Showing papers on "Phase space published in 2004"

Bubbling AdS space and 1/2 BPS geometries

Abstract: We consider all 1/2 BPS excitations of AdS × S configurations in both type-IIB string theory and M-theory. In the dual field theories these excitations are described by free fermions. Configurations which are dual to arbitrary droplets of free fermions in phase space correspond to smooth geometries with no horizons. In fact, the ten dimensional geometry contains a special two dimensional plane which can be identified with the phase space of the free fermion system. The topology of the resulting geometries depends only on the topology of the collection of droplets on this plane. These solutions also give a very explicit realization of the geometric transitions between branes and fluxes. We also describe all 1/2 BPS excitations of plane wave geometries. The problem of finding the explicit geometries is reduced to solving a Laplace (or Toda) equation with simple boundary conditions. We present a large class of explicit solutions. In addition, we are led to a rather general class of AdS5 compactifications of M-theory preserving = 2 superconformal symmetry. We also find smooth geometries that correspond to various vacua of the maximally supersymmetric mass-deformed M2 brane theory. Finally, we present a smooth 1/2 BPS solution of seven dimensional gauged supergravity corresponding to a condensate of one of the charged scalars.

Discrete phase space based on finite fields

Abstract: The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a $2N\ifmmode\times\else\texttimes\fi{}2N$ discrete phase space for a system with $N$ orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having $N$ elements. There exists such a field if and only if $N$ is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our $N\ifmmode\times\else\texttimes\fi{}N$ phase space also leads naturally to a method of constructing a complete set of $N+1$ mutually unbiased bases for the state space.

Relativistic electrons in the outer radiation belt: Differentiating between acceleration mechanisms

Abstract: [1] Many theoretical models have been developed to explain the rapid acceleration to relativistic energies of electrons that form the Earth's radiation belts. However, after decades of research, none of these models has been unambiguously confirmed by comparison to observations. Proposed models can be separated into two types: internal and external source acceleration mechanisms. Internal source acceleration mechanisms accelerate electrons already present in the inner magnetosphere (L < 6.6), while external source acceleration mechanisms transport and accelerate a source population of electrons from the outer to the inner magnetosphere. In principle, the two types of acceleration mechanisms can be differentiated because they imply that different radial gradients of electron phase space density expressed as a function of the three adiabatic invariants will develop. Model predictions can be tested by transforming measured electron flux (given as a function of pitch angle, energy, and position) to phase space density as a function of the three invariants, μ, K, and Φ. The transformation requires adoption of a magnetic field model. Phase space density estimates have, in the past, produced contradictory results because of limited measurements and field model errors. In this study we greatly reduce the uncertainties of previous work and account for the contradictions. We use data principally from the Polar High Sensitivity Telescope energetic detector on the Polar spacecraft and the Tsyganenko and Stern [1996] field model to obtain phase space density. We show how imperfect magnetic field models produce phase space density errors and explore how those errors modify interpretations. On the basis of the analysis we conclude that the data are best explained by models that require acceleration of an internal source of electrons near L* ∼ 5. We also suggest that outward radial diffusion from a phase space density peak near L* ∼ 5 can explain the observed correspondence between flux enhancements at geostationary orbit and increases in ULF wave power.

A Finite Difference Representation of Neutrino Radiation Hydrodynamics in Spherically Symmetric General Relativistic Spacetime

Abstract: We present an implicit finite difference representation for general relativistic radiation hydrodynamics in spherical symmetry. Our code, AGILE-BOLTZTRAN, solves the Boltzmann transport equation for the angular and spectral neutrino distribution functions in self-consistent simulations of stellar core collapse and postbounce evolution. It implements a dynamically adaptive grid in comoving coordinates. A comoving frame in the momentum phase space facilitates the evaluation and tabulation of neutrino-matter interaction cross sections but produces a multitude of observer corrections in the transport equation. Most macroscopically interesting physical quantities are defined by expectation values of the distribution function. We optimize the finite differencing of the microscopic transport equation for a consistent evolution of important expectation values. We test our code in simulations launched from progenitor stars with 13 solar masses and 40 solar masses. Half a second after core collapse and bounce, the protoneutron star in the latter case reaches its maximum mass and collapses further to form a black hole. When the hydrostatic gravitational contraction sets in, we find a transient increase in electron flavor neutrino luminosities due to a change in the accretion rate. The μ- and τ-neutrino luminosities and rms energies, however, continue to rise because previously shock-heated material with a nondegenerate electron gas starts to replace the cool degenerate material at their production site. We demonstrate this by supplementing the concept of neutrinospheres with a more detailed statistical description of the origin of escaping neutrinos. Adhering to our tradition, we compare the evolution of the 13 M⊙ progenitor star to corresponding simulations with the multigroup flux-limited diffusion approximation, based on a recently developed flux limiter. We find similar results in the postbounce phase and validate this MGFLD approach for the spherically symmetric case with standard input physics.

Quark imaging in the proton via quantum phase-space distributions

Abstract: We develop the concept of quantum phase-space (Wigner) distributions for quarks and gluons in the proton. To appreciate their physical content, we analyze the contraints from special relativity on the interpretation of elastic form factors, and examine the physics of the Feynman parton distributions in the proton's rest frame. We relate the quark Wigner functions to the transverse-momentum dependent parton distributions and generalized parton distributions, emphasizing the physical role of the skewness parameter. We show that the Wigner functions allow us to visualize quantum quarks and gluons using the language of classical phase space. We present two examples of the quark Wigner distributions and point out some model-independent features.

Disentanglement and decoherence by open system dynamics

Abstract: The destruction of quantum interference, decoherence, and the destruction of entanglement both appear to occur under the same circumstances. To address the connection between these two phenomena, we consider the evolution of arbitrary initial states of a two-particle system under open system dynamics described by a class of master equations which produce decoherence of each particle. We show that all initial states become separable after a finite time, and we produce the explicit form of the separated state. The result extends and amplifies an earlier result of Di\'osi. We illustrate the general result by considering the case in which the initial state is an Einstein-Podolsky-Rosen state (in which both the positions and momenta of a particle pair are perfectly correlated). This example clearly illustrates how the spreading out in phase space produced by the environment leads to certain disentanglement conditions becoming satisfied.

A new method for real radiation at next-to-next-to-leading order

Abstract: We propose a new method of computing real emission contributions to hard QCD processes. Our approach uses sector decomposition of the exclusive final-state phase space to enable extraction of all singularities of the real emission matrix elements before integration over any kinematic variable. The exact kinematics of the real emission process are preserved in all regions of phase space. Traditional approaches to extracting singularities from real emission matrix elements, such as phase space slicing and dipole subtraction, require both the determination of counterterms for double real emission amplitudes in singular kinematic limits and the integration of these contributions analytically to cancel the resulting singularities against virtual corrections. Our method addresses both of these issues. The implementation of constraints on the final-state phase space, including various jet algorithms, is simple using our approach. We illustrate our method using ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}$ jets at $O({\ensuremath{\alpha}}_{S}^{2})$ as an example.

Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices

Abstract: Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.

Cosmological Perturbations Through a Simple Bounce

Abstract: We present a study of a simple scalar field model that yields non‐singular cosmological solutions. We study both the qualitative dynamics of the homogeneous and isotropic background and the evolution of inhomogeneous linear perturbations. We calculate the spectrum of perturbations generated on super‐Hubble scales during the collapse phase from initial vacuum fluctuations on small scales and then evolve these numerically through the bounce. We show that the comoving curvature perturbation calculated during the collapse phase provides a good estimate of the resulting large scale adiabatic perturbation in the expanding phase while the Bardeen metric potential is dominated by what becomes a decaying mode after the bounce. We show that a power‐law collapse phase with scale factor proportional to (−t)2/3 can yield a scale‐invariant spectrum of adiabatic scalar perturbations in the expanding phase, but the amplitude of tensor perturbations places important constraints on the allowed initial conditions.

Sensitive dependence on initial conditions in transition to turbulence in pipe flow

Abstract: The experiments by Darbyshire & Mullin (1995) on the transition to turbulence in pipe flow show that there is no sharp border between initial conditions that trigger turbulence and those that do not. We here relate this behaviour to the possibility that the transition to turbulence is connected with the formation of a chaotic saddle in the phase space of the system. We quantify a sensitive dependence on initial conditions and find in a statistical analysis that in the transition region the distribution of turbulent lifetimes follows an exponential law. The characteristic mean lifetime of the distribution increases rapidly with Reynolds number and becomes inaccessibly large for Reynolds numbers exceeding about 2250. Suitable experiments to further probe this concept are proposed.

On quantum mechanics on noncommutative quantum phase space

Abstract: In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter and a momentum noncommutativity matrix parameter , is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraints on this particular transformation, we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on the physical system under study. This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β, representing the same particle in presence of a magnetic field . For the other examples, additional correction terms depending on β appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively feeled with opposite sign.

Solution representations for a wave equation with weak dissipation

Abstract: We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+tvt=0, x∈ℝn, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain Lp–Lq estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of μand that μ=2 is critical. Copyright © 2004 John Wiley & Sons, Ltd.

Time-Frequency Analysis of Sjöstrand's Class

Abstract: We investigate the properties an exotic symbol class of pseudodifferential operators, Sjostrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjostrand's fundamental results and to far-reaching generalizations

Bubbling AdS space and 1/2 BPS geometries

Abstract: We consider all 1/2 BPS excitations of $AdS \times S$ configurations in both type IIB string theory and M-theory. In the dual field theories these excitations are described by free fermions. Configurations which are dual to arbitrary droplets of free fermions in phase space correspond to smooth geometries with no horizons. In fact, the ten dimensional geometry contains a special two dimensional plane which can be identified with the phase space of the free fermion system. The topology of the resulting geometries depends only on the topology of the collection of droplets on this plane. These solutions also give a very explicit realization of the geometric transitions between branes and fluxes. We also describe all 1/2 BPS excitations of plane wave geometries. The problem of finding the explicit geometries is reduced to solving a Laplace (or Toda) equation with simple boundary conditions. We present a large class of explicit solutions. In addition, we are led to a rather general class of $AdS_5$ compactifications of M-theory preserving ${\cal N} =2$ superconformal symmetry. We also find smooth geometries that correspond to various vacua of the maximally supersymmetric mass-deformed M2 brane theory. Finally, we present a smooth 1/2 BPS solution of seven dimensional gauged supergravity corresponding to a condensate of one of the charged scalars.

Statistical stability in time reversal

Abstract: When a signal is emitted from a source, recorded by an array of transducers, time-reversed, and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency remote-sensing regime and show that, because of multiple scattering in an inhomogeneous or random medium, it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime, so we analyze time reversal with the parabolic or paraxial wave equation.

Geometric Optics on Phase Space

Abstract: Preliminary ToC: Optical Phase Space, Hamiltonian Systems and Lie Algebras.- Lie Groups of Optical Transformation.- The Paraxial Regime.- Hamiltonian Aberrations.

Nonsequential double ionization at the single-optical-cycle limit.

Abstract: We report differential measurements of ${\mathrm{A}\mathrm{r}}^{++}$ ion momentum distributions from nonsequential double ionization in phase-stabilized few-cycle laser pulses. The distributions depend strongly on the carrier-envelope (CE) phase. Via control over the CE phase one is able to direct the nonsequential double-ionization dynamics. Data analysis through a classical model calculation reveals that the influence of the optical phase enters via (i) the cycle dependent electric field ionization rate, (ii) the electron recollision time, and (iii) the accessible phase space for inelastic collisions. Our model indicates that the combination of these effects allows a look into single cycle dynamics already for few-cycle pulses.

Nonextensive statistical mechanics: A brief introduction

Abstract: Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{\rm BG} = -k \sum_{i = 1}^W p_i \ln p_i$ . It enables a successful thermal approach to ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be yes for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy $S_q = k\frac{1-\sum_{i = 1}^W p_i^q}{q-1}$ (with $q \in \cal R$ , and $S_1 = S_{\rm BG}$ ). This theory is sometimes referred in the literature as nonextensive statistical mechanics. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of stability (or experimental robustness) introduced by B. Lesche in 1982 and recently revisited by S. Abe.

The uncertainty relation for joint measurement of postion and momentum

Abstract: We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (ΔP)(ΔQ) ≥ Ch, where the 'uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision ΔQ of the position measurement, and the perturbation ΔP of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

Fractional generalization of Liouville equations.

Abstract: In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed.

Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

Abstract: The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary (finite) number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a “surface of no return.” The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold (NHIM). A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices

Abstract: Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem1,2,3,6,7,8,9—a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt2. By contrast, ‘non-KAM’ chaos switches on and off abruptly at critical values of the perturbation frequency6,7,8,9. This type of dynamics has wide-ranging implications in the theory of plasma physics10, tokamak fusion11, turbulence6,7,12, ion traps13, and quasicrystals6,8. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice22,23,24,25,26,27 with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns6,7,8,9,10,12,13,14,15,16 in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

Performance limitations of flat-histogram methods.

Abstract: Monte Carlo methods are well-suited for the simulation of large many body problems, since the complexity for a single Monte Carlo update step scales only polynomially and often linearly in the system size, while the config- uration space grows exponentially with the system size. The performance of a Monte Carlo method is then deter- mined by how many update steps are needed to efficiently sample the configuration space. For second order phase transitions in unfrustrated systems the problem of "crit- ical slowing down" - a rapid divergence of the number of Monte Carlo steps needed to obtain a subsequent un- correlated configuration - was solved more than a decade ago by cluster update algorithms (1). At first order phase transitions and in systems with many local minima of the free energy such as frustrated magnets or spin glasses, there is the similar problem of long tunneling times be- tween local minima. With energy barriersE scaling lin- early with the linear system size L, the tunneling times � at an inverse temperature � = 1/kBT scale exponentially with the system size, � � exp(��E) / exp(const × L). Several methods were developed to overcome this tun- neling problem, such as the multicanonical method (2), broad histograms (4), simulated and parallel tempering (3), and Wang-Landau sampling (5). The common aim of all these methods is to broaden the range of energies sam- pled within Monte Carlo simulations from the sharply peaked distribution of canonical sampling at fixed tem- perature in order to ease the tunneling through barriers. Ideally, all relevant energy levels are sampled equally often during a simulation, thus producing a "flat his- togram" in energy space. Some methods approach this goal by variations and generalizations of canonical dis- tributions (2, 3), while others (4, 5) discard the notion of temperature completely and instead are formulated in terms of the density of states. With a probability p(E) for a single configuration with energy E, the probability of sampling an arbitrary configuration with energy E is given as PE = �(E)p(E), where the density of states �(E) counts the number of states with energy E. Upon choos- ing p(E) / 1/�(E) instead of p(E) / exp( �E) one ob- tains a constant probability PE for visiting each energy level E, and hence a flat histogram. Wang and Landau (5) proposed a simple and elegant flat histogram algorithm that iteratively improves approximations to the initially unknown density of states �(E). Once �(E) is determined with sufficient accuracy, the Monte Carlo algorithm just performs a random walk in energy space. Within two years of publication this algorithm has been applied to a large number of problems (6, 7, 8) and extended to quantum systems (9). In this Letter we investigate the performance of flat histogram algorithms in general, and the Wang-Landau algorithm in particular, for three systems for which the density of states �(E) is known exactly on finite two- dimensional (2D) lattices: the Ising ferromagnet as the simplest example, the fully frustrated Ising model as a prototype for frustrated systems, and the ±J Ising spin glass. For each of these models we construct a perfect flat histogram method by simulating a random walk in configuration space where we employ the known density of states for these models to set p(E) / 1/�(E). As a measure of performance we use the average tun- neling timeto get from a ground state (lowest energy configuration) to an anti-ground state (configuration of highest energy), which is the relevant time scale for sam- pling the whole phase space (10). Since the number of energy levels in a d-dimensional system with linear size L scales with the number of spins N = L d , the tunneling time for a pure random walk in energy space is

Picturing qubits in phase space

Abstract: Focusing particularly on one-qubit and two-qubit systems, I explain how the quantum state of a system of n qubits can be expressed as a real function--a generalized Wigner function--on a discrete 2n ?? 2n phase space. The phase space is based on the finite field having 2n elements, and its geometric structure leads naturally to the construction of a complete set of 2n + 1 mutually conjugate bases.