Probability Distribution Of Particle Research Articles

Multidisciplinary benchmarks of a conservative spectral solver for the nonlinear Boltzmann equation

The Boltzmann equation describes the evolution of the phase-space probability distribution of classical particles under binary collisions. Approximations to it underlie the basis for several scholarly fields, including aerodynamics and plasma physics. While these approximations are appropriate in their respective domains, they can be violated in niche but diverse applications which require direct numerical solution of the original nonlinear Boltzmann equation. An expanded implementation of the Galerkin–Petrov conservative spectral algorithm is employed to study a wide variety of physical problems. Enabled by distributed precomputation, solutions of the spatially homogeneous Boltzmann equation can be achieved in seconds on modern personal hardware, while spatially-inhomogeneous problems are solvable in minutes. Several benchmarks are presented focusing on accuracy compared to both analytic theoretical predictions and other Boltzmann solvers. These benchmarks span several physical domains including weakly ionized plasma, gaseous fluids, and atomic-plasma interaction.

Dissipation-induced recurrence of non-Hermitian edge burst

In quantum mechanics, the Hermitian Hamiltonian is generally used to describe the ideal closed quantum system, but in reality, the physical system is closely related to the environment, and the open quantum system coupled to the environment can be described by the equivalent non-Hermitian Hamiltonian to a certain extent. Among them, the dissipation intensity is closely related to the dynamic properties of non-Hermitian quantum systems. Therefore, it is of great practical significance to study how dissipation affects particle loss. In this paper, the dynamic law related to dissipation intensity in a one-dimensional non-Hermitian system under open boundary condition is studied, and it is found that dissipation can induce the recurrence of edge burst. After the time-dependent evolution of the particles in the one-dimensional non-Hermitian dissipative lattice system with open boundary condition, there is an edge burst in the system, that is, there is a large probability of particle loss at the edge, and the edge burst disappears after increasing the intracell hopping. It is found that if the dissipation intensity increases or decreases, the edge burst will reappear. This kind of reappearance is different from the original edge burst, which is mainly manifested in the loss probability distribution of particles from the edge distribution to the bulk distribution, which is due to the difference in probability of particle motion direction between the two cases. Under the re-induced edge burst, the particles move leftward and rightward from their initial positions, and rebound from the left after having reached the boundary, forming a more obvious loss probability at the edge and gradually decreasing to the body area. In the original edge burst, the particles only move to the left with a greater probability, and are ‘trapped’ at the edge to completely dissipated, forming a distribution with an independent loss peak at the edge. The movement to the left is due to the non-Hermitian skin effect. The deeper reason for different movement directions is related to parity-time symmetry. Under the parameter near the parity-time symmetry breaking point, the loss probability of the particle is of unilateral distribution, and the loss probability of the particle moving to both sides is of bilateral distribution when it is far away. This is the description of the dissipation-induced edge burst recurrence phenomenon and its characteristics. In addition, this paper also studies the influence of impurity barrier on the probability distribution of particle loss in non-Hermitian dynamics. The results show that placing a small barrier on the non-dissipative <i>A</i>-site can obviously hinder the particle motion, and when the barrier increases to a certain height, its influence on the particle motion tends to be unchanged. And the barrier at the dissipative <i>B</i> lattice has little effect on the dynamics.

Full counting statistics in the many-body Hatano-Nelson model

We study non-hermitian many-body physics in the interacting Hatano-Nelson model with open boundary condition. The violation of reciprocity, resulting from an imaginary vector potential, induces the non-hermitian skin-effect and causes exponential localization for all single particle eigenfunctions in the non-interacting limit. Nevertheless, the density profile of the interacting system becomes only slightly tilted relative to the average filling. The Friedel oscillations exhibit a beating pattern due to the modification of the Fermi wavenumber. The probability distribution of particles over any finite interval is the normal distribution, whose mean scales with the imaginary vector potential and the variance is symmetric to the center of the chain. This is confirmed by several numerically exact methods even for relatively small systems. These features are expected to be generic not only for fermions, which naturally repel each other due to Pauli's exclusion principle, but for interacting bosons as well. Our findings indicate that many-body effects can significantly alter and conceal the single particle properties and the skin effect in non-hermitian systems.

Moving Target Tracking Algorithm Based on Improved Resampling Particle Filter in UWB Environment

In this paper, a moving target tracking (MTT) algorithm based on the improved resampling particle filter (IRPF) was put forward for the reduced accuracy of particle filter (PF) due to the lack of particle diversity resulting from traditional resampling methods. In this algorithm, the influences of the likelihood probability distribution of particles on the PF accuracy were firstly analyzed to stratify the adaptive regions of particles, and a particle diversity measurement index based on stratification was proposed. After that, a threshold was set for the particle diversity after resampling. If the particle diversity failed to reach the set threshold, all new particles would be subjected to a Gaussian random walk in a preset variance matrix to improve the particle diversity. Finally, the performance of related algorithms was tested in both simulation environment and actual indoor ultrawideband (UWB) nonline-of-sight (NLOS) environment. The experimental results revealed that the nonlinear target state estimation accuracy was maximally and minimally improved by 12.83% and 1.97%, respectively, in the simulation environment, and the root mean square error (RMSE) of MTT was reduced from 17.131 cm to 10.471 cm in actual UWB NLOS environment, indicating that the IRPF algorithm can enhance the target estimation accuracy and state tracking capability, manifesting better filter performance.

Study on the scattering characteristics of Ka millimeter wave by dust storms

As to study the scattering characteristics of millimeter wave by sand particles, the integrated detection on dust storms over the Taklimakan Desert was the first time to conduct with the Ka band millimeter wave radar, the Grimm 180, the Laser Particle Size Analyzer and the visibility meter. The complex permittivity of sand particle (εm), the probability distribution function of sand particles, the number of sand particles, the attenuation coefficient and the reflectivity factors were calculated by the observations of these instruments; moreover, the power spectrums of the dust storm, sunny day and precipitation were comparatively analyzed. The results showed that: the εm of 35 GHz millimeter wave by sand particles is 3.0078-j0.0987. The probability distribution of sand particles conforms to lognormal distribution, with the mean of -2.352 and the standard deviation of 0.422. Furthermore, when the visibility drops from 3000 to 0 m, the number of sand particles rises from 2 × 105 to 1 × 106 in L · min−1 and the attenuation coefficient increases from 1 to 8 dB · km−1. The peak and the average values of power spectrums in dust storm are lower than those in precipitation, which indicates that the scattering ability of sand particles is less than that of precipitation particles. During the dust storms, the reflectivity factors range between -40 and 20 dBZ. These findings provide theoretical basis for the further study of the scattering properties of sand particles in the future.

Transient anomalous diffusion in heterogeneous media with stochastic resetting

We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatial-dependent diffusion coefficient with a power-law dependence on particles’ positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles’ positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusion process and obtain an exact expression for the mean first-passage time. Our findings show that this system exhibits non-Gaussian distributions, transient anomalous diffusion (sub- or superdiffusion) and stationary states that simultaneously depend on the media heterogeneity and the resetting rate. We further demonstrate that the media heterogeneity non-trivially affect the mean first-passage time, yielding an optimal resetting rate for which this quantity displays a minimum.

Research on Simulation System of Rubber Particle Mat Based on Web Technology

The current design of rubber granular floor mats is limited to a single fixed plane sample drawing, and it consumes a lot of time, manpower and material resources in obtaining customer design requirements. In response to this situation, the use of Web development technology to realize the free combination of particles in proportion, generate simulation application scenarios, provide interface operations, and finally complete the order process. By constructing a particle probability distribution model, establishing a particle position coordinate matrix, and developing a particle mat simulation system, based on the sample quality of the sampled data, it is compared with the real artificial mechanical product. Finally, by mixing and matching 9 different color values according to the industrial production ratio, randomly combining 10 groups and comparing with the real products of the same color ratio, quantifying the color difference, contrast, and particle position offset to obtain the production simulation floor mat sample and the actual product. The finished product is quite close. The system generates simulated particle mat maps and real sample maps with a high degree of simulation, and provides an interface that allows users to directly match the particle combination program that suits their needs, saving labor cycles and improving work efficiency.

Pauli crystals in harmonic trap and on a sphere

We study configurations maximizing N-particle probability distributions of non-interacting particles obeying Fermi statistics as discussed in [1]. We consider two different types of particles' confinement: (i) trapping in a harmonic potential and (ii) constraining to the surface of a sphere. In such settings the most probable arrangements of particles correspond to characteristic geometric patterns resulting from the combined effect of confinement and the Pauli exclusion principle. We discuss the geometry of these optimal configurations as well as two different methods of extracting these shapes from a number of single shot pictures of the system. The post-selection method which does not assume a priori any information on the optimal configurations is illustrated in detail.

Thermodynamics of a Brownian particle in a nonconfining potential.

We consider the overdamped Brownian dynamics of a particle starting inside a square potential well which, upon exiting the well, experiences a flat potential where it is free to diffuse. We calculate the particle's probability distribution function (PDF) at coordinate x and time t,P(x,t), by solving the corresponding Smoluchowski equation. The solution is expressed by a multipole expansion, with each term decaying t^{1/2} faster than the previous one. At asymptotically large times, the PDF outside the well converges to the Gaussian PDF of a free Brownian particle. The average energy, which is proportional to the probability of finding the particle inside the well, diminishes as E∼1/t^{1/2}. Interestingly, we find that the free energy of the particle, F, approaches the free energy of a freely diffusing particle, F_{0}, as δF=F-F_{0}∼1/t, i.e., at a rate faster than E. We provide analytical and computational evidence that this scaling behavior of δF is a general feature of Brownian dynamics in nonconfining potential fields. Furthermore, we argue that δF represents a diminishing entropic component which is localized in the region of the potential, and which diffuses away with the spreading particle without being transferred to the heat bath.

A study on the leeway drift characteristic of a typical fishing vessel common in the Northern South China Sea

Leeway drift characteristics of drifting objects are essential to accurately forecast the drifting trajectory of lost persons, boats and other objects. Traditionally, leeway coefficients of the drifting objects are calculated using the observed wind, current and drift speed data from field experiments. Based on the combined samples of four drift experiments for a new type of 20 m length fishing vessel common in the northern South China Sea, this study determine the wind-induced drift characteristic through leeway speed, downwind component of leeway (DWL), crosswind component of leeway (CWL) coefficients, and explore how the fishing vessels drift in the surface water with the wind effect. Both the combined samples’ mean value and standard deviation (std) of the three rates (containing leeway rate, absolute DWL rate, and absolute CWL rate) show a decreasing trend as wind speed at 10 m increasing. Changes of the coefficients along with wind speed should be considered in the future calculation of drift simulation. Sign changes of CWL is observed in low wind speed, and the sharp variations of wind speed or wind direction. The forecast search area and the probability distribution of drift particles are sensitive to the wind condition and the frequencies of sign change respectively.

Magnetic-moment probability distribution of a quantum charged particle in thermodynamic equilibrium

We consider a quantum charged particle moving in the $xy$ plane under the action of a uniform perpendicular constant magnetic field, in the presence of a parabolic binding potential. The magnetic-moment operator has a continuous spectrum, and its eigenfunctions in the momentum representation are expressed in terms of modified Bessel functions. The probability distribution of the magnetic moment in the thermodynamic equilibrium state is calculated. At zero temperature, it has a simple exponential form in the diamagnetic region, with a sharp jump to zero at the origin. With an increase of temperature, a paramagnetic wing of the distribution becomes more and more pronounced. In the high-temperature regime, the diamagnetic and paramagnetic wings of the distribution have almost identical forms, described with a high precision by simple exponential functions with very large extensions. Therefore, diamagnetic and paramagnetic contributions almost cancel each other. The remaining non-zero diamagnetic value is due to a small asymmetry of the distribution nearby the origin, where some nonexponential fine structure is observed. The total width of the distribution function strongly depends on the strength of the binding potential. Strong fluctuations of the magnetic moment (described in terms of the variance) are discovered in all temperature regimes.

Lifetime prediction of carbon black filled elastomers based on the probability distribution of particle using an inelastic and hyperelastic material model

In this work, lifetime of carbon black filled elastomers is predicted based on a micro-flaw statistics lifetime concept. Within this concept two material models, the hyperelastic Yeoh model and the inelastic material model of rubber phenomenology (MORPH), are compared. The Yeoh model is chosen as a demonstrator for any hyperelastic material model. This model is not able to reproduce the characteristic material behaviour such as permanent set, softening effects and hysteresis and can only reproduce the nonlinearity of the stress-strain response. To overcome these deficiencies the hyperelastic models are typically fitted to the softening state of the material at a maximal (cyclic) load level. However, the results show that it is not foreseeable, which load level should be used for fitting the model to get a reasonable lifetime prediction. This is even more influenced by the restriction that only one parameter set has to be chosen to represent the inhomogeneous softening state of the material when arbitrary loads are applied. The above-mentioned characteristic properties of filled elastomers can be reproduced by the MORPH model. The lifetime prediction using this inelastic material model shows very good agreement with the measured values. Furthermore, the variable softening within the model adumbrate a good lifetime prediction quality of components with complex geometries.

New fatigue test sample: Lifetime prediction of carbon black filled elastomers based on the probability distribution of particles

In this paper, a new test specimen for fatigue testing is described. Most of the durability tests for elastomers are performed on dumbbell test specimen. Due to the load, the dumbbell shows stress concentration in the transition area. The newly concave test sample was developed through specific geometric changes in the dumbbell sample. These changes have minimized the stress concentration at transition area. The position of the fracture planes of the dumbbell and the concave test specimens are an indication of the stress concentration effect, since the concave did not occur any cracks in the transition area.The lifetime prediction of the dumbbell and the newly developed concave test sample at different load levels shows very good accordance with the measured values.

Mortal Brownian motion: Three short stories

Mortality introduces an intrinsic time scale into the scale-invariant Brownian motion. This fact has important consequences for different statistics of Brownian motion. Here we are telling three short stories, where spontaneous death, such as radioactive decay, puts a natural limit to “lifetime achievements” of a Brownian particle. In story 1 we determine the probability distribution of a mortal Brownian particle (MBP) reaching a specified point in space at the time of its death. In story 2 we determine the probability distribution of the area A = [Formula: see text] x(t)dt of an MBP on the line. Story 3 addresses the distribution of the winding angle of an MBP wandering around a reflecting disk in the plane. In stories 1 and 2 the probability distributions exhibit integrable singularities at zero values of the position and the area, respectively. In story 3 a singularity at zero winding angle appears only in the limit of very high mortality. A different integrable singularity appears at a nonzero winding angle. It is inherited from the recently uncovered singularity of the short-time large-deviation function of the winding angle for immortal Brownian motion.

Theoretical Estimation of Donor Strength of Common Conjugated Units for Organic Electronics.

Donor strength of commonly used conjugated building blocks in organic electronics were investigated with density functional theory. The donor molecules were coupled to thiophene-incorporated acceptor groups, and electronic structure calculations were performed for the energies of frontier orbitals, total charge on donors, and particle probability distribution. A novel approach is also developed to analyze the large set of data on frontier orbitals. The electron donating ability of a donor was determined by comparing the highest occupied molecular orbital energies of the calculated structures. The effect of selected acceptor group and chosen functional method were also investigated to accurately determine the donor strength of each conjugated building block. Ethylenedioxythiophene, propylenedioxythiophene, and triphenylamine derivatives were found to be best donors among the conjugated units investigated. Such a comparative analysis of donor strengths is believed to be useful for researchers in designing novel organic semiconductors for organic electronic applications.

Lifetime prediction of simple shear loaded filled elastomers based on the probability distribution of particles

A new method of the end-of-life prediction for filled elastomers has been introduced and applied on fatigue under uniaxial deformations in [1]. The method considers measured inhomogeneity size frequencies. The dimension of these inhomogeneities is assumed to be equal to the sizes of the initial cracks responsible for the final end-of-life of the elastomer. For an end-of-life prediction, the initial crack size information is inserted into a standard fracture mechanic law. Within a Monte Carlo simulation, several spatial realization steps of statistical initial crack size distributions over the deformation field of the investigated component or specimen are performed. Within each realization step, the minimum number of cycles until end-of-life is extracted and supplies a failure probability plot. Performing a large number of realizations provides the complete failure probability information. In this work, that concept is applied on fatigue under simple shear. The predicted failure probability values match the experimental findings from dynamic fatigue tests for the considered Styrene Rubber.

Non-Gaussianity of the van Hove function and dynamic-heterogeneity length scale.

Non-Gaussian nature of the probability distribution of particles' displacements in the supercooled temperature regime in glass-forming liquids are believed to be one of the major hallmarks of glass transition. It has already been established that this probability distribution, which is also known as the van Hove function, shows universal exponential tail. The origin of such an exponential tail in the distribution function is attributed to the hopping motion of particles observed in the supercooled regime. The non-Gaussian nature can also be explained if one assumes that the system has heterogeneous dynamics in space and time. Thus exponential tail is the manifestation of dynamic heterogeneity. In this work we directly show that non-Gaussianity of the distribution of particles' displacements occur over the dynamic heterogeneity length scale and the dynamical behavior course grained over this length scale becomes homogeneous. We study the non-Gaussianity of the van Hove function by systematically coarse graining at different length scales and extract the length scale of dynamic heterogeneity at which the shape of the van Hove function crosses over from non-Gaussian to Gaussian. The obtained dynamic heterogeneity scale is found to be in very good agreement with the scale obtained from other conventional methods.

Wavelet multi-resolution analysis on particle dynamics in a horizontal pneumatic conveying

The particle velocities are measured by the high-speed particle image velocimetry (PIV) in the acceleration and fully developed regimes of a horizontal pneumatic conveying. Based on the measured particle fluctuation velocities, continuous wavelet transform and one-dimensional orthogonal wavelet decomposition were applied to reveal particle dynamics in terms of time frequency analysis, the contribution from wavelet level to the particle fluctuation energy, spatial correlation and probability distribution of wavelet levels. The time frequency characteristics of particle fluctuation velocity suggest that the small-scale particle motions are suppressed and tend to transfer into large scale particle motions from acceleration regime to fully developed regime. In the near bottom part of pipe, the fluctuation energy of axial particle motion is mainly contributed from the wavelet levels of relatively low frequency, however, in the near top part of pipe, wavelet levels of relatively high frequency make comparable contribution to the axial particle fluctuation energy in the suspension flow regime, and this contribution decreases as particles are accelerated along the pipe. The low frequency wavelet levels exhibit large spatial correlation, and this spatial correlation increases as the particles flow from acceleration regime to fully developed regime. The skewness factor and kurtosis factor of wavelet level suggest that the deviation of Gaussian probability distribution is associated with the central frequency of wavelet level, and the deviation from Gaussian distribution is more evident as increasing central frequency. The higher wavelet levels can be linked to small sale particle motions, which lead to irregular particle fluctuation velocity.

Trigonometric Functions Above and Below of the Double-Slit Barrier

The Double-Slit Experiment with individual electrons represents the central mystery of quantum mechanics and was voted by readers of Physics World as the most beautiful experiment in physics. In our trigonometric model we have divided the Young-Feynman double-slit interferometer into two zones: Aristotelian mixing chamber (the space between the entrance slit S0 and above the double-slit barrier with slits S1 and S2 where particles have been reflected (Euclid) and/or randomly deviated – parenklisis of Epicurus) and Plato’s projection chamber (the space below the double-slit barrier and above the projection screen). Aristotelian mixing chamber represents the classical hidden variable: individual particles have been reflected and/or randomly deviated in this zone and later pass through the double-slit barrier in such a way that they create probability distributions of single particles described by the very well known trigonometric relations. The double-slit barrier serves as a “self-organizing filter” for reflected and/or deviated individual particles and creates an order in the population densities among individual particles. For the experimental evaluation of this trigonometric model we propose to insert an internal and external “sfumato mirrors” for the quantitative manipulation of probability distributions of individual particles. The concept “sfumato” was inspired by the Old Master Leonardo da Vinci (see his picture Mona Lisa) (“sfumato” means “smoke”). Sfumato mirrors represent the classical observable variable and their influence on the resulting probability densities of individual particles can be trigonometrically described. Sfumato mirrors must not absorb any particles. In another experimental modification we can insert below one slit of the double-slit barrier an “absorption chiaroscuro mask” to fully remove those penetrating particles from the system and thus manipulate with the probability distribution observed with one opened slit (“chiaroscuro” means “light-dark”). It could be of interest to observe the resulting probability distribution of individual particles registered on the projection screen from one opened slit S1 or S2. The absorption chiaroscuro mask must not reflect particles back to the Aristotelian mixing chamber or to the Plato´s projection chamber. The results of these experiments will decide if we are just going up in a blind alley”.

A Simple Pile-up Model for Time Series Analysis

Abstract In this paper, a simple pile-up model is presented. This model calculates the probability of having n counts if N particles collide with a sensor during an exposure time. Through some approximations, an analytic expression depending on only one parameter is obtained. This parameter characterizes the pile-up magnitude, and depends on features of the instrument and the source. The statistical model obtained permits the determination of probability distributions of measured counts from the probability distributions of incoming particles, which is valuable for time series analysis. Applicability limits are discussed, and an example of the improvement that can be achieved in the statistical analysis considering the proposed pile-up model is shown by analyzing real data.