Angular Probability Density Function Research Articles

Phase transitions on a class of generalized Vicsek-like models of collective motion.

Systems composed of interacting self-propelled particles (SPPs) display different forms of order-disorder phase transitions relevant to collective motion. In this paper, we propose a generalization of the Vicsek model characterized by an angular noise term following an arbitrary probability density function, which might depend on the state of the system and thus have a multiplicative character. We show that the well established vectorial Vicsek model can be expressed in this general formalism by deriving the corresponding angular probability density function, as well as we propose two new multiplicative models consisting of bivariate Gaussian and wrapped Gaussian distributions. With the proposed formalism, the mean-field system can be solved using the mean resultant length of the angular stochastic term. Accordingly, when the SPPs interact globally, the character of the phase transition depends on the choice of the noise distribution, being first order with a hybrid scaling for the vectorial and wrapped Gaussian distributions, and second order for the bivariate Gaussian distribution. Numerical simulations reveal that this scenario also holds when the interactions among SPPs are given by a static complex network. On the other hand, using spatial short-range interactions displays, in all the considered instances, a discontinuous transition with a coexistence region, consistent with the original formulation of the Vicsek model.

One-dimensional displacement of active matter on curved substrates

We analytically find the diffusion of overdamped active Brownian particles (ABPs) constrained to move along curved one-dimensional channels. The autonomous motion of these particles is achieved by a projection of their internal propulsion force along the channels' long section. In particular, the diffusion of ABPs moving on one-dimensional channels with a form of a circle, an ellipse, and a limacon of second order is analysed. To characterise the effect of substrate's geometry and self-propulsion on their diffusion, analytical expressions for the ABPs short- and long-time variances, as well as their steady angular probability density functions are offered. Curvature effects are found to reduce the time an ABP reaches its steady state. Our theoretical results are validated using Brownian dynamics simulations. This model may be relevant for experiments dealing with catalytic driven systems, bacteria, and tumour cell dispersion in one-dimensional channels.

Improving AoA Localization Accuracy in Wireless Acoustic Sensor Networks with Angular Probability Density Functions.

Advances in energy efficient electronic components create new opportunities for wireless acoustic sensor networks. Such sensors can be deployed to localize unwanted and unexpected sound events in surveillance applications, home assisted living, etc. This research focused on a wireless acoustic sensor network with low-profile low-power linear MEMS microphone arrays, enabling the retrieval of angular information of sound events. The angular information was wirelessly transmitted to a central server, which estimated the location of the sound event. Common angle-of-arrival localization approaches use triangulation, however this article presents a way of using angular probability density functions combined with a matching algorithm to localize sound events. First, two computationally efficient delay-based angle-of-arrival calculation methods were investigated. The matching algorithm is described and compared to a common triangulation approach. The two localization algorithms were experimentally evaluated in a m by m room, localizing white noise and vocal sounds. The results demonstrate the superior accuracy of the proposed matching algorithm over a common triangulation approach. When localizing a white noise source, an accuracy improvement of up to 114% was achieved.

A spectral decomposition approach for the mechanical statistical characterization of distributed fiber-reinforced tissues

We discuss a spectral decomposition formulation for the mechanical statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with distributed fibers. We consider a generalized angular probability density function (PDF) of the reinforcement built upon the local eigenvalue and eigenvector system of the Cauchy–Green deformation tensor. We focus our analysis on material models dependent on the fourth pseudo-invariant of the deformation, I4, and on exponential forms of the fiber strain energy function. Within such a spectral reference system, we derive the closed-form expression of the PDF for I4 by generalizing the multi-value random variable transformation procedure recently developed in Gizzi et al. 2016. Our formulation bypasses the cumbersome extension–contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors. Accordingly, we identify analytically the support of the fibers in pure extension for significant loading conditions. We can readily compute any statistics of the fourth pseudo-invariant and we can derive the direct definition of the average second Piola–Kirchhoff stress tensor according to the second order approximation.

Complex Dynamics of Photo-Switchable Guest Molecules in All-Optical Poling Close to the Glass Transition: Kinetic Monte Carlo Modeling.

We study theoretically the kinetics of noninteracting photoswitchable guest molecules (model azo-dye) dispersed at low concentration in host (model polymer matrix) in the all-optical poling process close to the glass transition temperature Tg. We modify kinetic Monte Carlo model used in our previous studies of nonlinear optical processes in host-guest systems. The polymer matrix is simulated using the bond-fluctuation model. The kinetics of multiple trans-cis-trans cycles is formulated in terms of transition probabilities which depend on local free volume in the matrix and its dynamics. Close to Tg, the buildup of polar order, monitored in terms of angular probability density functions, follows a power-law in time while the evolution of the nonlinear susceptibilities related to second harmonic generation effect follows the stretched-exponential law. This complex dynamics of guest molecules implies the presence of dynamic heterogeneities of the matrix in space and time which spread the complexity from the matrix to the otherwise simple dynamics of noninteracting guest molecules. A qualitative physical picture of mosaic-like states-intertwined areas of free- and hindered angular motion of guest molecules-is proposed and the role of related short and longer scales in space for the promotion of complex dynamics of guest molecules is discussed. A brief comparison of the theory to available experimental data is given.

Brownian self-driven particles on the surface of a sphere.

We present the dynamics of overdamped Brownian self-propelled particles moving on the surface of a sphere. The effect of self-propulsion on the diffusion of these particles is elucidated by determining their angular (azimuthal and polar) mean-square displacement. Short- and long-times analytical expressions for their angular mean-square displacement are offered. Finally, the particles' steady marginal angular probability density functions are also elucidated.

Nanoscale Spin-Transfer Torque MRAM Etching Using Various Gases

The magnetic tunnel junction (MTJ)-related materials were etched using He, Ar, and CO/NH3 in an inductively coupled plasma etching system. When the MTJ materials were etched with He, the etch selectivities and the etch profiles of the etched MTJ patterns were significantly improved. The sputter etch characteristics were also investigated using the Stopping and Range of Ions in Matter (SRIM) simulation program. The results showed higher sputter etch selectivity of CoFeB over W using He than those obtained using Ar. Also He exhibited higher energy and narrower angular probability density function of the sputtered atoms. Also, by using He, similar to Ar, magnetic properties of the MTJ were preserved compare to CO/NH3 due to no oxidation of magnetic materials.

Underlying Low-Order Dynamics of Nonlinear Interaction among Northern Hemisphere Teleconnection Patterns and Its Association with the AO/NAM

Abstract Studies on the nonlinear natures of the spatiotemporal structure of the Arctic Oscillation/Northern Hemisphere annular mode (AO/NAM) in the context of nonlinear interaction among the North Atlantic Oscillation (NAO), the Pacific–North American pattern (PNA), and the stratospheric polar vortex (SPV) are performed. The non-Gaussianity of the multivariate probability density function (PDF) in the phase space spanned by their indices is examined first. Five local maxima potentially related to circulation regimes are identified from the so-called angular PDF. One opposite pair of these regimes is found to correspond to the positive and negative phases of the AO/NAM. Since the authors are not sure that, due to uncertainty as suggested by statistical tests, some of the above regimes are non-Gaussian, the nonlinearity of phase-space tendency is employed as an assistant measure to identify them as nonlinear modes. It seems phase-space tendency traditionally estimated from time difference failed to be effective because of its dependence on Δt. To overcome this drawback a low-order stochastic dynamical model is established empirically from the indices. The investigation on the basic deterministic dynamics of this model suggests that the existence of regimes, such as those associated with the AO/NAM, can primarily be explained by its nonlinear deterministic part. However, two problems still remain unsolved: 1) one of the local maxima was almost not identified and 2) life cycles of the basic deterministic dynamics are too long to be related to the low-frequency variability. By introducing a multilevel approach of modeling, further insight into the residual noise of the above stochastic model can address these two issues quite well.

Unsteady motion of a spherical bubble in a complex fluid: Mathematical modelling and simulation

The nonlinear response of an oscillatory bubble in a complex fluid is studied. The bubble is immersed in a Newtonian liquid, which may have a dilute volume fraction of anisotropic additives such as fibers or few ppm of macromolecules. The constitutive equation for the fluid is based on a Maxwell model with an extensional viscosity for the viscous contribution. The model is considered new in the study of bubble dynamics in complex fluids. The numerical computation solves a system of three first order ordinary differential equations, including the one associated with the solution of the convolution integral, using a fifth order Runge–Kutta scheme with appropriated time steps. Asymptotic solutions of governing equation are developed for small values of the pressure forcing amplitude and for small values of the elastic parameter. A study of the bubble collapse radius is also presented. We compare the results predicted by our model with other model in the literature and a good agreement is observed. The calculated asymptotic solutions are also used to test the results of the numerical simulations. In addition, the orientation of the additives is considered. The angular probability density function is assumed to be a normal distribution. The results show that the model based on the fully aligned additives with the radial direction overestimates the tendency of the additives to stabilize the bubble motion, since the effect of extensional viscosity occurs due to the particle resistance to the movement throughout its longitudinal direction.

A pedestrian navigation system using a map-based angular motion model for indoor and outdoor environments

By incorporating known floor-plans in sequential Bayesian positioning estimators, such as particle filters, long-term positioning accuracy can be achieved as long as the map is sufficiently accurate and the environment sufficiently constrains pedestrians’ motion. Instead of using binary decisions to eliminate particles when crossing a wall, map-based angular probability density functions (PDFs) are used in this article that are capable of weighting the possible headings of the pedestrian according to local infrastructure. In addition, we will include outdoor maps by processing satellite images of the region. We will show that the angular PDFs will help to obtain better performance in critical multi-modal navigation scenarios and in the outdoor area when including maps.

Compound-Poisson-process method for the multiple scattering of charged particles.

The theory of multiple scattering for electrons and other changed particles is presented here within a mathematically unifying compound-Poisson-process framework. The compound Poisson model of scattering phenomena allows an arbitrary differential cross section for single scattering, is not restricted to any particular range for the number of collisions experienced by the particle, and is applicable to both the planar coordinate description of scattering (small angle approximation) as well as the spherical coordinate description (exact angle description). An efficient method for computing the angular probability density function after a large number of collisions is derived and then applied in a few example computations. This method is based on the decomposition of the compound Poisson process into hard and soft scattering.

Multiple Flow Regimes in the Northern Hemisphere Winter. Part II: Sectorial Regimes and Preferred Transitions

Abstract This paper presents an observational analysis of recurrent flow patterns in the Northern Hemisphere (NH) winter, based on a 37-year series of daily 700-mb height anomalies. Large-scale anomaly patterns that appear repeatedly and persist beyond synoptic time scales are identified by searching for local maxima of probability density in a phase subspace, which is spanned by the leading empirical orthogenal functions (EOFs). By using an angular probability density function (PDF), we focus on the shape, not magnitude, of the anomaly patterns. The PDF estimate is nonparametric; that is, our algorithm makes no a priori assumption on symmetry with respect to the climatological mean as in one-point correlation and rotated EOF analyses. The local density maxima are searched by iterative bump hunting. Based on observed partial decoupling between the Pacific (PAC) and the Atlantic-Eurasian (ATL) sectors, the classification algorithm is applied separately to each of the two. Seven PAC and six ATL patterns are...

On Kernel Methods of Estimating Marginal Radial and Angular Probability Density Functions

AbstractSpatial point patterns which possess a natural origin are considered. Two ways of estimating the marginal radial and angular probability density functions associated with the stochastic process underlying such a pattern are presented. These methods are based on one and two‐dimensional kernel functions respectively. The angular density estimate can be used to detect angular trend and to test for angular uniformity within a particular sector about the origin. The two methods of estimation produce essentially the same results. That based on the one‐dimensional kernel is recommended because it is computationally simpler.

Finding the approximate angular probability density function of wave arrival by using a directional antenna

The first-order approximate angular probability density function of wave arrival can be obtained by plotting the mean-square value of signal level versus the pointing angle of a mobile radio directional antenna during an entire test run. The accuracy of this first-order approximation is dependent on the beamwidth of the antenna. As the beamwidth of the antenna becomes narrower, the approximate density function becomes more accurate. There are also higher order approximations for obtaining a more accurate angular probability density function without narrowing the directional antenna beamwidth. These higher order approximations are related to the moments of the signal strength received by the same directional antenna used in obtaining the first-order approximation. All of these are theoretical predictions. An experimental verification using a first-order approximate angular probability density function in finding theoretical level crossing rates of a mobile radio signal is given. The theoretical level crossing rates are in fairly good agreement with the experimental ones as long as the angular probability density function is not far from a uniform distribution.