Showing papers on "Phase space published in 2000"

Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states

Abstract: Jaynes' information theory formalism of statistical mechanics is applied to the stationary states of open, non-equilibrium systems. The key result is the construction of the probability distribution for the underlying microscopic phase space trajectories. Three consequences of this result are then derived : the fluctuation theorem, the principle of maximum entropy production, and the emergence of self-organized criticality for flux-driven systems in the slowly-driven limit. The accumulating empirical evidence for these results lends support to Jaynes' formalism as a common predictive framework for equilibrium and non-equilibrium statistical mechanics.

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

Abstract: In a rst part we discuss the well-known prob- lem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but al- most universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approx- imate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear pic- ture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the prob- lem, the 2D sphere, is presented. This provides a better visualization of the dynamics. In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth fac- tor of Nearby Orbits (MEGNO), that is suitable to inves- tigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic poten- tial it is shown to be eective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physi- cal times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynam- ics in the phase space than other wide used tools. Finally, we discuss the structure of the phase space as- sociated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always conned to narrow laments, with a Lyapunov time about three characteristic periods.

The Gaussian-BGK model of Boltzmann equation with small Prandtl number

Abstract: In this paper we prove the entropy inequality for the Gaussian-BGK model of Boltzmann equation This model, also called ellipsoidal statistical model, was introduced in order to fit realistic values of the transport coefficients (Prandtl number, second viscosity) in the Navier-Stokes approxima- tion, which cannot be achieved by the usual relaxation towards isotropic Maxwellians introduced in standard BGK models Moreover, we introduce new entropic kinetic models for polyatomic gases which suppress the internal energy variable in the phase space by using two distribution functions (one for particles mass and one for their internal energy) This reduces the cost of their numerical solution while keeping a kinetic description well adapted to desequilibrium regions

Collisions near threshold in atomic and molecular physics

Abstract: We review topics of current interest in the physics of electronic, atomic and molecular scattering in the vicinity of thresholds. Starting from phase space arguments, we discuss the modifications of the Wigner law that are required to deal with scattering by Coulomb, dipolar and dispersion potentials, as well as aspects of threshold behaviour observed in ultracold atomic collisions. We employ the tools of quantum defect and semiclassical theories to bring out the rich variety of threshold behaviours. The discussion is then turned to recent progress in understanding threshold behaviour of many-body break-ups into both charged and neutral species, including both Wannier double ionization and three-body recombination in ultracold gases. We emphasize the dominant role that hyperspherical coordinate methods have played in understanding these problems. We assess the effects of external fields on scattering, and the corresponding modification of phase space that alters the Wigner law. Threshold laws in low dimensions and examples of their applications to specific collision processes are discussed.

On the source location of radiation belt relativistic electrons

Abstract: Observations from the High Sensitivity Telescope (HIST) on Polar made around January and May 1998 are used to constrain the source location of outer radiation belt relativistic electrons. Phase space densities calculated as a function of the three adiabatic invariants show positive radial gradients for L 4, peaks in the radial dependence of the phase space density are suggestive of a local electron source that may be nonadiabatic acceleration or pitch angle scattering. However, discrepancies in the results obtained with different magnetic field models and at different local times make this a tentative conclusion.

Cohomology in the pure spinor formalism for the superstring

Abstract: Inst. de Fis. Teorica Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, Sao Paulo, SP

Statistical mechanics treatment of the evolution of dislocation distributions in single crystals

Abstract: A statistical mechanics framework for the evolution of the distribution of dislocations in a single crystal is established. Dislocations on various slip systems are represented by a set of phase-space distributions each of which depends on an angular phase space coordinate that represents the line sense of dislocations. The invariance of the integral of the dislocation density tensor over the crystal volume is proved. From the invariance of this integral, a set of Liouville-type kinetic equations for the phase-space distributions is developed. The classically known continuity equation for the dislocation density tensor is established as a macroscopic transport equation, showing that the geometric and crystallographic notions of dislocations are unified. A detailed account for the short-range reactions and cross slip of dislocations is presented. In addition to the nonlinear coupling arising from the long-range interaction between dislocations, the kinetic equations are quadratically coupled via the short-range reactions and linearly coupled via cross slip. The framework developed here can be used to derive macroscopic transport-reaction models, which is shown for a special case of single-slip configuration. The boundary value problem of dislocation dynamics is summarized, and the prospects of development of physical plasticity models for single crystals are discussed.

Mixed quantum-classical surface hopping dynamics

Abstract: An algorithm is presented for the exact solution of the evolution of the density matrix of a mixed quantum-classical system in terms of an ensemble of surface hopping trajectories. The system comprises a quantum subsystem coupled to a classical bath whose evolution is governed by a mixed quantum-classical Liouville equation. The integral solution of the evolution equation is formulated in terms of a concatenation of classical evolution segments for the bath phase space coordinates separated by operators that change the quantum state and bath momenta. A hybrid Molecular Dynamics–Monte Carlo scheme which follows a branching tree of trajectories arising from the action of momentum derivatives is constructed to solve the integral equation. We also consider a simpler scheme where changes in the bath momenta are approximated by momentum jumps. These schemes are illustrated by considering the computation of the evolution of the density matrix for a two-level system coupled to a low dimensional classical bath.

Absence of thermodynamic phase transition in a model glass former

Abstract: The glass transition can be viewed simply as the point at which the viscosity of a structurally disordered liquid reaches a universal threshold value1. But this is an operational definition that circumvents fundamental issues, such as whether the glass transition is a purely dynamical phenomenon2. If so, ergodicity gets broken (the system becomes confined to some part of its phase space), but the thermodynamic properties of the liquid remain unchanged across the transition, provided they are determined as thermodynamic equilibrium averages over the whole phase space. The opposite view3,4,5,6 claims that an underlying thermodynamic phase transition is responsible for the pronounced slow-down in the dynamics at the liquid–glass boundary. Such a phase transition would trigger the dynamic standstill, and then be masked by it. Here we perform Monte Carlo simulations of a two-dimensional system of polydisperse hard disks far within its glassy phase. The approach7 allows for non-local moves in a way that preserves micro-reversibility. We find no evidence for a thermodynamic phase transition up to very high densities; the glass is thus indistinguishable from the liquid on purely thermodynamic grounds.

Passivity and microlocal spectrum condition

Abstract: In the setting of vector-valued quantum fields obeying a linear wave-equation in a globally hyperbolic, stationary spacetime, it is shown that the two-point functions of passive quantum states (mixtures of ground- or KMS-states) fulfill the microlocal spectrum condition (which in the case of the canonically quantized scalar field is equivalent to saying that the two-pnt function is of Hadamard form). The fields can be of bosonic or fermionic character. We also give an abstract version of this result by showing that passive states of a topological *-dynamical system have an asymptotic pair correlation spectrum of a specific type.

Gauge Field Theory Coherent States (GCS) : III.

Abstract: In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge eld theory by Ashtekar, Lewandowski, Marolf, Mour~ ao and Thiemann. In this paper we establish the \Ehrenfest Property" of these states which are labelled by a point (A;E), a connection and an electric eld, in the classical phase space. By this we mean that i) The expectation value of all elementary quantum operators ^ O with respect to the coherent state with label (A;E) is given to zeroth order in h by the value of the corresponding classical function O evaluated at the phase space point (A;E) and ii) The expectation value of the commutator between two elementary quantum operators [ ^ O1; ^ O2]=(i h) divided by i h with respect to the coherent state with label (A;E) is given to zeroth order in h by the value of the Poisson bracket between the corresponding classical functionsfO1;O2g evaluated at the phase space point (A;E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the innitesimal quantum dynamics of quantum general relativity is to zeroth order in h indeed given by classical general relativity.

Microscopic expressions for the thermodynamic temperature

Abstract: We show that arbitrary phase space vector fields can be used to generate phase functions whose ensemble averages give the thermodynamic temperature. We describe conditions for the validity of these functions in periodic boundary systems and the molecular dynamics (MD) ensemble, and test them with a short-ranged potential MD simulation.

Calculating Free Energies Using Scaled-Force Molecular Dynamics Algorithm

Abstract: One common objective of molecular simulations in chemistry and biology is to calculate the free energy difference between different states of the system of interest. Examples of problems that have such an objective are calculations of receptor-ligand or protein-drug interactions, associations of molecules in response to hydrophobic, and electrostatic interactions or partition of molecules between immiscible liquids. Another common objective is to describe evolution of the system towards a low energy (possibly the global minimum energy), 'native' state. Perhaps the best example of such a problem is folding of proteins or short RNA molecules. Both types of problems share the same difficulty. Often, different states of the system are separated by high energy barriers, which implies that transitions between these states are rare events. This, in turn, can greatly impede exploration of phase space. In some instances this can lead to 'quasi non-ergodicity', whereby a part of phase space is inaccessible on timescales of the simulation. A host of strategies has been developed to improve efficiency of sampling the phase space. For example, some Monte Carlo techniques involve large steps which move the system between low-energy regions in phase space without the need for sampling the configurations corresponding to energy barriers (J-walking). Most strategies, however, rely on modifying probabilities of sampling low and high-energy regions in phase space such that transitions between states of interest are encouraged. Perhaps the simplest implementation of this strategy is to increase the temperature of the system. This approach was successfully used to identify denaturation pathways in several proteins, but it is clearly not applicable to protein folding. It is also not a successful method for determining free energy differences. Finally, the approach is likely to fail for systems with co-existing phases, such as water-membrane systems, because it may lead to spontaneous mixing. A similar difficulty may be encountered in any method relying on global modifications of phase space.

Generalized Gaussian moment thermostatting: A new continuous dynamical approach to the canonical ensemble

Abstract: A new method for generating the canonical ensemble via continuous dynamics is presented. The new method is based on controlling the fluctuations of an arbitrary number of moments of the multidimensional Gaussian momentum distribution function. The equations of motion are non-Hamiltonian, and hence have a nonvanishing phase space compressibility. By applying the statistical mechanical theory of non-Hamiltonian systems recently introduced by the authors [M. E. Tuckerman, C. J. Mundy, and G. J. Martyna, Europhys. Lett. 45, 149 (1999)], the equations are shown to produce the correct canonical phase space distribution function. Reversible integrators for the new equations of motion are derived based on a Trotter-type factorization of the classical Liouville propagator. The new method is applied to a variety of simple one-dimensional example problems and is shown to generate ergodic trajectories and correct canonical distribution functions of both position and momentum. The new method is further shown to lead to rapid convergence in molecular dynamics based calculations of path integrals. The performance of the new method in these examples is compared to that of another canonical dynamics method, the Nose–Hoover chain method [G. J. Martyna, M. L. Klein, and M. E. Tuckerman, J. Chem. Phys. 97, 2635 (1992)]. The comparison demonstrates the improvements afforded by the new method as a molecular dynamics tool. Finally, when employed in molecular dynamics simulations of biological macromolecules, the new method is shown to provide better energy equipartitioning and temperature control and to lead to improved spatial sampling over the Nose–Hoover chain method in a realistic application.

On the perpendicular scale of electron phase-space holes

Abstract: The perpendicular scale of electron phase-space holes is investigated using electric eld data from the Po- lar Plasma Wave Instrument. We show that the electron phase-space holes are roughly spherical fore=!p > 1, and become more oblate (with the perpendicular scale larger than the parallel scale) with decreasing e=!p. A scaling argument based upon electron gyrokinetic theory is pro- posed as a possible explanation for the observed scaling. The data indicate that the ratio of the parallel dimen- sion (Lk) to the perpendicular dimension (L?) is such that Lk=L?' (1+ 2= 2) 1=2 . Our results provide a connection between the Geotail measurements in the deep magnetotail, where e=!p 1, and the FAST measurements in the low altitude auroral zone, wheree=!p 1.

Time dependent quantum propagation in phase space

Abstract: Numerical solutions of the quantum time-dependent integro-differential Schrodinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble.

First experimental evidence for chaos-assisted tunneling in a microwave annular billiard

Abstract: We report on first experimental signatures for chaos-assisted tunneling in a two-dimensional annular billiard. Measurements of microwave spectra from a superconducting cavity with high frequency resolution are combined with electromagnetic field distributions experimentally determined from a normal conducting twin cavity with high spatial resolution to resolve eigenmodes with properly identified quantum numbers. Distributions of quasidoublet splittings serve as basic observables for the tunneling between whispering gallery-type modes localized to congruent, but distinct tori which are coupled weakly to irregular eigenstates associated with the chaotic region in phase space.

Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics

Abstract: Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov–Poisson system, which is the non-relativistic case, or of the Vlasov–Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E, L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E0 needs to be prescribed, where E0 is a cut-off energy above which no particles exist.

Measurement of a negative value for the Wigner function of radiation

Abstract: Following a proposal by two of us [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)], we have measured the Wigner function at the origin of phase space for a single photon field. Its value is negative, exhibiting the nonclassical nature of this state. The experiment is based on the absorption-free detection of the microwave field stored in a superconducting cavity [G. Nogues et al., Nature (London) 400, 239 (1999)]. Extension to a measurement of the Wigner function over the complete phase space is discussed.

Correlated histogram representation of Monte Carlo derived medical accelerator photon-output phase space

Abstract: A method is provided to represent the calculated phase space of photons emanating from medical accelerators used in photon teletherapy The method reproduces the energy distributions and trajectories of the photons originating in the bremsstrahlung target and of photons scattered by components within the accelerator head The method reproduces the energy and directional information from sources up to several centimeters in radial extent, so it is expected to generalize well to accelerators made by different manufacturers The method is computationally both fast and efficient, with overall sampling efficiency of 80% or higher for most field sizes The computational cost is independant of the number of beams used in the treatment plan

BRST Cohomology and Phase Space Reduction in Deformation Quantization

Abstract: In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a “quantum” Chevalley–Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is “sufficiently nice”, e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.

The method of loop molecules and the topology of the Kovalevskaya top

Abstract: A method for calculating topological invariants of the foliation of a phase space into invariant Liouville tori in the case of integrable Hamiltonian systems with two degrees of freedom is put forward. The structure of this foliation is completely described for the Kovalevskaya integrable case in rigid body dynamics.

Chiral invariant phase space event generator. I: Nucleon antinucleon annihilation at rest

Abstract: The CHIPS model and its first implementation within the GEANT4 simulation software package are considered Hadron production in the process of nucleon-antinucleon annihilation is used as the basic example showing the structure of the model and corresponding software modeling algorithms Model calculations of multiplicities and spectra of secondary hadrons in the annihilation process are compared with experimental data

Quantum transport through ballistic cavities: soft vs hard quantum chaos

Abstract: We study transport through a two-dimensional billiard attached to two infinite leads by numerically calculating the Landauer conductance and the Wigner time delay. In the generic case of a mixed phase space we find a power-law distribution of resonance widths and a power-law dependence of conductance increments apparently reflecting the classical dwell time exponent, in striking difference to the case of a fully chaotic phase space. Surprisingly, these power laws appear on energy scales below the mean level spacing, in contrast to semiclassical expectations.

On the second-order corrections to the quantum canonical equilibrium density matrix

Abstract: We consider the equilibrium state of a quantum system weakly coupled to a quantum bath within second order perturbation theory. It was previously shown by Romero-Rochin and Oppenheim [Physica A 155, 52 (1989)] that the equilibrium state deviates from the canonical form, e−βHs/Zs (Hs is the free system Hamiltonian and Zs the canonical partition function). We reproduce this result via a different derivation, starting from the non-Markovian, rather than the Markovian, quantum Master equation. Our derivation sheds new light on the mechanism that stabilizes the deviation from the canonical form and shows that it involves an interplay between a static distortion to the equilibrium state and dynamical system–bath correlations. We show that this deviation is a necessary consequence of translational invariance and vanishes when the rotating-wave-approximation is applied. The deviation is also shown to vanish for a two-level system off-diagonally coupled to a heat bath or when the Lamb shifts are neglected. Two ways for numerically evaluating the second order deviations are described. Finally, the deviations from canonical equilibrium are given an illuminating geometrical interpretation in terms of the phase space Wigner distribution.

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

Abstract: We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.