Phase Space Measurements Research Articles

Thermodynamics of itinerant magnets in a classical spin-fluctuation model

Thermodynamics of itinerant magnets is studied using a classical model with one parameter characterizing the degree of itinerancy. Monte Carlo simulations for bcc and fcc lattices are compared with the mean-field approximation and with the Onsager cavity field approximation extended to itinerant systems. The qualitative features of thermodynamics are similar to the known results of the functional integral method. It is found that magnetic short-range order is weak and almost independent on the degree of itinerancy, and the mean-field approximation describes the thermodynamics reasonably well. Ambiguity of the phase space measure for classical models is emphasized. The Onsager cavity field method is extended to itinerant systems, which involves the renormalization of both the Weiss field and the on-site exchange interaction. The predictions of this approximation are in excellent agreement with Monte Carlo results.

Benchmarking of 3D space charge codes using direct phase space measurements from photoemission high voltage dc gun

We present a comparison between space charge calculations and direct measurements of the transverse phase space of space charge dominated electron bunches from a high voltage dc photoemission gun followed by an emittance compensation solenoid magnet. The measurements were performed using a double-slit emittance measurement system over a range of bunch charge and solenoid current values. The data are compared with detailed simulations using the 3D space charge codes GPT and Parmela3D. The initial particle distributions were generated from measured transverse and temporal laser beam profiles at the photocathode. The beam brightness as a function of beam fraction is calculated for the measured phase space maps and found to approach within a factor of 2 the theoretical maximum set by the thermal energy and the accelerating field at the photocathode.

On the resonance eigenstates of an open quantum baker map

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, |zmin| ⩽ |z| ⩽ |zmax|. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius r. We prove that if the moduli converge to r = |zmax| then the sequence of eigenstates is associated with a fixed phase space measure ρmax. The same holds for sequences with eigenvalue moduli converging to |zmin|, with a different limit measure ρmin. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius |zmin| < r < |zmax| there is no unique limit measure, and we identify some families of eigenstates with precise self-similar properties.

Spectral and entropic characterizations of Wigner functions: Applications to model vibrational systems

The Wigner function for the pure quantum states is used as an integral kernel of the non-Hermitian operator K, to which the standard singular value decomposition (SVD) is applied. It provides a set of the squared singular values treated as probabilities of the individual phase-space processes, the latter being described by eigenfunctions of KK(+) (for coordinate variables) and K(+)K (for momentum variables). Such a SVD representation is employed to obviate the well-known difficulties in the definition of the phase-space entropy measures in terms of the Wigner function that usually allows negative values. In particular, the new measures of nonclassicality are constructed in the form that automatically satisfies additivity for systems composed of noninteracting parts. Furthermore, the emphasis is given on the geometrical interpretation of the full entropy measure as the effective phase-space volume in the Wigner picture of quantum mechanics. The approach is exemplified by considering some generic vibrational systems. Specifically, for eigenstates of the harmonic oscillator and a superposition of coherent states, the singular value spectrum is evaluated analytically. Numerical computations are given for the nonlinear problems (the Morse and double well oscillators, and the Henon-Heiles system). We also discuss the difficulties in implementation of a similar technique for electronic problems.

Erratum: Beam emittance measurement with laser wire scanners in the International Linear Collider beam delivery system [Phys. Rev. ST Accel. Beams10, 112801 (2007)

Accurate measurement of the beam phase-space is essential for the next generation of electron accelerators. A scheme for beam optics optimization and beam matrix reconstruction algorithms for the diagnostics section of the beam delivery system of the International Linear Collider based on laser-wire beam profile monitors are discussed. Possible modes of operation of the laser-wire system together with their corresponding performance are presented. Based on these results, prospects for reconstructing the ILC beam emittance from representative laser-wire beam size measurements are evaluated.

Beam emittance measurement with laser wire scanners in the International Linear Collider beam delivery system

Accurate measurement of the beam phase space is essential for the next generation of electron accelerators. A scheme for beam optics optimization and beam matrix reconstruction algorithms for the diagnostics section of the beam delivery system of the International Linear Collider (ILC) based on laser-wire beam profile monitors are discussed. Possible modes of operation of the laser-wire system together with their corresponding performance are presented. Based on these results, prospects for reconstructing the ILC beam emittance from representative laser-wire beam size measurements are evaluated.

On physics at the Planck scale: Space as a network

It is shown that the one-dimensional quantum field theory can be modeled as a chain of classical oscillators in a thermal bath provided that the Gibbs measure is identified with the phase-space volume measure, the chain being in a nonequilibrium state. Quantized strings and p-branes are also modeled by ordered systems of oscillators. The model of a one-dimensional superspace is constructed. It is shown that the Ramond-Neveu-Schwarz superstring is modeled by a helix formed of a bosonic-string in a multidimensional space. The physical 3D space is represented by a superstring structure (3D “network”), which is described by some Lagrangian. Thus, the unified description of all interactions including gravity is achieved because the superstring excitations involve all fields. In view of the discrete character of the initial structure, the theory is free of ultraviolet divergences. The essential element of the model is the occurrence of the cosmological constant in the gravity equations. A black hole model giving reasonable values for its temperature and entropy is proposed.

Unicity of KAM tori

The classical KAM theorem establishes the persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. A long-standing question is to what extent the perturbed tori are unique. Using the fact that at the level of tori there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine torus union of full measure.

On the geometry and entropy of non-Hamiltonian phase space

We analyse the equilibrium statistical mechanics of canonical, non-canonical andnon-Hamiltonian equations of motion, throwing light on the peculiar geometric structure ofphase space. Some fundamental issues regarding time translation and phase space measureare clarified. In particular, we emphasize that a phase space measure should be defined bymeans of the Jacobian of the transformation between different kinds of coordinates sincesuch a determinant is different from zero in the non-canonical case even if thephase space compressibility is null. Instead, the Jacobian determinant associatedwith phase space flows is unity whenever non-canonical coordinates lead to avanishing compressibility, so its use for defining a measure may not always be correct.To better illustrate this point, we derive a mathematical condition for definingnon-Hamiltonian phase space flows with zero compressibility. The Jacobian determinantassociated with the time evolution in phase space is very useful for analysing timetranslation invariance. The proper definition of a phase space measure is particularlyimportant when defining the entropy functional in the canonical, non-canonical, andnon-Hamiltonian cases. We show how the use of relative entropies can circumvent somesubtle problems that are encountered when dealing with continuous probabilitydistributions and phase space measures. Finally, a maximum (relative) entropyprinciple is formulated for non-canonical and non-Hamiltonian phase space flows.

A device for automated phase space measurement of ion beams

Equipment for automated phase–space measurements was developed at the VERA Lab. The measurement of the beam’s intensity distribution, as well as its relative position with respect to the reference orbit is performed at two locations along the beam line. The device basically consists of moveable slits and a beam profile monitor, which are both coordinated and controlled by an embedded controller. The operating system of the controller is based on Linux with real-time extension. It controls the movement of the slits and records the data synchronized to the movement of the beam profile monitor. The data is sent via TCP/IP to the data acquisition system of VERA where visualization takes place. The duration of one phase space measurement is less than 10s, which allows for using the device during routine beam tuning.

Experimental characterization of the transverse phase space of a 60-MeV electron beam through a compressor chicane

Space charge and coherent synchrotron radiation may deteriorate electron beam quality when the beam passes through a magnetic bunch compressor. This paper presents the transverse phase-space tomographic measurements for a compressed beam at 60 MeV, around which energy the first stage of magnetic bunch compression takes place in most advanced linacs. Transverse phase-space bifurcation of a compressed beam is observed at that energy, but the degree of the space charge-induced bifurcation is appreciably lower than the one observed at 12 MeV.

Observation of coherent nonlinear interactions in the ion velocity distribution function

Using laser induced fluorescence (LIF) and higher order spectral analysis, we present the first measurements of phase space resolved coherent nonlinear interactions among the components of low frequency density fluctuations, (ω⩽ωci), in a linearly magnetized device. The bicoherence calculations employing the two point correlation technique suggest that there are two different coherent nonlinear wave-wave interactions in the measured spectrum. The first one, having a short correlation length and existing for slow moving ions, for which ∣υi‖∣⩽∣υith∣, is an interaction between fluctuations below the electron drift frequency, ω*. The second one is the strongest for fast moving ions, for which ∣υi‖∣⩾∣υith∣, and is a mode coupling between the azimuthal drift wave modes, m=1 and m=2. Combining these bispectral results with earlier linear analysis based on the power spectra of the fluctuations, we suggest that the nonlinear coupling observed between the spectral components below ω* for the case of slow moving ions is associated with the anomalous kinetic component. For slow moving ions, as we increase the neutral collision frequencies, the nonlinear interaction observed for spectral components below ω* decreases and the harmonic mode coupling for ω* takes over.

Comparison of particle-in-cell simulation with experiment for the transport system of the superconducting electron cyclotron resonance ion source VENUS

The three-dimensional, particle-in-cell code WARP has been enhanced to allow end-to-end beam dynamics simulations of the VENUS beam transport system from the extraction region, through a mass-analyzing magnet, and up to a two-axis emittance scanner. This article presents the first results of comparisons between the simulation and experimental data. A helium beam (He+ and He2+) is chosen as an initial comparison beam due to its simple mass spectrum. Although a number of simplifications are made for the initial extracted beam, aberration characteristics appear in simulations that are also present in experimental phase-space current-density measurements. Further, measurements of phase-space tilt indicate that simulations must have little or no space-charge neutralization along the transport system to best agree with experiment. In addition, recent measurements of triangular beam structure immediately after the source are presented. This beam structure is related to the source magnetic confinement fields and will need to be taken into account as the initial beam approximations are lifted.

Enhancements in epilepsy forewarning via phase-space dissimilarity.

The authors extend the recent application of phase-space dissimilarity measures for scalp EEG data in two directions. First, a forewarning window of up to 8 hours was used, thereby providing more forewarning time of the seizure event. This window was limited to a maximum of 1 hour in their previous work. Second, they combined information from two channels via a multichannel phase-space to improve the quality and confidence limits of the forewarning. Combining these two enhancements, they obtained two-channel results that were superior to the single-channel ones.

Phase space flows for non-Hamiltonian systems with constraints

In this paper, non-Hamiltonian systems with holonomic constraints are treated by a generalization of Dirac's formalism. Non-Hamiltonian phase space flows can be described by generalized antisymmetric brackets or by general Liouville operators which cannot be derived from brackets. Both situations are treated. In the first case, a Nosé-Dirac bracket is introduced as an example. In the second one, Dirac's recipe for projecting out constrained variables from time translation operators is generalized and then applied to non-Hamiltonian linear response. Dirac's formalism avoids spurious terms in the response function of constrained systems. However, corrections coming from phase space measure must be considered for general perturbations.

Phase-space overlap measures. II. Design and implementation of staging methods for free-energy calculations

We consider staged free-energy calculation methods in the context of phase-space overlap relations, and argue that the selection of work-based methods should be guided by consideration of the phase-space overlap of the systems of interest. Stages should always be constructed such that work is performed only into a system that has a phase-space subset relation with the starting system. Thus multiple stages are required if the systems of interest are not such that one forms a phase-space subset with the other. Three two-stage methods are possible, termed umbrella sampling, overlap sampling, and funnel sampling. The last is appropriate for cases in which the subset relation holds, but only in the extreme, meaning that one system's important phase space constitutes a very small portion of the others. Umbrella sampling is most suitable for nonoverlap systems, and overlap sampling is appropriate for systems exhibiting partial phase-space overlap. We review recently introduced metrics that characterize phase-space overlap, showing that the performance of the single- and two-stage methods is consistent with the phase-space picture. We also demonstrate that a recently introduced bias-detection measure is effective in identifying inaccuracy in single- and multistage calculations. The examples used are the chemical-potential calculation for a Lennard-Jones liquid at moderate and at high densities, the same for model water at ambient conditions, and a process of charging a neutral ion in water.

Phase-space overlap measures. I. Fail-safe bias detection in free energies calculated by molecular simulation

We consider ways to quantify the overlap of the parts of phase space important to two systems, labeled A and B. Of interest is how much of the A-important phase space lies in that important to B, and how much of B lies in A. Two measures are proposed. The first considers four total-energy distributions, formed from all combinations made by tabulating either the A-system or the B-system energy when sampling either the A or B system. Measures for A in B and B in A are given by two overlap integrals defined on pairs of these distributions. The second measure is based on information theory, and defines two relative entropies which are conveniently expressed in terms of the dissipated work for free-energy perturbation (FEP) calculations in the A-->B and B-->A directions, respectively. Phase-space overlap is an important consideration in the performance of free-energy calculations. To demonstrate this connection, we examine bias in FEP calculations applied to a system of independent particles in a harmonic potential. Systems are selected to represent a range of overlap situations, including extreme subset, subset, partial overlap, and nonoverlap. The magnitude and symmetry of the bias (A-->B vs B-->A) are shown to correlate well with the overlap, and consequently with the overlap measures. The relative entropies are used to scale the amount of sampling to obtain a universal bias curve. This result leads to develop a simple heuristic that can be applied to determine whether a work-based free-energy measurement is free of bias. The heuristic is based in part on the measured free energy, but we argue that it is fail-safe inasmuch as any bias in the measurement will not promote a false indication of accuracy.

Ensemble theory for force networks in hyperstatic granular matter

An ensemble approach for force networks in static granular packings is developed. The framework is based on the separation of packing and force scales, together with an a priori flat measure in the force phase space under the constraints that the contact forces are repulsive and balance on every particle. In this paper we will give a general formulation of this force network ensemble, and derive the general expression for the force distribution P(f). For small regular packings these probability densities are obtained in closed form, while for larger packings we present a systematic numerical analysis. Since technically the problem can be written as a noninvertible matrix problem (where the matrix is determined by the contact geometry), we study what happens if we perturb the packing matrix or replace it by a random matrix. The resulting P(f) 's differ significantly from those of normal packings, which touches upon the deep question of how network statistics is related to the underlying network structure. Overall, the ensemble formulation opens up a different perspective on force networks that is analytically accessible, and which may find applications beyond granular matter.

Experimental characterization of an ultrafast Thomson scattering x-ray source with three-dimensional time and frequency-domain analysis

We present a detailed comparison of the measured characteristics of Thomson backscattered x rays produced at the Picosecond Laser-Electron Interaction for the Dynamic Evaluation of Structures facility at Lawrence Livermore National Laboratory to predicted results from a newly developed, fully three-dimensional time and frequency-domain code. Based on the relativistic differential cross section, this code has the capability to calculate time and space dependent spectra of the x-ray photons produced from linear Thomson scattering for both bandwidth-limited and chirped incident laser pulses. Spectral broadening of the scattered x-ray pulse resulting from the incident laser bandwidth, perpendicular wave vector components in the laser focus, and the transverse and longitudinal phase spaces of the electron beam are included. Electron beam energy, energy spread, and transverse phase space measurements of the electron beam at the interaction point are presented, and the corresponding predicted x-ray characteristics are determined. In addition, time-integrated measurements of the x rays produced from the interaction are presented and shown to agree well with the simulations.

Machine failure forewarning via phase-space dissimilarity measures.

We present a model-independent, data-driven approach to quantify dynamical changes in nonlinear, possibly chaotic, processes with application to machine failure forewarning. From time-windowed data sets, we use time-delay phase-space reconstruction to obtain a discrete form of the invariant distribution function on the attractor. Condition change in the system's dynamic is quantified by dissimilarity measures of the difference between the test case and baseline distribution functions. We analyze time-serial mechanical (vibration) power data from several large motor-driven systems with accelerated failures and seeded faults. The phase-space dissimilarity measures show a higher consistency and discriminating power than traditional statistical and nonlinear measures, which warrants their use for timely forewarning of equipment failure.