Showing papers on "Phase space published in 1997"

A priori derivation of the lattice Boltzmann equation

Abstract: The lattice Boltzmann equation (LBE) is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided.

Homogenization limits and Wigner transforms

Abstract: We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical examples like (semi)classical limits of Schrodinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. © 1997 John Wiley & Sons, Inc.

Semiclassical Description of Nonadiabatic Quantum Dynamics

Abstract: A semiclassical approach is presented that allows us to extend the usual Van Vleck--Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. Based on Schwinger's theory of angular momentum, the formulation employs an exact mapping of the discrete quantum variables onto continuous degrees of freedom. The resulting dynamical problem is evaluated through a semiclassical initial-value representation of the time-dependent propagator. As a first application we have performed semiclassical simulations for a spin-boson model, which reproduce the exact quantum-mechanical results quite accurately.

Quantum Chaos on Graphs

Abstract: We quantize graphs (networks) which consist of a finite number of bonds and nodes. We show that their spectral statistics is well reproduced by random matrix theory. We also define a classical phase space for the graph, where the dynamics is mixing and the periodic orbits (loops on the graph) proliferate exponentially. An exact trace formula for the quantum spectrum is developed and used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in quantum chaos and related fields.

Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps

Abstract: Phase space representations are very useful in quantum mechanics since they allow the calculation of correlation functions of operators as classical-like integrals, and are also helpful for the study of the transition to classical physics. The oldest of such representations is due to Wigner [1], who used it as a convenient tool to calculate quantum corrections to classical statistical mechanics. It was shown by Moyal [2] that the quantum average of a Weyl-ordered (symmetric-ordered) function of the momentum and position operators could be expressed as a classical-like average of the corresponding classical function (in which the operators are replaced by c numbers), with the Wigner distribution acting as a weight measure in phase space [3]. The uncertainty principle forbids, however, the interpretation of this function as a probability distribution, since it is not possible to determine simultaneously the momentum and the position of a particle. In fact, it is easy to find examples of states for which the Wigner distribution assumes negative values. This fact may lead to the idea that it does not correspond to any directly measured quantity. Up until now, this notion has been upheld by the fact that the different schemes proposed so far to determine the Wigner distribution of a quantum system rely either on tomographic reconstructions from data obtained in homodyne measurements [4,5] or on convolutions obtained by photon counting [6]. It is the purpose of this Letter to show that the Wigner function can be directly measured, through a very simple scheme, especially suitable to experiments in cavity quantum electrodynamics and in ion traps. This is especially important in view of recent experimental results concerning the production and detection of coherent superpositions of localized mesoscopic states [7,8]. In these experiments, the existence of coherence is inferred from partial information obtained about the system. A method yielding more complete knowledge onf the quantum states involved would be highly desirable. Quantum tomography schemes for determining the vibrational state of a trapped ion were proposed in [5]. In cavity QED, information on the field must be obtained from probe atoms, since the high Q of the cavity and the weak intensities involved do not allow the direct measurement of the field. A method for realizing the “quantum endoscopy” of a field in a cav-

Semiclassical-limit molecular dynamics on multiple electronic surfaces

Abstract: In this paper, we present a new approach to treating many-body molecular dynamics on coupled electronic surfaces. The method is based on a semiclassical limit of the quantum Liouville equation. The formal result is a set of coupled classical-like partial differential equations for generalized distribution functions which describe both the nuclear probability densities on the coupled surfaces and the coherences between the electronic states. The Hamiltonian dynamics underlying the evolution of these distributions is augmented by nonclassical source and sink terms, which allow the flow of probability between the coupled surfaces and the corresponding formation and decay of electronic coherences. The formal results are shown analytically to reproduce the well-known Rabi and Landau–Zener results in appropriate limits. In addition, a direct numerical solution of the phase space partial differential equations is performed, and the results compared with exact quantum solutions for a model curve-crossing problem,...

Phase space approach to theories of quantum dissipation

Abstract: Six major theories of quantum dissipative dynamics are compared: Redfield theory, the Gaussian phase space ansatz of Yan and Mukamel, the master equations of Agarwal, Caldeira-Leggett/Oppenheim-Romero-Rochin, and Louisell/Lax, and the semigroup theory of Lindblad. The time evolving density operator from each theory is transformed into a Wigner phase space distribution, and classical-quantum correspondence is investigated via comparison with the phase space distribution of the classical Fokker-Planck (FP) equation. Although the comparison is for the specific case of Markovian dynamics of the damped harmonic oscillator with no pure dephasing, certain inferences can be drawn about general systems. The following are our major conclusions: (1) The harmonic oscillator master equation derived from Redfield theory, in the limit of a classical bath, is identical to the Agarwal master equation. (2) Following Agarwal, the Agarwal master equation can be transformed to phase space, and differs from the classical FP eq...

Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics

Abstract: A detailed description of fractional kinetics is given in connection to islands’ topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered.

Modified nonequilibrium molecular dynamics for fluid flows with energy conservation

Abstract: The nonequilibrium molecular dynamics generated by the SLLOD algorithm [so called due to its association with the DOLLS tensor algorithm (D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic, New York, 1990)] for fluid flow is considered. It is shown that, in the absence of time-dependent boundary conditions (e.g., shearing boundary conditions via explicit cell dynamics or Lees–Edwards boundary conditions), a conserved energy, H exists for the equations of motion. The phase space distribution generated by SLLOD dynamics can be explicitly derived from H. In the case of a fluid confined between two immobile boundaries undergoing planar Couette flow, the phase space distribution predicts a linear velocity profile, a fact which suggests the flow is field driven rather than boundary driven. For a general flow in the absence of time-dependent boundaries, it is shown that the SLLOD equations are no longer canonical in the laboratory momenta, and a modified form of the SLLOD dynamics is presented which is valid arbitrarily far from equilibrium for boundary conditions appropriate to the flow. From an analysis of the conserved energy for the new SLLOD equations in the absence of time-dependent boundary conditions, it is shown that the correct local thermodynamics is obtained. In addition, the idea of coupling each degree of freedom in the system to a Nose–Hoover chain thermostat is presented as a means of efficiently generating the phase space distribution.

The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

Abstract: The motion of a collisionless plasma is modeled by solutions to the Vlasov–Maxwell system. The Cauchy problem for the relativistic Vlasov–Maxwell system is studied in the case when the phase space distribution function f = f(t,x,v) depends on the time t, \(\) and \(\). Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows.

The N- and 1-Time Classical Descriptions of N-Body Relativistic Kinematics and the Electromagnetic Interaction

Abstract: Given N relativistic scalar free particles described by N mass-shell first class constraints in their 8N-dimensional phase space, their N-time description is obtained by means of a series of canonical transformations to a quasi-Shanmugadhasan basis adapted to the constraints. Then the same system is reformulated on spacelike hypersurfaces: the restriction to the family of hyperplanes orthogonal to the total timelike momentum gives rise to a covariant intrinsic 1-time formulation called the "rest-frame instant form" of dynamics. The relation between the N- and 1-time descriptions, the mass spectrum of the system and the way to introduce mutual interactions among the particles are studied. Then the 1-time description of the isolated system of N charged scalar particles plus the electromagnetic field is obtained. The use of Grassmann variables to describe the charges together with the determination of the field and particle Dirac observables leads to a formulation without infinite self-energies and with mutual Coulomb interactions extracted from classical electromagnetic field theory. A comparison with the Feshbach–Villars Hamiltonian formulation of the Klein–Gordon equation is made. Finally a 1-time covariant formulation of relativistic statistical mechanics is found.

Computational Physics: Problem Solving with Computers

Abstract: Partial table of contents: GENERALITIES. Computing Software Basics. Errors and Uncertainties in Computations. APPLICATIONS. Data Fitting. Deterministic Randomness. Monte Carlo Applications. Differentiation. Differential Equations and Oscillations. Anharmonic Oscillations. Unusual Dynamics of Nonlinear Systems. Differential Chaos in Phase Space. Matrix Computing and Subroutine Libraries. Bound States in Momentum Space. Computing Hardware Basics: Memory and CPU. High-Performance Computing: Profiling and Tuning. Object-Oriented Programming: Kinematics. Thermodynamic Simulations: The Ising Model. Fractals. PARTIAL DIFFERENTIAL EQUATIONS. Heat Flow. Waves on a String. NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS. Solitons, The KdeV Equation. Confined Electronic Wave Packets. Appendices. Glossary. References. Index.

Laser cooling of internal degrees of freedom. II

Abstract: Theoretical progress in the cooling of internal degrees of freedom of molecules using shaped laser pulses is reported. The emphasis is on general concepts and universal constraints. Several alternative definitions of cooling are considered, including reduction of the von Neumann entropy, −tr{ρlogρ} and increase of the Renyi entropy, tr{ρ2}. A distinction between intensive and extensive considerations is used to analyse the cooling process in open systems. It is shown that the Renyi entropy increase is consistent with an increase in the system phase space density and an increase in the absolute population in the ground state. The limitations on cooling processes imposed by Hamiltonian generated unitary transformations are analyzed. For a single mode system with a ground and excited electronic surfaces driven by an external field it is shown that it is impossible to increase the ground state population beyond its initial value. A numerical example based on optimal control theory demonstrates this result....

Phase Space Bounds for Quantum Mechanics on a Compact Lie Group

Abstract: Let K be a compact, connected Lie group and KC its complexification. I consider the Hilbert space HL 2 (KC ; t )of holomorphic functions introduced in (H1), where the parameter t is to be interpreted as Planck's constant. In light of (L-S), the complex group KC may be identified canonically with the cotangent bundle of K. Using this identification I associate to each F 2H L 2 ( K C ; t )a "phase space probability density." The main result of this paper is Theorem 1, which provides an upper bound on this density which holds uniformly over all F and all points in phase space. Specifically, the phase space probability density is at most at (2t) n , where n = dim K and at is a constant which tends to one exponentially fast as t tends to zero. At least for small t, this bound cannot be significantly improved. With t regarded as Planck's constant, the quantity (2t) n is precisely what is ex- pected on physical grounds. Theorem 1 should be interpreted as a form of the Heisenberg uncertainty principle for K, that is, a limit on the concentration of states in phase space. The theorem supports the interpretation of the Hilbert space HL 2 (KC ; t )as the phase space representation of quantum mechanics for a particle with configuration space K. The phase space bound is deduced from very sharp pointwise bounds on functions in HL 2 (KC ; t )(Theorem 2). The proofs rely on precise calculations involving the heat kernel on K and the heat kernel on KC=K.

Adiabatically changing the phase-space density of a trapped bose gas

Abstract: We show that the degeneracy parameter of a trapped Bose gas can be changed adiabatically in a reversible way, both in the Boltzmann regime and in the degenerate Bose regime. We have performed measurements on spin-polarized atomic hydrogen in the Boltzmann regime, demonstrating reversible changes of the degeneracy parameter (phase-space density) by more than a factor of 2. This result is in good agreement with theory. By extending our theoretical analysis to the quantum degenerate regime we predict that, starting close enough to the Bose-Einstein phase transition, one can cross the transition by an adiabatic change of the trap shape. [S0031-9007(97)02357-0] PACS numbers: 03.75.Fi, 67.65.+z, 32.80.Pj

Flavor oscillations from a spatially localized source: A simple general treatment

Abstract: A unique description avoiding confusion is presented for all flavor oscillation experiments in which particles of a definite flavor are emitted from a localized source. The probability for finding a particle with the wrong flavor must vanish at the position of the source for all times. This condition requires flavor-time and flavor-energy factorizations which determine uniquely the flavor mixture observed at a detector in the oscillation region, i.e., where the overlaps between the wave packets for different mass eigenstates are almost complete. Oscillation periods calculated for ``gedanken'' time-measurement experiments are shown to give the correct measured oscillation wavelength in space when multiplied by the group velocity. Examples of neutrino propagation in a weak field and in a gravitational field are given. In these cases the relative phase is modified differently for measurements in space and time. Energy-momentum (frequency-wave number) and space-time descriptions are complementary, equally valid, and give the same results. The two identical phase shifts obtained describe the same physics; adding them together to get a factor of 2 is double counting.

Quantum Optical Phase

Abstract: Simple physical ideas lead us to identify the phase of a single mode of the electromagnetic field as the quantity conjugate to the photon number. This in turn leads to the form of the phase probability distribution. The phase operator cannot be represented in the conventional infinite dimensional Hilbert space but can be expressed by means of a finite-dimensional state space together with a suitable limiting procedure. We introduce the Hermitian optical phase operator and describe its most important properties.

Phase space structure of triatomic molecules

Abstract: The bifurcation structure is investigated for a Hamiltonian for the three coupled nonlinear vibrations of a highly excited triatomic molecule. The starting point is a quantum Hamiltonian used to fit experimental spectra. This Hamiltonian includes 1:1 Darling–Dennison resonance coupling between the stretches, and 2:1 Fermi resonance coupling between the stretches and bend. A classical Hamiltonian is obtained using the Heisenberg correspondence principle. Surfaces of section show a pronounced degree of chaos at high energies, with a mixture of chaotic and regular dynamics. The large-scale bifurcation structure is found semianalytically, without recourse to numerical solution of Hamilton’s equations, by taking advantage of the fact that the spectroscopic Hamiltonian has a conserved polyad quantum number, corresponding to an approximate constant of the motion of the molecule. Bifurcation diagrams are analyzed for a number of molecules including H2O, D2O, NO2, ClO2, O3, and H2S.

A q-deformation of the harmonic oscillator

Abstract: The q-deformed harmonic oscillator is studied in the light of q-deformed phase space variables. This allows a formulation of the corresponding Hamiltonian in terms of the ordinary canonical variables x and p. The spectrum shows unexpected features such as degeneracy and an additional part that cannot be reached from the ground state by creation operators. The eigenfunctions show lattice structure, as expected.

Persistence and stability of relative equilibria

Abstract: We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.