Average Phase Space Research Articles

Noise-induced chaos and phase space flux

We study the effect of additive noise on near-integrable second-order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits. The noise is represented by a type of Shinozuka stochastic process capable of arbitrarily closely approximating Gaussian noise with any specified spectrum. We derive a formula for the flux factor applicable for any asymptotic mean stationary excitation. This derivation shows that, to first order, the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit. It also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux. This is true for both random and nonrandom excitations and indicates that, for the dynamical systems considered here, these two classes of excitation play substantially equivalent roles in the promotion of chaos.

Nosé–Hoover chains: The canonical ensemble via continuous dynamics

Nosé has derived a set of dynamical equations that can be shown to give canonically distributed positions and momenta provided the phase space average can be taken into the trajectory average, i.e., the system is ergodic [S. Nosé, J. Chem. Phys. 81, 511 (1984), W. G. Hoover, Phys. Rev. A 31, 1695 (1985)]. Unfortunately, the Nosé–Hoover dynamics is not ergodic for small or stiff systems. Here a modification of the dynamics is proposed which includes not a single thermostat variable but a chain of variables, Nosé–Hoover chains. The ‘‘new’’ dynamics gives the canonical distribution where the simple formalism fails. In addition, the new method is easier to use than an extension [D. Kusnezov, A. Bulgac, and W. Bauer, Ann. Phys. 204, 155 (1990)] which also gives the canonical distribution for stiff cases.

On the continuum equations for dispersed particles in nonuniform flows

The continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow are derived from a kinetic equation for the transport of the average phase space density 〈W(v,x,t)〉 for particles with velocity v and position x at time t. The crucial feature of this equation is the form given for the phase space diffusion current j representing the net acceleration of a particle from interactions with turbulent eddies. This is based on Kraichnan’s Lagrangian history direct interaction approximation which gives j=−[(∂/∂v)⋅μ+(∂/∂x)⋅λ+γ]〈W(v,x,t)〉, where μ, λ, and γ are dispersion tensors dependent upon displacements in the velocity and position of a particle about v,x in times of order of the time scale of the fluctuating aerodynamic driving force. Most important these tensors are affected by spatial variations in the mean carrier flow velocity and external force as well as inhomogeneities in the carrier phase turbulence; γ is zero for homogeneous turbulence. This form for j is invariant under a random Galilean transformation. The dispersed phase momentum and energy equations deduced from the kinetic equation contain local gradient forms for the net fluctuating interphase force (per unit volume) and its rate of working. The former contains in general an asymmetric stress component (which adds to the particle Reynolds stress) as well as a body force dependent upon inhomogeneities in the turbulence. Conditions are examined under which the mass and momentum equations reduce to a convection–diffusion equation. The convection velocity in this case is the local carrier flow velocity plus a drift velocity proportional to local gradients of the turbulence (zero if the particles follow the flow). The diffusion coefficient is linearly related to the pressure/mean density of the dispersed phase via the particle response time. It is shown to be influenced by the local mean shearing of the carrier flow. Conditions are derived when this contribution can be ignored, the diffusion coefficient reducing to the form for homogeneous turbulence.

On a kinetic equation for the transport of particles in turbulent flows

The work described is part of a long-term study in pursuit of a kinetic equation for particles suspended in a turbulent flow. This equation represents the transport of the average particle phase space density and can be used to derive the continuum equations and constitutive relations for a two-fluid model of a dispersed particle flow and to establish the correct form of boundary conditions. A key role in this study is played by the equation of state of the dispersed particles, which is derived here for a Langevin equation of motion with a random driving force not limited to white noise. A suitable form of kinetic equation is then sought that reproduces this equation of state and predicts Gaussian spatial diffusion in the long-term limit with the correct form for the diffusion coefficient in homogeneous stationary turbulence. It is shown that both these requirements are automatically satisfied by constructing forms for the phase space diffusion coefficient that preserve invariance to a random Galilean transformation. The most general form is shown to be an expansion in successively higher-order cumulants of the aerodynamic driving force. When this force is a Gaussian process all terms apart from the first term contract to zero, i.e., the phase space diffusion current is of the form −[μ⋅(∂/∂v) +λ⋅(∂/∂x)]〈W〉 , where 〈W〉 is the average phase space density of particles with velocity v and position x at time t and μ and λ are phase space diffusion tensors whose components are memory integrals involving the correlation of the aerodynamic force along a particle trajectory. With both this form and the more general form of phase space diffusion current, the kinetic equation contracts to the classical Fokker–Planck equation in the white noise limit. Finally with the Gaussian form for the diffusion current, the form of the average phase space density is examined for the important case of a point source diffusing in an unbounded homogeneous stationary flow.

Canonical ensembles from chaos

We present methods to compute thermodynamic properties of classical systems which involve extending the phase space by two degrees of freedom. These two additional degrees of freedom are used to replicate the coupling of the original system to the infinite degrees of freedom of a heat bath. In the extended phase space, the trajectories are ergodic. This feature enables one to replace phase space averages by time averages, which are extremely simple to compute. We examine phase space patterns, thermal distributions, correlations, ergodicity, Lyapunov exponents, mixing, and rate of convergence in an analysis of several simple systems.

Semiclassical percolation approach to electronic states in simple fluids.

Recent microscopic studies have underscored the relevance of classical percolation ideas to electron mobility in simple fluids. We show that Wigner phase-space averaging can be used to construct a semiclassical potential on which these concepts can be defined rigorously. The conduction-band threshold energy ${\mathit{V}}_{0}$ is then determined by solving a well-defined classical continuum percolation problem. Application of our model predicts the density dependence of ${\mathit{V}}_{0}$ quantitatively for the noble-gas fluids.

Trajectory studies of unimolecular reactions of Si2H4 and SiH2 on a global potential surface fitted to a b i n i t i o and experimental data

The unimolecular decomposition dynamics of Si2H4 have been investigated using classical trajectory methods on a global potential-energy surface fitted to the results of ab initio calculations and the available experimental data. The required phase-space averages are computed using Metropolis sampling techniques. It is found that unless the parameters of the Markov walk are adjusted for each different type of atom present, extremely long Markov walks are required to adequately cover the phase space of the system. Microcanonical rate coefficients for the decomposition of Si2H4 into all open channels are reported at energies in the range 5.0<E<9.0 eV. The most important dissociation channel over this energy range is three-center elimination of molecular hydrogen leading to H2 Si=Si. At energies below 7.0 eV, the other channels are, in order of importance, Si–Si bond rupture, four-center H2 elimination, and simple Si–H bond rupture. At or above 8.0 eV, four-center H2 elimination replaces Si–Si bond rupture as the second most important decomposition channel. The energy dependence of the rate coefficients is well described by an RRK expression. Three-center H2 elimination involves a simultaneous rupture of both Si–H bonds whereas the four-center elimination is found to proceed by a hydrogen atom transfer process followed by H2 elimination. Except for a small propensity to form H2 with excess rotational energy, the energy partitioning among the products is nearly statistical. A comparison study of the decomposition of Si2H4 complexes formed by the recombination of two SiH2 molecules shows that the rates for both three- and four-center H2 elimination are in agreement with those computed using a statistical distribution of the same internal energy. The rate for Si–Si bond rupture, however, is significantly larger for Si2H4 complexes formed by SiH2 recombination than for Si2H4 molecules with the same internal energy randomly distributed. The decomposition dynamics of SiH2 on the global surface are also reported.

Diffusion of H atoms on a Si(111) surface with partial hydrogen coverage: Monte Carlo variational phase-space theory with tunneling correction

The diffusion of hydrogen atoms on a partially hydrogen-covered Si(111) surface has been studied by using Monte Carlo techniques with a potential-energy surface based on the available ab initio results and experimental data. The potential describes two kinds of binding sites, a covalent Si–H bond (top site) and an interstitial threefold bonding site (open site). Classical jump frequencies between the top and open sites were calculated using Monte Carlo variation phase-space theory with importance sampling at 300, 600, 900, and 1200 K. A new approach for treating tunneling through two-dimensional diffusional barriers is presented and used to calculate the phonon-assisted tunneling rates. This method assumes continuum-to-continuum WKB tunneling with classical Monte Carlo phase space averaging. Thermal diffusion coefficients are calculated using the jump frequencies. The diffusional barriers between the two binding sites on the equilibrium surface are 2.79 and 0.65 eV for top-to-open site and open-to-top site jumps, respectively. The calculated classical jump frequencies give Arrhenius parameters of A=1.3×1014 and 9.9×1013 s−1 Ea=2.72 and 0.59 eV for top-to-open and open-to-top site jumps, respectively. Monte Carlo techniques were used to compute the minimum energy path; the dynamical barrier is 2.64 eV for top-to-open site jumps. Tunneling rates were calculated at 300 K and estimated at higher temperatures. Due, in part, to the small width of the barrier, the tunneling rate at 300 K is 257 times larger than the classical value. Tunneling is important at room temperature, but its importance relative to the classical rate decreases with increasing temperature. The results indicate that surface phonons significantly enhance the tunneling rate.

Simulation of phase space hole growth and the development of intermittent plasma turbulence

An unstable, highly intermittent state of turbulence is observed to evolve from a quiescent homogeneous plasma that is linearly stable. This intermittent state, which consists of isolated phase space density holes, produces a pronounced non-Gaussian distribution of fluctuation amplitudes. The skewness becomes increasingly more negative with time. The plasma is nonlinearly unstable because the holes grow in amplitude by accelerating to regions of higher average phase space density. The instability can be interpreted as a collection of colliding, growing holes and, in its early stages, is consistent with theoretical predictions for the clump instability. A series of runs with a single isolated hole indicates that an isolated hole grows for any finite electron-ion drift velocity. The isolated hole growth rate is consistent with theoretical predictions. The implications of the instability to laboratory and space plasma are discussed.

Phase-space procedure for calculating multitime correlation functions of the electromagnetic field

It is pointed out that a method recently described by Hillery and Zubairy can be generalized to yield the multitime correlation function in any order for an interacting quantum field in the form of a phase-space average. The correlations are expressible as $c$-number integrals with the help of the coherent-state propagator, but the associated phase-space density is complex.

Relativistic canonical formalism and the invariant single-particle distribution function in the general theory of relativity

The relativistic canonical formalism is used to construct an eight-dimensional phase space and an invariant distribution function, and integral and differential operations in the phase space and statistical averages, associated with the field of geodesic observers, are introduced. Liouville's theorem is proved.

Massive neutrinos in large-scale gravitational clustering

view Abstract Citations (71) References (36) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Massive neutrinos in large-scale gravitational clustering Melott, A. L. Abstract A large-scale simulation is made of the plane-symmetric collapse of massive neutrinos. It is found that a condensation forms in the central plane with high mass density, a velocity dispersion much lower than the mean value, and a phase space density which is nearly equal over a large region to the maximum of the original distribution. The coarse-grain average phase space density over the central megaparsec is comfortably higher than that for an extended galactic halo. It is concluded that there is no obstacle to the formation of massive neutrino galactic halos in the neutrino variant of the adiabatic theory of galaxy formation, because a population of neutrinos arises which may easily be bound into galactic halos. The calculations, with the exception of the thermal broadening, will apply to any system of collisionless particles. Publication: The Astrophysical Journal Pub Date: January 1983 DOI: 10.1086/160574 Bibcode: 1983ApJ...264...59M Keywords: Galactic Structure; Gravitational Collapse; Neutrinos; Particle Theory; Galactic Evolution; Halos; Space Density; Astrophysics full text sources ADS |

On chemical bond definitions; comparisons to computer study on dimer formation

Dimer mole fractions are calculated for the hard disk square well dilute gas from three alternative definitions of chemical bond (negative energy, proximity, and ’’exact,’’ including both stable and metastable dimers). Resulting theoretical predictions, based on phase–space averages are confronted with computer experiment, in which dimers can be identified dynamically from the history of interparticle collisions. Proximity and negative energy definitions give (respectively) upper and lower bounds on the molecular dynamics dynamical dimer mole fractions which fall closest to the exact prediction. It is concluded that for the purposes of both theoretical studies and computer experiments on clustering and nucleation it may be advantageous to use proximity and negative energy definitions simultaneously.

Quantum mechanics as a generalized stochastic theory in phase space

In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.

Theory of intercollisional interference effects

We extend our previous kinetic theory of intercollisional interference effects to include vibrational perturbations in collisions, and so obtain expressions for the pressure shifts of the interference minima in H 2 -rare gas mixtures. These shifts are found to be different from those of the isotropic Raman Q lines in the same systems. The use of phase space averages for the analysis of Raman line shifting is discussed.

Eine phasenraumdarstellung für spinsysteme

We start with the definition of two mapping operators, one of them is the projection operator onto coherent spin states. With the help of these operators we derive a mapping theorem which defines a correspondence between the operators in spin space andc-number functions of a certain class. It is shown that this correspondence is one-to-one. The quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase space average of classical statistical mechanics. We also derive a product theorem which allows us to transcribe the equations of motion for operators into equivalent equations for thec-number functions. As an illustration of the theory, some examples are discussed.

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space

In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto $c$-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.

Time Correlations and Conditional Distribution Functions for Classical Ensembles of Free Rotors

Explicit expressions are obtained for the classical rotational time-correlation functions for free rotation of the following: the linear, spherical-top, and symmetric-top molecule. This is done by deriving the eigenfunctions and eigenvalues of the potential-free rotational Liouville operator for each case. Conditional distribution functions for orientations and angular momenta are then constructed, and correlation functions are computed by evaluating phasespace averages with the distribution functions. The results obtained agree with those of Sears for the linear molecule and Steele for the spherical top, but differ somewhat from the calculation of Agrawal and Yip for the symmetric top molecule. The reason for this discrepancy is discussed briefly.

The rotational wigner function

The quantum mechanics of angular momentum is studied using the Wigner phase-space distribution function. To accomplish this in a way which allows one to retain the bilinear form of the distribution function originally conceived by Wigner, a new set of generalized variables is introduced, viz. the Cayley-Klein (C.K.) coordinates and momenta. The equations relating the four C.K. coordinates to the three Euler angles give rise to a constraint which is relaxed in a natural way by introducing a fourth “radial” variable ϱ whose conjugate momentum is p ϱ . Physical quantities in the classical mechanics of rotational systems are re-expressed in the C.K. representation; and the canonical transformation between the C.K. and Eulerian variables is established. In a similar manner, after introducing the basic quantum mechanical operators for the C.K. coordinates and momenta, the wavefunctions and operators for rotational systems are re-expressed in these variables. With the aid of the Weyl correspondence to link the operators of quantum mechanics with the dynamical functions of classical mechanics, the phase-space formulation is established. Rotational Wigner distribution functions are constructed, their properties investigated, and their form is explicitly given for several quantum states. Expectation values are then calculated according to the procedure identical to the one in classical statistical mechanics for calculating phase space averages. The equation of motion for the distribution function is derived. It turns out to be the sine expansion of the Poisson-bracket operator expressed in the C.K. variables and operating on the Hamiltonian and distribution function. This equation reduces to Liouville's equation in the classical limit. The Heisenberg picture of the phase-space formulation is introduced, and the equations of motion for the body-fixed components of the angular momentum are studied. In the classical limit Euler's equations of motion are recovered. Finally, the thermodynamic properties of a system in equilibrium are investigated. The temperature dependence of the ensemble Wigner function is shown to be given by a cosine expansion of the Poisson bracket operator expressed in the C.K. variables, and operating on the Hamiltonian and distribution function. An ansatz is then made on the form of the distribution function, specifically, a power series in Planck's constant with coefficients which are functions of the coordinates and momenta. Substitution of this postulated form into the phase-space Bloch equation allows an explicit determination of these functions. Integration of the Wigner function over all momenta gives the configurational distribution function. A further integration over the coordinates gives the partition function. This procedure is carried out in detail for the asymmetric top, and the first quantum corrections to both the configurational distribution function and partition function are obtained. Upon setting appropriate moments of inertia equal to each other, these quantum corrections reduce to those of the symmetrical and spherical tops.

Weyl Transformation and the Classical Limit of Quantum Mechanics

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.