Particle Distribution Function Research Articles

Backward test particle simulation of nonlinear cyclotron wave-particle interactions in the radiation belts

Wave-particle interaction plays a crucial role in the dynamics of the Earth’s radiation belts. Cyclotron resonance between coherent whistler mode electromagnetic waves and energetic electrons of the radiation belts is often called a coherent instability. Coherent instability leads to wave amplification/generation and particle acceleration/scattering. The effect of wave on particle’s distribution function is a key component of the instability. In general, whistler wave amplitude can grow over threshold of quasi-linear (linear) diffusion theory which analytically tracks the time-evolution of a particle distribution. Thus, a numerical approach is required to model the nonlinear wave induced perturbations on particle distribution function. A backward test particle model is used to determine the energetic electrons phase space dynamics as a result of coherent whistler wave instability. The results show the formation of a phase space features with much higher resolution than is available with forward scattering models. In the nonlinear regime the formation of electron phase space holes upstream of a monochromatic wave is observed. The results validate the nonlinear phase trapping mechanism that drives nonlinear whistler mode growth. The key differences in phase-space perturbations between the linear and nonlinear scenarios are also illustrated. For the linearized equations or for low (below threshold) wave amplitudes in the nonlinear case, there is no formation of a phase-space hole and both models show features that can be characterized as linear striations or ripples in phase-space.

Decomposition of nonlinear collision operator in quantum Lattice Boltzmann algorithm

We propose a quantum algorithm to tackle the quadratic nonlinearity in the Lattice Boltzmann (LB) collision operator. The key idea is to build the quantum gates based on the particle distribution functions (PDF) within the coherence time for qubits. Thus, both the operator and a state vector are linear functions of PDFs, and upon quantum state evolution, the resulting PDFs will have quadraticity. To this end, we decompose the collision operator for a DmQn lattice model into a product of operators, where n is the number of lattice velocity directions. After decomposition, the operators with constant entries remain unchanged throughout the simulation, whereas the remaining will be built based on the statevector of the previous time step. Also, we show that such a decomposition is not unique. Compared to the second-order Carleman-linearized LB, the present approach reduces the circuit width by half and circuit depth by exponential order. The proposed algorithm has been verified through the one-dimensional flow discontinuity and two-dimensional Kolmogrov-like flow test cases.

Non-destructive measurement of the dispersion of High-density polyethylene/Polystyrene blends using near-infrared diffuse reflectance spectroscopy based on deep learning

The dispersion of polymer blends significantly influences their overall performance. However, current methods for characterizing dispersion are destructive and time-consuming. In this study, a near-infrared (NIR) regression model was first developed for the measurement of cumulative particle size distribution function (CDF) of dispersed particles in polymer blends, thus indirectly and quantitively characterizing dispersion. High-density Polyethylene/Polystyrene (HDPE/PS) with varying dispersed particle sizes were prepared by adjusting the content of PS, flow field intensity during processing, and viscosity ratio between HDPE and PS. Scanning electron microscopy (SEM) was used to provide reference particle size values for calibrating the NIR data. Deep-learning methods were used for the model building. Compared to traditional partial least-square regression, deep learning methods demonstrated superior fitting ability, The best-performing model, an MLP with three hidden layers, achieved an coefficient of determination (R2) of approximately 0.93 on the independent test set. The predicted CDFs closely matched the actual values, with no significant deviations observed. This study provides a non-destructive and time-saving new method for characterizing the dispersion of polymer blends.

The Migdal jump under the quantum Hall regime

In two-dimensional electron systems at large values of the Wigner-Seits parameter rs and in the quantum Hall effect mode, the distribution function of particles over Landau levels was calculated. It turned out that at small filling factors, the tail of the distribution function and the magnitude of the Migdal jump are qualitatively different from the case of a Fermi liquid in a zero magnetic field. Due to the presence of the cyclotron energy gap, the Fermi-liquid distortion of the distribution function is significantly suppressed.

Prototype of Circular Split Ring Resonator-Based Sensor for Estimating Soil Moisture as a Function of Soil Particle Distribution

Prototype of Circular Split Ring Resonator-Based Sensor for Estimating Soil Moisture as a Function of Soil Particle Distribution

Lateral Distribution Function Estimation of Electrons and Muons using Nishimura-Kamata-Greisen Function

Abstract The density of charged particles in extensive air showers reaching the surface of the Earth was calculated by estimating the lateral distribution function (LDF) of different primary particles at high energies. LDF simulation was performed using the AIRES system (version 19.04.10), a set of programs and subroutines designed to simulate ultra-high-energy air showers resulting from the interaction of cosmic rays with the Earth’s atmosphere. This system includes algorithms for fast simulation and output data management, and can simulate particle showers realistically and manage the related data efficiently. Its aim is to contribute to research on high-energy cosmic ray interactions for two charged particles, like muons and electrons, at very high energies (1016, 1018, and 1019 eV) and taking into account the effect of particles, such as protons, helium nuclei, and iron nuclei, and primary energies and zenith angles (0°, 20°, and 40°). The LDF was also calculated using the Nishimura–Kamata–Greisen function, and good agreement was found with the results produced by the AIRES system for high-energy muons and electrons created by primary particles.

Nonlinear Coupling of Kinetic Alfvén Waves and Ion Acoustic Waves in the Inner Heliosphere

We study the nonlinear coupling of kinetic Alfvén waves with ion acoustic waves applicable to the Earth’s radiation belt and near-Sun streamer belt solar wind using dynamical equations in the form of modified Zakharov systems. Numerical simulations show the formation of magnetic field filamentary structures associated with density humps and dips which become turbulent at later times, redistributing the energy to higher wavenumbers. The magnetic power spectra exhibit an inertial range Kolmogorov-like spectral index value of −5/3 for k ⊥ ρ i < 1, followed by a steeper dissipation range spectra with indices ∼ −3 for the radiation belt case and ∼ −4 for the near-Sun streamer belt solar wind case, here k ⊥ and ρ i represent the wavevector component perpendicular to the background magnetic field and the ion thermal gyroradius, respectively. Applying quasilinear theory in terms of the Fokker–Planck equation in the region of wavenumber turbulent spectra, we find the particle distribution function flattening in the superthermal tail population which is the signature of particle energization and plasma heating.

Correlation functions of weakly inhomogeneous plasma

This article focuses on the kinetic theory of inhomogeneous plasma and explores the interaction between a high-frequency electric field and weakly inhomogeneous plasma. Particularly, it examines the impact of an external variable field on the kinetic and high-frequency properties of the plasma, including kinetic equations, correlation functions, and distribution functions of charged particles. The study derives expressions for the pair (two-particle) correlation function and the corresponding distribution function, taking into ac- count the spatial inhomogeneity of the plasma and electric field, as well as the collisions between charged particles. The results were obtained using the kinetic equation for the spatial-temporal spectral density of fluctuations and the method of successive approximations (separation of slow motions and fast oscillations). The field amplitude is considered a slowly varying function of time and coordinates. The calculations neglect the contribution of the magnetic component of the electromagnetic field, which is applicable to longitudinal electric fields. The results obtained in this article are primarily of theoretical interest, they reveal the picture of the interaction of a weakly inhomogeneous plasma with a high-frequency electric field and can be used in the construction of a kinetic theory of an inhomogeneous plasma located in high-frequency electromagnetic fields. Note that for charged particles of the same sign, the correlation function is negative, and for particles of different signs it is positive. In addition, the correlation function is exponentially small when the distance between the particles is greater than the Debye radius. In all calculations, the contribution of the magnetic component of the electromagnetic field is neglected, which is quite true for the longitudinal electric field.

Impact of medium anisotropy on quarkonium dissociation and regeneration

Quarkonium production in ultra-relativistic collisions plays a crucial role in probing the existence of hot QCD matter. This study explores quarkonia states dissociation and regeneration in the hot QCD medium while considering momentum anisotropy. The net quarkonia decay width (ΓD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{D}$$\\end{document}) arises from two essential processes: collisional damping and gluonic dissociation. The quarkonia regeneration includes the transition from octet to singlet states within the anisotropic medium. Our study utilizes a medium-modified potential that incorporates anisotropy via particle distribution functions. This modified potential gives rise to collisional damping for quarkonia due to the surrounding medium, as well as the transition of quarkonia from singlet to octet states due to interactions with gluons. Furthermore, we employ the detailed balance approach to investigate the regeneration of quarkonia within this medium. Our comprehensive analysis spans various temperature settings, transverse momentum values, and anisotropic strengths. Notably, we find that, in addition to medium temperatures and heavy quark transverse momentum, anisotropy significantly influences the dissociation and regeneration of various quarkonia states.

Model for plasma transport due to drift waves based on mappings including finite Larmor radius

Abstract Turbulence in plasmas is modeled by fluctuating electric fields that determine particle motion through the E × B drift velocity which can lead to chaotic behavior. When applied to an ensemble of plasma particles in the chaotic regime their transport is studied: the anomalous plasma transport. The fluctuations are modeled by a spectrum of traveling waves with a wide frequency span which converts the equations of motion into an iterative mapping. The statistical properties of transport are derived. When the waves amplitude is small the particle orbits are regular, but as it is increased the behavior becomes increasingly chaotic. The effect of finite Larmor radius and the presence of a background plasma flow is also studied. We show that when a thermal population of particles is considered, the transport becomes non-local, as evidenced by a non-Gaussian particle distribution function (PDF). We have analyzed two different kinds of sheared flows: (1) a monotonic velocity shear and (2) a non-monotonic shear. The presence of transport barriers associated with the sheared velocity is also studied. We also present the application of the same techniques to study the transport of energetic particles that are born with a monoenergetic distribution.

Axisymmetric phase-field-based lattice Boltzmann model for incompressible two-phase flow with phase change

In this work, a phase-field-based lattice Boltzmann equation (LBE) model for axisymmetric two-phase flow with phase change is proposed. Two sets of discrete particle distribution functions are employed to match the conserved Allen–Cahn equation and the hydrodynamic equations with phase change effect, respectively. Since phase change occurs at the interface, the divergence-free condition of the velocity field is no longer satisfied due to mass transfer, and the conserved Allen–Cahn equation needs to be equipped with a source term dependent on the phase change model. To deal with these, a novel source term in the hydrodynamic LBE is delicately designed to recover the correct target governing equations. Meanwhile, the LBE for the Allen–Cahn equation is modified with a discrete force term to model mass transfer. In particular, an additional correction term is added into the hydrodynamic LBE to reduce the spurious velocity and improve numerical stability. Several axisymmetric benchmark multiphase problems with phase change, including bubble growing in superheated liquid, D2 law, film boiling, bubble rising in superheated liquid under gravity, and droplet impact on a hot surface, have been conducted to test the performance of the proposed model. Numerical results agree well with analytical solutions and available published data in the literature.

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Improvement of the model for calculating the protection coefficient of filtering respirators from harmful aerosol

Research objective. Improvement of the model for calculating the protection coefficient of filtering respirators from harmful solid aerosol. Materials and methods. For the experimental determination of the protection coefficient of the filtering respirator according to the standard DSTU EN 149:2017 «Personal respiratory protective devices. Filtering half masks to protect against aerosols. Requirements, testing, marking» and the application of the method DSTU EN 13274-1:2005 «Personal respiratory protective devices. Methods of testing. Part 1. Determination of the penetration coefficient and the overall penetration coefficient», lightweight filtering half masks of «Standard» FFP2 type were used on volunteers. Results. The protection coefficient of the filtering respirators used on volunteers in laboratory conditions was calculated at the specified concentration of test aerosol in the ambient air by the standard testing method described in DSTU EN 13274-1:2005 «Personal respiratory protective devices. Methods of testing. Part 1. Determination of the penetration coefficient and the overall penetration coefficient» and takes into account the measurement error from the instruments not exceeding 10%. A comparison of the obtained indicators of the protection coefficient of filtering respirators in laboratory conditions and with the developed theoretical model, taking into account particle deposition in the sampling probe and submask space during the exhalation process, showed a good agreement of the indicators by the Student's t-test criterion. Scientific novelty. The scientific novelty lies in the development of a model for calculating the protection coefficient of filtering respirators based on the assessed mass distribution function of aerosol particles in the breathing zone; penetration function through the gap between the face and the half mask, taking into account the inhalation and exhalation process, as well as particle deposition in the sampling probe. Practical significance of the work lies in the development of a formula for the theoretical determination of the protection coefficient of filtering respirators, which is used to calculate the professional risk of pneumoconiosis among users

Shakhov-type extension of the relaxation time approximation in relativistic kinetic theory and second-order fluid dynamics

We present a relativistic Shakhov-type generalization of the Anderson-Witting relaxation time model for the Boltzmann collision integral to modify the ratio of momentum diffusivity to thermal diffusivity. This is achieved by modifying the path on which the single particle distribution function fk approaches local equilibrium f0k by constructing an intermediate Shakhov-type distribution fSk similar to the 14-moment approximation of Israel and Stewart. We illustrate the effectiveness of this model in case of the Bjorken expansion of an ideal gas of massive particles and the damping of longitudinal waves through an ultrarelativistic ideal gas.

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence.

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equationfor the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

Transmission of waves and particles through the interface: Reversibility and coherence

We examine the transmission of quantum particles (phonons, electrons, and photons) across interfaces, identifying universal patterns in diverse physical scenarios. Starting with classical wave equations, we quantize them and derive kinetic equations. Those are matching conditions for the distribution functions of particles at the interface. We note the time irreversibility of the derived kinetic equations — an essential feature for accurately describing irreversible processes like heat transport. We identify the juncture in our derivation where the time symmetry of wave equations is disrupted, it is the assumption of the non-coherence of incident waves. Consequently, we infer that non-coherent transmission through the interface exhibits time irreversibility. We propose an experiment to validate this hypothesis.

Two-Dimensional System of Moment Equations and Macroscopic Boundary Conditions Depending on the Velocity of Movement and the Surface Temperature of a Body Moving in Fluid

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven.

Particle Scattering in the Presence of Moderate to Strong Adiabatic Focusing

AbstractStarting from the Vlasov‐Boltzmann equation, we derive corrections to the quasi linear Fokker‐Planck pitch angle scattering coefficient caused by a non‐uniformity of the background magnetic field. Integration of the perturbed particle distribution function is performed along a modified unperturbed trajectory that takes into account the effects of focusing and mirroring on pitch angle and gyrophase. We calculate the changes to the quasi‐linear scattering rate owing to a variations in the field strength and the gyrophase drift that affects the resonance between particles and magnetic fluctuations. It is shown that the scattering coefficient has an oscillating behavior in pitch angle when the focusing lengthscale is comparable to the correlation length of the turbulence and the particle's Larmor radius. The applicability of analytic results are demonstrated with a simple model of charged particles trapped in a magnetic field of a planet. It is suggested that the pitch angle diffusion coefficient could become negative for the case of very strong focusing.

Development of a cascaded lattice Boltzmann model for two‐layer shallow water flows

SummaryMany environmental phenomena, such as flows in rivers or in coastal region can be characterised by means of the ‘shallow approach’. A multi‐layer scheme allows to extend it to density layered shallow water flows (e.g., gravity currents). Although a variety of models allowing numerical investigation of single and multi‐layer shallow water flows, based on continuum and particle approaches, have been widely discussed, there are still some computational aspects that need further investigation. Focusing on the Lattice Boltzmann models (LBM), available multi‐layer models generally use the standard linear collision operator (CO). In this work we adopt a multi relaxation time (MRT) cascaded collision operator to develop a two‐layered liquid Lattice‐Boltzmann model (CaLB‐2). Specifically, the model solves the shallow water equations, taking into account two separate sets of particle distribution function (PDF), one for each layer, solved separately. Layers are connected through coupling terms, defined as external forces that model the mutual actions between the two layers. The model is validated through comparisons with experimental and numerical results from test cases available in literature. First results are very promising, highlighting a good correspondence between simulation results and literature benchmarks.

Kinetic theory formulation of the P - and CP -odd terms in the photon self-energy in a medium

In an optically active medium, such as a plasma that contains a neutrino background, the left-handed and right-handed polarization photon modes acquire different dispersion relations. We study the propagation of photons in such a medium, which is otherwise isotropic, within the framework of the covariant collissionless Boltzmann equation incorporating a term that parametrizes the optical activity. Using the linear response approximation, we obtain the formulas for the components of the photon polarization tensor, expressed in terms of integrals over the momentum distribution function of the background particles. The main result here is the formula for the P- and CP-breaking component of the photon polarization tensor in terms of the parameter involved in the new term we consider in the Boltzmann equation to describe the effects of optical activity. We discuss the results for some particular cases, such as long-wavelength and nonrelativistic limits, for illustrative purposes. We also discuss the generalizations of the P- and CP-breaking term we included in the Boltzmann equation. In particular we consider the application to a plasma with a neutrino background and establish contact with calculations of the photon self-energy in those systems in the framework of thermal field theory. Published by the American Physical Society 2024