organization. This loop is understood as a limit cycle of the dynamical system. This universality guarantees the stability of total cluster size.

Edge-ray principle of nonimaging optics

Abstract: The edge-ray principle of nonimaging optics states that nonimaging devices can be designed by the mapping of edge rays from the source to the edge of the target. However, in most nonimaging reflectors, including the compound parabolic concentrator (CPC), at least part of the radiation undergoes multiple reflections, some rays even appear to be reflected infinitely many times, and closer examination reveals that some edge rays of the source are not mapped onto the edge of the target even though the CPC is indeed ideal in two dimensions. Using a topological approach, we refine the formulation of the edge-ray principle to ensure its validity for all configurations. We present two different versions of the general principle. The first involves the boundaries of the different zones corresponding to a different number of reflections. The second version is stated in terms of only a single reflection, but it involves the addition of an auxiliary region of phase space. We discuss the use of the edge-ray principle as a design procedure for nonimaging devices. The CPC is used to illustrate all steps of the argument.

Generalized weighting scheme for δf particle‐simulation method

Abstract: An improved nonlinear weighting scheme for the δf method of kinetic particle simulation is derived. The method employs two weight functions to evolve δf in phase space. It is valid for quite general, non‐Hamiltonian dynamics with arbitrary sources. In the absence of sources, only one weight function is required and the scheme reduces to the nonlinear algorithm developed by Parker and Lee [Phys. Fluids B 5, 77 (1993)] for sourceless simulations. (It is shown that their original restriction to Hamiltonian dynamics is unnecessary.) One‐dimensional gyrokinetic simulations are performed to show the utility of this two‐weight scheme. A systematic kinetic theory is developed for the sampling noise due to a finite number of marker trajectories. The noise intensity is proportional to the square of an effective charge qeff=q(w/D), where w ∼δf/f is a typical weight and D is the dielectric response function.

Multistate quantum Fokker–Planck approach to nonadiabatic wave packet dynamics in pump–probe spectroscopy

Abstract: The quantum Fokker–Planck equation of Caldeira and Leggett is generalized to a multistate system with anharmonic potentials and a coordinate dependent nonadiabatic coupling. A rigorous procedure for calculating the dynamics of nonadiabatic transitions in condensed phases and their monitoring by femtosecond pump–probe spectroscopy is developed using this equation. Model calculations for a harmonic system with various nonadiabatic coupling strengths and damping rates are presented. Nuclear wave packets in phase space related to electronic coherence are shown to provide an insight into the mechanism of nonadiabatic transitions. The Green’s function expression for these wave packets is used to explore possible algorithms for incorporating electronic dephasing in molecular dynamics simulations of curve crossing processes.

Statistical and phase properties of the binomial states of the electromagnetic field.

Abstract: We investigate the nonclassical properties of the single-mode binomial states of the quantized electromagnetic field. We concentrate our analysis on the fact that the binomial states interpolate between the coherent states and the number states, depending on the values of the parameters involved. We discuss their statistical properties, such as squeezing (second and fourth order) and sub-Poissonian character. We show how the transition between those two fundamentally different states occurs, employing quasiprobability distributions in phase space, and we provide, at the same time, an interesting picture for the origin of second-order quadrature squeezing. We also discuss the phase properties of the binomial states using the Hermitian-phase-operator formalism.

All electroweak four fermion processes in electron-positron collisions

Abstract: This paper studies the electroweak production of all possible four fermion states in e+ e- collisions. Since the methods employed to evaluate the complete matrix elements and phase space are very general, all four fermion final states in which the charged particles are detected can be considered. Also all kinds of experimental cuts can be imposed. With the help of the constructed event generator a large number of illustrative results is obtained, which show the relevance of backgrounds to a number of signals. For LEP 200 the W-pair signal and its background are discussed, for higher energies also Z-pair and single W and Z signals and backgrounds are presented.

Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach

Abstract: We develop the operator formalism to show how systematically the fractional Fourier transformation of a wavefunction, recently introduced by Namias (1980), can be derived from the rotation of the corresponding Wigner distribution function in phase space. In this formalism, the phase factor obtained by McBride and Kerr (1987) is seen to come from the caustics of the harmonic oscillator Green function. Then the idea is generalized to the case of an arbitrary area-preserving linear transformation in phase space, and a concept of the special affine Fourier transformation (SAFT) is introduced. An explicit form of the integral representation of the SAFT is given, and some simple examples are presented.

Planck’s law and the light quantum hypothesis

Abstract: The phase space of a light quantum in a given volume is subdivided into “cells” of magnitudeh3. The number of possible distributions of the light quanta of a macroscopically defined radiation over these cells gives the entropy and with it all thermodynamic properties of the radiation.

Hamiltonian formalism for nonlocal Lagrangians

Abstract: A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case.

Scars in Groups of Eigenstates in a Classically Chaotic System

Abstract: A general method to construct wave functions highly localized on a given periodic orbit, using the information contained in the short term quantum dynamics of the system, is presented. The relationship with the Husimi's quasiprobability distribution in phase space is also discussed.

Edge-ray principle of nonimaging optics

Abstract: Some form of the edge-ray principle is used to design most nonimaging systems. (Only systems based on geometrical optics are considered.) For proving certain statements of this principle, the optical system is considered to be surrounded by an enclosure that any ray emerging from the system must intersect. A phase space, of topology Sphere × Disk, corresponding to the intersection of rays with the enclosure is introduced, and the system gives rise to a mapping f among points of this space. The proofs hold only if f is continuous, which is not the case for all real systems. Discontinuities in f may be caused by (1) tangential incidence of a ray with a surface, (2) incidence where the radius of curvature of a surface is zero, (3) transition from refraction to total internal reflection, and (4) intersection of different types of optical surface. Although the condition of the continuity of f is used in the proofs, even among systems in which continuity is absent it is difficult in practice to find counterexamples to the edge-ray principle formulated.

A nonlinear probabilistic method for predicting vessel capsizing in random beam seas

Abstract: The capsizing of vessels in random beam seas is investigated using a single degree-of-freedom model which is limited to the roll motion. Several factors, including sea wave spectrum, nonlinear righting moment characteristics, and nonlinear damping, are taken into account in the analysis. A nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics (specifically, the Melnikov function and the phase space area flux) and random vibrations. Conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics (significant wave height and characteristic wave period) and the vessel parameters (damping and stiffness coefficients). Extensive numerical simulations are carried out to demonstrate the validity of the analytical results. It is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing.

Meson phase space density in heavy ion collisions from interferometry

Abstract: The interferometric analysis of meson correlations provides a measure of the average phase space density of the mesons in the final state. This quantity is a useful indicator of the statistical properties of the system, and it can be extracted with a minimum of model assumptions. Values obtained from recent measurements are consistent with the thermal value, but do not rule out superradiance effects.

Movability of localized excitations in nonlinear discrete systems: A separatrix problem.

Abstract: We analyze the effect of internal degrees of freedom on the movability properties of localized excitations on nonlinear Hamiltonian lattices by means of properties of a local phase space which is at least of dimension six. We formulate generic properties of a movability separatrix in this local phase space. We prove that due to the presence of internal degrees of freedom of the localized excitation it is generically impossible to define a Peierls-Nabarro potential in order to describe the motion of the excitation through the lattice. The results are verified analytically and numerically for Fermi-Pasta-Ulam chains.