Full Distribution Function Research Articles

Informed constrained spherical deconvolution (iCSD).

Diffusion-weighted (DW) magnetic resonance imaging (MRI) is a noninvasive imaging method, which can be used to investigate neural tracts in the white matter (WM) of the brain. However, the voxel sizes used in DW-MRI are relatively large, making DW-MRI prone to significant partial volume effects (PVE). These PVEs can be caused both by complex (e.g. crossing) WM fiber configurations and non-WM tissue, such as gray matter (GM) and cerebrospinal fluid. High angular resolution diffusion imaging methods have been developed to correctly characterize complex WM fiber configurations, but significant non-WM PVEs are also present in a large proportion of WM voxels. In constrained spherical deconvolution (CSD), the full fiber orientation distribution function (fODF) is deconvolved from clinically feasible DW data using a response function (RF) representing the signal of a single coherently oriented population of fibers. Non-WM PVEs cause a loss of precision in the detected fiber orientations and an emergence of false peaks in CSD, more prominently in voxels with GM PVEs. We propose a method, informed CSD (iCSD), to improve the estimation of fODFs under non-WM PVEs by modifying the RF to account for non-WM PVEs locally. In practice, the RF is modified based on tissue fractions estimated from high-resolution anatomical data. Results from simulation and in-vivo bootstrapping experiments demonstrate a significant improvement in the precision of the identified fiber orientations and in the number of false peaks detected under GM PVEs. Probabilistic whole brain tractography shows fiber density is increased in the major WM tracts and decreased in subcortical GM regions. The iCSD method significantly improves the fiber orientation estimation at the WM-GM interface, which is especially important in connectomics, where the connectivity between GM regions is analyzed.

DYNAMICAL MODELING OF TIDAL STREAMS

I present a new framework for modeling the dynamics of tidal streams. The framework consists of simple models for the initial action-angle distribution of tidal debris, which can be straightforwardly evolved forward in time. Taking advantage of the essentially one-dimensional nature of tidal streams, the transformation to position–velocity coordinates can be linearized and interpolated near a small number of points along the stream, thus allowing for efficient computations of a stream's properties in observable quantities. I illustrate how to calculate the stream's average location (its "track") in different coordinate systems, how to quickly estimate the dispersion around its track, and how to draw mock stream data. As a generative model, this framework allows one to compute the full probability distribution function and marginalize over or condition it on certain phase-space dimensions as well as convolve it with observational uncertainties. This will be instrumental in proper data analysis of stream data. In addition to providing a computationally efficient practical tool for modeling the dynamics of tidal streams, the action-angle nature of the framework helps elucidate how the observed width of the stream relates to the velocity dispersion or mass of the progenitor, and how the progenitors of "orphan" streams could be located. The practical usefulness of the proposed framework crucially depends on the ability to calculate action-angle variables for any orbit in any gravitational potential. A novel method for calculating actions, frequencies, and angles in any static potential using a single orbit integration is described in the Appendix.

Schrödinger method asN-body double and UV completion of dust

We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time-evolution of density and peculiar velocity demands solving the full Vlasov hierarchy for the moments of the phase space distribution function. In the presence of long-range interaction no consistent truncation of the hierarchy is known apart from the pressure- less fluid (dust) model which is incapable of describing virialization due to the occurrence of shell-crossing singularities and the inability to generate vorticity and higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function without pathologies in a controlled way and therefore can serve as theoretical N-body double and as a replacement for the dust model. We argue that the coarse-grained Wigner probability distribution obtained from a wave function fulfilling the Schr\"odinger-Poisson equation (SPE) is the sought-after function. We show that its evolution equation approximates the Vlasov equation and therefore also the dust equations before shell-crossing, but cures the shell-crossing singularities and is able to describe regions of multi-streaming and virialization. This feature was already employed in simulations by Widrow & Kaiser (1993). The coarse-grained Wigner ansatz allows to calculate all higher moments from density and velocity analytically, thereby incorporating nonzero higher cumulants in a self-consistent manner. On this basis we are able to show that the Schr\"odinger method (ScM) automatically closes the corresponding hierarchy such that it suffices to solve the SPE in order to directly determine density and velocity and all higher cumulants.

Towards a better understanding of galaxy clusters

AbstractObservational data on clusters of galaxies holds relevant information that can be used to determine the relative plausibility of different models for the large-scale evolution of the Universe, or estimate the joint posterior probability distribution function of the parameters that pertain to each model. Within the next few years, several surveys of the sky will yield large galaxy cluster catalogues. In order to make use of the vast amount of information they will contain, their selection functions will have to be properly understood. We argue this, as well as the estimation of the full joint posterior probability distribution function of the most relevant cluster properties, can be best achieved in the framework of bayesian statistics.

A LARGE CATALOG OF ACCURATE DISTANCES TO MOLECULAR CLOUDS FROM PS1 PHOTOMETRY

Distance measurements to molecular clouds are important, but are often made separately for each cloud of interest, employing very different different data and techniques. We present a large, homogeneous catalog of distances to molecular clouds, most of which are of unprecedented accuracy. We determine distances using optical photometry of stars along lines of sight toward these clouds, obtained from PanSTARRS-1. We simultaneously infer the reddenings and distances to these stars, tracking the full probability distribution function using a technique presented in Green et al. (2014). We fit these star-by-star measurements using a simple dust screen model to find the distance to each cloud. We thus estimate the distances to almost all of the clouds in the Magnani et al. (1985) catalog, as well as many other well-studied clouds, including Orion, Perseus, Taurus, Cepheus, Polaris, California, and Monoceros R2, avoiding only the inner Galaxy. Typical statistical uncertainties in the distances are 5%, though the systematic uncertainty stemming from the quality of our stellar models is about 10%. The resulting catalog is the largest catalog of accurate, directly-measured distances to molecular clouds. Our distance estimates are generally consistent with available distance estimates from the literature, though in some cases the literature estimates are off by a factor of more than two.

Influence of finite-sized detection solid angle on bidirectional reflectance distribution function measurements

This paper deals with limitations and often overlooked sources of error introduced in compact double-beam goniophotometers. It is shown that relative errors in measured radiance factor, comparable to the total measurement uncertainty, can be introduced if recommended corrections are not carried out. Two different error sources are investigated, both related to the size of the detection solid angle. The first is a geometrical error that occurs when the size of the illuminated area and detector aperture are comparable to the distance between them. The second is a convolution error due to variations in radiant flux over the detector aperture, which is quantified by simulating the full 3D bidirectional reflectance distribution function (BRDF) of a set of samples with different degrees of anisotropic reflectance. The evaluation is performed for a compact double-beam goniophotometer using different detection solid angles, and it is shown that both error sources introduce relative errors of 1%-3%, depending on viewing angle and optical properties of the sample. Commercially available compact goniophotometers, capable of absolute measurements, are becoming more and more common, and the findings in this paper are therefore important for anyone using or planning to use this type of instrument.

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. IV. DISTRIBUTION OF ANGULAR MOMENTUM

It has been shown in previous work that DARKexp, which is a theoretically derived, maximum entropy, one shape parameter model for isotropic collisionless systems, provides very good fits to simulated and observed dark-matter halos. Specifically, it fits the energy distribution, N(E), and the density profiles, including the central cusp. Here, we extend DARKexp N(E) to include the distribution in angular momentum, L^2, for spherically symmetric systems. First, we argue, based on theoretical, semi-analytical, and simulation results, that while dark-matter halos are relaxed in energy, they are not nearly as relaxed in angular momentum, which precludes using maximum entropy to uniquely derive N(E,L^2). Instead, we require that when integrating N(E,L^2) over squared angular momenta one retrieves the DARKexp N(E). Starting with a general expression for N(E,L^2) we show how the distribution of particles in L^2 is related to the shape of the velocity distribution function, VDF, and velocity anisotropy profile, \beta(r). We then demonstrate that astrophysically realistic halos, as judged by the VDF shape and \beta(r), must have linear or convex distributions in L^2, for each separate energy bin. The distribution in energy of the most bound particles must be nearly flat, and become more tilted in favor of radial orbits for less bound particles. These results are consistent with numerical simulations and represent an important step towards deriving the full distribution function for spherically symmetric dark-matter halos.

Universal time fluctuations in near-critical out-of-equilibrium quantum dynamics.

Out-of-equilibrium quantum systems display complex temporal patterns. Such time fluctuations are generically exponentially small in the system volume and therefore can be safely ignored in most of the cases. However, if one consider small quench experiments, time fluctuations can be greatly enhanced. We show that time fluctuations may become stronger than other forms of equilibrium quantum fluctuations if the quench is performed close to a critical point. For sufficiently relevant operators the full distribution function of dynamically evolving observable expectation values becomes a universal function uniquely characterized by the critical exponents and the boundary conditions. At regular points of the phase diagram and for nonsufficiently relevant operators the distribution becomes Gaussian. Our predictions are confirmed by an explicit calculation on the quantum Ising model.

Comparison of sampling strategies and sparsifying transforms to improve compressed sensing diffusion spectrum imaging

Diffusion Spectrum Imaging enables to reconstruct the ensemble average propagator (EAP) at the expense of having to acquire a large number of measurements. Compressive sensing offers an efficient way to decrease the required number of measurements. The purpose of this work is to perform a thorough experimental comparison of three sampling strategies and six sparsifying transforms to show their impact when applied to accelerate compressive sensing-diffusion spectrum imaging. We propose a novel sampling scheme that assures uniform angular and random radial q-space samples. We also compare and implement six discrete sparse representations of the EAP and thoroughly evaluate them on synthetic and real data using metrics from the full EAP, kurtosis, and orientation distribution function. The discrete wavelet transform with Cohen-Daubechies-Feauveau 9/7 wavelets and uniform angular sampling in combination with random radial sampling showed to be better than other tested techniques to accurately reconstruct the EAP and its features. It is important to jointly optimize the sampling scheme and the sparsifying transform to obtain accelerated compressive sensing-diffusion spectrum imaging. Experiments on synthetic and real human brain data show that one can robustly recover both radial and angular EAP features while undersampling the acquisition to 64 measurements (undersampling factor of 4).

Injection of thermal and suprathermal seed particles into coronal shocks of varying obliquity

Context: Diffusive shock acceleration in the solar corona can accelerate solar energetic particles to very high energies. Acceleration efficiency is increased by entrapment through self-generated waves, which is highly dependent on the amount of accelerated particles. This, in turn, is determined by the efficiency of particle injection into the acceleration process. Aims: We present an analysis of the injection efficiency at coronal shocks of varying obliquity. We assessed injection through reflection and downstream scattering, including the effect of a cross-shock potential. Both quasi-thermal and suprathermal seed populations were analysed. We present results on the effect of cross-field diffusion downstream of the shock on the injection efficiency. Methods: Using analytical methods, we present applicable injection speed thresholds that were compared with both semi-analytical flux integration and Monte Carlo simulations, which do not resort to binary thresholds. Shock-normal angle and shock-normal velocity were varied to assess the injection efficiency with respect to these parameters. Results: We present evidence of a significant bias of thermal seed particle injection at small shock-normal angles. We show that downstream isotropisation methods affect the theta-dependence of this result. We show a non-negligible effect caused by the cross-shock potential, and that the effect of downstream cross-field diffusion is highly dependent on boundary definitions. Conclusions: Our results show that for Monte Carlo simulations of coronal shock acceleration a full distribution function assessment with downstream isotropisation through scatterings is necessary to realistically model particle injection. Based on our results, seed particle injection at quasi-parallel coronal shocks can result in significant acceleration efficiency, especially when combined with varying field-line geometry.

Effect of doubling the ensemble size on the performance of ensemble prediction in the warm season using MOGREPS implemented at the KMA

Using the Met Office Global and Regional Ensemble Prediction System (MOGREPS) implemented at the Korea Meteorological Administration (KMA), the effect of doubling the ensemble size on the performance of ensemble prediction in the warm season was evaluated. Because a finite ensemble size causes sampling error in the full forecast probability distribution function (PDF), ensemble size is closely related to the efficiency of the ensemble prediction system. Prediction capability according to doubling the ensemble size was evaluated by increasing the number of ensembles from 24 to 48 in MOGREPS implemented at the KMA. The initial analysis perturbations generated by the Ensemble Transform Kalman Filter (ETKF) were integrated for 10 days from 22 May to 23 June 2009. Several statistical verification scores were used to measure the accuracy, reliability, and resolution of ensemble probabilistic forecasts for 24 and 48 ensemble member forecasts. Even though the results were not significant, the accuracy of ensemble prediction improved slightly as ensemble size increased, especially for longer forecast times in the Northern Hemisphere. While increasing the number of ensemble members resulted in a slight improvement in resolution as forecast time increased, inconsistent results were obtained for the scores assessing the reliability of ensemble prediction. The overall performance of ensemble prediction in terms of accuracy, resolution, and reliability increased slightly with ensemble size, especially for longer forecast times.

Prethermalization revealed by the relaxation dynamics of full distribution functions

We detail the experimental observation of the non-equilibrium many-body phenomenon prethermalization. We study the dynamics of a rapidly and coherently split one-dimensional Bose gas. An analysis based on the use of full quantum mechanical probability distributions of matter wave interference contrast reveals that the system evolves toward a quasi-steady state. This state, which can be characterized by an effective temperature, is not the final thermal equilibrium state. We compare the evolution of the system to an integrable Tomonaga–Luttinger liquid model, and show that the system dephases to a prethermalized state rather than undergoing thermalization toward a final thermal equilibrium state.

Characterization of suprathermal electron population in He dc glow discharges by optical emission and probe diagnostics

Evidence of the strong contribution of electrons with energies up to the cathode fall potential was found in low-pressure He dc glow discharge plasmas (3–8 mTorr), in agreement with expectations from the estimates of the mean free path of electron–neutral inelastic collisions. A simple gridded probe, of the retarding field analyzer type, was applied to the characterization of the full electron energy distribution function (EEDF) in He plasmas with significant contribution from suprathermal electrons (up to 3.5%). The inferred EEDFs were cross-checked with results from the He line ratio diagnostic and good agreement was found at several values of pressure and plasma current.

Uncertainty in modeled upper ocean heat content change

This paper examines the uncertainty in the change in the heat content in the ocean component of a general circulation model. We describe the design and implementation of our statistical methodology. Using an ensemble of model runs and an emulator, we produce an estimate of the full probability distribution function (PDF) for the change in upper ocean heat in an Atmosphere/Ocean General Circulation Model, the Community Climate System Model v. 3, across a multi-dimensional input space. We show how the emulator of the GCM’s heat content change and hence, the PDF, can be validated and how implausible outcomes from the emulator can be identified when compared to observational estimates of the metric. In addition, the paper describes how the emulator outcomes and related uncertainty information might inform estimates of the same metric from a multi-model Coupled Model Intercomparison Project phase 3 ensemble. We illustrate how to (1) construct an ensemble based on experiment design methods, (2) construct and evaluate an emulator for a particular metric of a complex model, (3) validate the emulator using observational estimates and explore the input space with respect to implausible outcomes and (4) contribute to the understanding of uncertainties within a multi-model ensemble. Finally, we estimate the most likely value for heat content change and its uncertainty for the model, with respect to both observations and the uncertainty in the value for the input parameters.

Multimode Dynamics and Emergence of a Characteristic Length Scale in a One-Dimensional Quantum System

We study the nonequilibrium dynamics of a coherently split one-dimensional Bose gas by measuring the full probability distribution functions of matter-wave interference. Observing the system on different length scales allows us to probe the dynamics of excitations on different energy scales, revealing two distinct length-scale-dependent regimes of relaxation. We measure the crossover length scale separating these two regimes and identify it with the prethermalized phase-correlation length of the system. Our approach enables a direct observation of the multimode dynamics characterizing one-dimensional quantum systems.

Many-body localization and thermalization in the full probability distribution function of observables

We investigate the relation between thermalization following a quantum quench and many-body localization in quasi-particle space in terms of the long-time full distribution function of physical observables. In particular, expanding on our recent work (Canovi et al 2011 Phys. Rev. B 83 094431), we focus on the long-time behavior of an integrable XXZ chain subject to an integrability-breaking perturbation. After a characterization of the breaking of integrability and the associated localization/delocalization transition using the level spacing statistics and the properties of the eigenstates, we study the effect of integrability breaking on the asymptotic state after a quantum quench of the anisotropy parameter, looking at the behavior of the full probability distribution of the transverse and longitudinal magnetization of a subsystem. We compare the resulting distributions with those obtained in equilibrium at an effective temperature set by the initial energy. We find that, while the long-time distribution functions appear to always agree qualitatively with the equilibrium ones, quantitative agreement is obtained only when integrability is fully broken and the relevant eigenstates are diffusive in quasi-particle space.

Single-particle dissipation in a time-dependent Hartree-Fock approach studied from a phase-space perspective

We study dissipation and relaxation processes within the time-dependent Hartree-Fock approach using the Wigner distribution function. On the technical side we present a geometrically unrestricted framework which allows us to calculate the full six-dimensional Wigner distribution function. With the removal of geometrical constraints, we are now able to extend our previous phase-space analysis of heavy-ion collisions in the reaction plane to unrestricted mean-field simulations of nuclear matter on a three-dimensional Cartesian lattice. From the physical point of view we provide a quantitative analysis on the stopping power in TDHF. This is linked to the effect of transparency. For the medium-heavy ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}$ system we examine the impact of different parametrizations of the Skyrme force, energy-dependence, and the significance of extra time-odd terms in the Skyrme functional.

Modeling Transmission Time of Silicon Nanophotonic Waveguides

We propose a simple computational model for silicon nanophotonic waveguides which simulates the full spectral distribution function of propagation time (group delay), including the role of interfaces and roughness. Experimental validation is shown using complex field measurements of light transmission on fabricated waveguides.

Assessing the impact of uncertainty on flood risk estimates with reliability analysis using 1-D and 2-D hydraulic models

Abstract. This paper addresses the use of reliability techniques such as Rosenblueth's Point-Estimate Method (PEM) as a practical alternative to more precise Monte Carlo approaches to get estimates of the mean and variance of uncertain flood parameters water depth and velocity. These parameters define the flood severity, which is a concept used for decision-making in the context of flood risk assessment. The method proposed is particularly useful when the degree of complexity of the hydraulic models makes Monte Carlo inapplicable in terms of computing time, but when a measure of the variability of these parameters is still needed. The capacity of PEM, which is a special case of numerical quadrature based on orthogonal polynomials, to evaluate the first two moments of performance functions such as the water depth and velocity is demonstrated in the case of a single river reach using a 1-D HEC-RAS model. It is shown that in some cases, using a simple variable transformation, statistical distributions of both water depth and velocity approximate the lognormal. As this distribution is fully defined by its mean and variance, PEM can be used to define the full probability distribution function of these flood parameters and so allowing for probability estimations of flood severity. Then, an application of the method to the same river reach using a 2-D Shallow Water Equations (SWE) model is performed. Flood maps of mean and standard deviation of water depth and velocity are obtained, and uncertainty in the extension of flooded areas with different severity levels is assessed. It is recognized, though, that whenever application of Monte Carlo method is practically feasible, it is a preferred approach.

Droplet size distribution in homogeneous isotropic turbulence

We study the physics of droplet breakup in a statistically stationary homogeneous and isotropic turbulent flow by means of high resolution numerical investigations based on the multicomponent lattice Boltzmann method. We verified the validity of the criterion proposed by Hinze [AIChE J. 1, 289 (1955)] for droplet breakup and we measured the full probability distribution function of droplets radii at different Reynolds numbers and for different volume fractions. By means of a Lagrangian tracking we could follow individual droplets along their trajectories, define a local Weber number based on the velocity gradients, and study its cross-correlation with droplet deformation.