Particle Distribution Function Research Articles

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

From QFT to Boltzmann: freeze-in in the presence of oscillating condensates

Scalar dark matter (DM), and axions in particular, have an irreducible abundance of particles produced by freeze-in due to portal interactions with the Standard Model plasma in the early Universe. In addition, vacuum misalignment and other mechanisms can lead to the presence of a cold, oscillating condensate. Therefore, generically, the evolution of the DM in both forms, condensate and particles, needs to be studied simultaneously. In non-equilibrium quantum field theory, the condensate and particles are described by one- and two-point functions, respectively. The fundamental coupled equations of motion (EoMs) of these objects are non-local. To simplify the EoMs and bring them into a familiar form for relic abundance calculations, we perform a Markovianization process for a quasi-harmonically oscillating homogeneous condensate, leading to local EoMs for the particle distribution function and the envelope function of condensate oscillation. This reduces the dynamics to a pair of coupled Boltzmann equations, and we derive explicitly the form of the collision operators for all particle and condensate interactions.

Skewness and kurtosis of solar wind proton distribution functions: The normal inverse-Gaussian model and its implications

Context.In the solar wind (SW), the particle distribution functions are generally not Gaussian. They present nonthermal features that are related to underlying acceleration and heating processes. These processes are critical in the overall dynamics of this expanding astrophysical fluid.Aims.The Proton Alpha Sensor (PAS) on board Solar Orbiter commonly observes skewed proton distributions, with a more populated high-energy side in the magnetic field direction than the Gaussian distribution. Our objectives are: (1) to identify a theoretical statistical function that adequately models the observed distributions and (2) to use its statistical interpretation to constrain the acceleration and heating processes.Methods.We analyzed the 3D velocity distribution functions (VDFs) measured by PAS and compared them to model statistical functions.Results.We show that the normal inverse Gaussian (NIG), a type of hyperbolic statistical distribution, provides excellent fits of skewed and leptokurtic proton distributions. NIG can model both the core distribution and the beam, if present. We propose an interpretation that is inspired by the mathematical formulation of the NIG. It assumes that the acceleration or heating mechanism can be modeled as a drifting diffusion process in velocity space, controlled (or subordinated) by the time of interaction of the particles with “accelerating structures”. The probability function of the interaction time is an inverse Gaussian (IG), obtained by considering a random drift across structures of a given size. The control of the diffusion by interaction times that follow an IG probability function formally defines the NIG distribution. Following this model, we show that skewness and kurtosis can be used to estimate the kinetic and thermal energy gains provided by the interaction with structures. For example, in the case studies presented here, the analyzed populations would have gained kinetic energy representing approximately two to four times their thermal energy, with an increase in velocity – due to acceleration – of from one-tenth to one-third of the observed flow velocity. We also show that the model constrains the initial temperature of the populations.Conclusions.Overall, the NIG model offers excellent fits of the observed proton distributions. Combining the skewness and the kurtosis, it also leads to constraints in the part of acceleration and heating due to the interactions with structures in the formation of the proton populations. We suggest that these effects add to the classical thermal evolution of the bulk velocity and temperature resulting from SW expansion.

Plasma properties and discharging of dust particles in an Ar/C2H2 plasma afterglow

A global (volume averaged) model is developed for an argon-acetylene plasma afterglow. The model is used to study the electron and ion densities, electron temperature and densities of argon metastable atoms in the afterglow plasma. The calculated time dependence for the electron density is found to be in agreement with the existing experimental data. These calculated plasma characteristics are used to investigate the dust charge distribution function (DCDF) for particles with radii of 10–200 nm. The DCDF is found by solving numerically the master equation describing dust discharging as a one-step stochastic process and is also calculated as a Gaussian distribution with mean dust charge and variance, which are functions of time. The time dependences for mean dust charge, variance and dust charging time are obtained and analysed. If the electronegativity of the plasma in the steady-state is low, negative ions do not affect much discharging of dust particles in the afterglow, while at large electronegativity their role is essential. In the case of low electronegativity, discharging of dust particles is mainly due to deposition of positive ions with small and moderate masses (less than the mass of C10H6 + ions). Increasing electronegativity, the effect of heavy positive ions on dust discharging in the late afterglow is important. Secondary electron emission from dust surface at collisions of metastable atoms appears to be negligible.

Conditions under which a natural iterative method for calculating the orientation distribution of rodlike particles decreases the free energy at each step.

It is shown that under certain conditions the iterative method suggested by the calculus of variations for calculating a cylindrically symmetric orientation distribution function (ODF) for rodlike particles strictly decreases the free energy at each full step. This monotonic behavior has strong implications for convergence of the sequence, or a subsequence, of the calculated ODFs. The result is valid not only for the reference system of hard core particles with indistinguishable ends, but also for free energy functions of similar type. The effect of an applied field is also permitted. Since the behavior of the iteration is intrinsic to the general form of the free energy function it applies to rod mixtures and may extend to a wider class of free energy functions. Outside the conditions that guarantee a monotonically decreasing free energy, we find that the iteration can fail to converge.

Quasilinear Simulation of the Development of Weibel Turbulence in Anisotropic Collisionless Plasma

A spectral quasilinear approach to the problem of TEM-Weibel instability in an anisotropic collisionless plasma is developed, which takes into account only the integral nonlinear interaction of modes through the joint variation of the spatially averaged particle velocity distribution induced by these modes. Within this approximation, a closed system of equations is obtained for the one- and two-dimensional evolution of spatial modes (harmonics) of the distribution function of particles and the electromagnetic field under conditions when the plasma anisotropy axis, the wave vector, and the magnetic field of the modes are orthogonal to each other. The numerical solution of this system of equations is compared with the available results of one-dimensional analytical quasilinear theory in the region of its applicability, as well as with the results of two-dimensional simulation by the particle-in-cell method, which also takes into account the direct four-wave interaction of modes. It is established that in the simplest cases of one-dimensional and axially symmetric two-dimensional problems for a bi-Maxwellian plasma, quasilinear phenomena play the leading role at a quite long stage of nonlinear development of turbulence. It is noted that at a later stage of decay of turbulence and in a more general formulation of the problem, in particular, in the presence of an external magnetic field, the direct nonlinear interaction of modes can manifest itself along with quasilinear phenomena. Based on the analysis carried out, the contribution of certain nonlinear effects to the evolution of the spatial spectrum of Weibel turbulence is revealed, and the properties of this turbulence are studied, including the self-similar character and qualitatively different stages of the dynamics of unstable modes.

Nonlinear equilibria and transport processes in burning plasmas

In this work, we put forward a general phase space transport theory in axisymmetric tokamak plasmas based upon the concept of zonal state (ZS). Within this theoretical framework, the ZS corresponds to a renormalized plasma nonlinear equilibrium consisting of phase space zonal structures (PSZS) and zonal electromagnetic fields (ZFs) which evolve self-consistently with symmetry breaking fluctuations and sources/collisions. More specifically, our approach involves deriving governing equations for the evolution of particle distribution functions (i.e, PSZS), which can be used to compute the corresponding macro-/meso-scale evolving magnetized plasma equilibrium adopting the Chew Goldberger Low description, separating the spatiotemporal microscale structures. The nonlinear physics of ZFs and of geodesic acoustic modes (GAMs)/energetic particle driven GAMs is then analyzed to illustrate the applications of our theory.

Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function f 0(r, p) at cylindrical radius r from the jet axis (assumed to lie along the z-axis) is given by convolving the particle momentum spectrum with the Green’s function , which describes the monoenergetic spectrum solution in which as r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for . The flow velocity u = u(r) e z is along the axis of the jet (the z-axis). u is independent of z, and u(r) is a monotonic decreasing function of r. The scattering time in the shear flow region 0 < r < r 2, and , where s > 0 in the region r > r 2 is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r, p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution that particles observed at (r, p) originated from r → ∞ with momentum . The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.

On the Partition Temperature of Massless Particles in High-Energy Collisions

Although partition temperature derived using the Darwin–Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase-space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed.

Analytical expression of Weibel electromagnetic instability growth rate in strongly coupled quantum plasma

The Weibel instability is one of the basic beam-plasma electromagnetic instabilities which plays an important role in the energy traveling mechanism of relativistic electrons generated by the laser-plasma interaction in the Inertial Confinement Fusion Plasma. In this paper, the Weibel instability is investigated in strongly coupled quantum plasmas, considering the Kappa distribution function for beam-plasma particles. An analytical expression is derived for the Weibel instability growth rate as the quantum and temperature anisotropy parameters for different values of the spectral index, k, under the low-frequency wave condition. The obtained results show that in the limit of low-frequency wave, the growth rate of instability decreases by increasing quantum parameter, and increases by increasing the values of special index and the temperature anisotropy parameter.

On global weak solutions of the Vlasov-Poisson equations with external magnetic field

We consider the first mixed problem for the system of Vlasov-Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.

A hybrid lattice Boltzmann method for gaseous detonations

This article is dedicated to the construction of a robust and accurate numerical scheme based on the lattice Boltzmann method (LBM) for simulations of gaseous detonations. This objective is achieved through careful construction of a fully conservative hybrid lattice Boltzmann scheme tailored for multi-species reactive flows. The core concept is to retain LBM low dissipation properties for acoustic and vortical modes by using the collide and stream algorithm for the particle distribution function, while transporting entropic and species modes via a specifically designed finite-volume scheme. The proposed method is first evaluated on common academic cases, demonstrating its ability to accurately simulate multi-species compressible and reactive flows with discontinuities: the convection of inert species, a Sod shock tube with two ideal gases and a steady one-dimensional inviscid detonation wave. Subsequently, the potential of this novel approach is demonstrated in one- and two-dimensional inviscid unsteady gaseous detonations, highlighting its ability to accurately recover detonation structures and associated instabilities for high activation energies. To the authors' knowledge, this study is the first successful simulation of detonation cellular structures capitalizing on the LBM collide and stream algorithm.

An exact solution to nonstationary Smoluchowski's coagulation equation complicated by Ostwald ripening

The concluding stage of a phase transformation, governed by particle coagulation and Ostwald ripening processes, is considered. An exact solution to the non-stationary integrodifferential kinetic equation is constructed in a parametric form. Namely, the particle-volume distribution function, time, number of particles, volume of the condensed phase, and the average particle radius are derived as functions of liquid supersaturation. The analytical solution is found (i) for a constant collision-frequency function, and (ii) using the averaging of this function over all possible combinations of particle volumes for various coagulation mechanisms. It is shown that the particle-volume distribution function and average radius of particles are substantially different in cases of constant and averaged collision-frequency functions for shear coagulation and coagulation under gravity.

Secondary Proton Beam Instability in a Turbulent Solar Wind at 50 R ⊙

We investigate a secondary proton beam instability coexisting with the ambient solar wind turbulence at 50 R ☉. Three-dimensional hybrid numerical simulations (particle ions and a quasi-neutralizing electron fluid) are carried out with the plasma parameters in the observed range. In the turbulent background, the particle distribution function, in particular the slope of the “bump-on-tail” responsible for the instability, is time-dependent and inhomogeneous. The presence of the turbulence substantially reduces the growth rate and saturation level of the instability. We derive magnetic power spectra from the observational data and perform a statistical analysis to evaluate the average turbulence intensity at 50 R ☉. This information is used to link the observed frequency spectrum to the wavenumber spectrum in the simulations. We verify that Taylor’s frozen-in hypothesis is valid for this purpose to a sufficient extent. To reproduce the typical magnetic power spectrum of the instability observed concurrently with the background turbulence, an artificial spacecraft probe is run through the simulation box. The thermal-ion instabilities are often seen as power elevations in the kinetic range of scales above an extrapolation of the turbulence spectrum from larger scales. We show that the elevated power in the simulations is much higher than the background level. Therefore, the turbulence at the average intensity does not obscure the secondary proton beam instability, as opposed to the solar wind at 1 au, in which the ambient turbulence typically obscures thermal-ion instabilities.

Generation of O-Mode in the Presence of Ion-Cyclotron Drift Wave Turbulence in a Nonuniform Plasma

This study aims to investigate the effect of ion-cyclotron drift wave turbulence on the generation of ordinary mode (O-mode) in the presence of density and temperature gradients. For this, a Vlasov plasma is considered where a resonant, and non-resonant modes are considered to be present in the system. Here, the non-resonant mode is a perturbation caused by O-mode in a quasi-steady state of plasma, which is characterised by the presence of low frequency ion-cyclotron resonant mode waves. The interaction between these waves is studied by the Vlasov-Maxwell set of equations and a modified Maxwellian-type distribution function for particles that includes the external force field and associated density and temperature gradient parameters . The study analyses the growth rate of electromagnetic O-mode at the expense of ion-cyclotron drift wave energy and the associated impact of the density and temperature gradient. This model uses the linear response theory on weakly turbulent plasma, evaluates the responses due to turbulent and perturbed fields, and obtains the nonlinear dispersion relation for O-mode.

Phase space dynamics of unmagnetized plasmas: Collisionless and collisional regimes

Eulerian electrostatic kinetic simulations of unmagnetized plasmas (kinetic electrons and motionless protons) with high-frequency equilibrium perturbations have been employed to investigate the phase space free energy transfer across spatial and velocity scales, associated with the resonant interaction of electrons with the self-induced electric field. Numerical runs cover a wide range of collisionless and weakly collisional plasma regimes. An analysis technique based on the Fourier–Hermite transform of the particle distribution function allows to point out how kinetic processes trigger the free energy cascade, which is instead inhibited at finer scales when collisions are turned on. Numerical results are presented and discussed for the cases of linear wave Landau damping, nonlinear electron trapping, and bump-on-tail and two-stream instabilities. A more realistic situation of turbulent Langmuir fluctuations is also discussed in detail. Fourier–Hermite transform shows a free energy spread, highly conditioned by collisions, which involves velocity scales more quickly than the spatial scales, even when nonlinear effects are dominant. This results in anisotropic spectra whose slopes are compatible with theoretical expectations. Finally, an exact conservation law has been derived, which describes the time evolution of the free energy of the system, taking into account the collisional dissipation.

Galactic Bar Resonances with Diffusion: An Analytic Model with Implications for Bar–Dark Matter Halo Dynamical Friction

The secular evolution of disk galaxies is largely driven by resonances between the orbits of “particles” (stars or dark matter) and the rotation of non-axisymmetric features (spiral arms or a bar). Such resonances may also explain kinematic and photometric features observed in the Milky Way and external galaxies. In simplified cases, these resonant interactions are well understood: for instance, the dynamics of a test particle trapped near a resonance of a steadily rotating bar is easily analyzed using the angle-action tools pioneered by Binney, Monari, and others. However, such treatments do not address the stochasticity and messiness inherent to real galaxies—effects that have, with few exceptions, been previously explored only with complex N-body simulations. In this paper, we propose a simple kinetic equation describing the distribution function of particles near an orbital resonance with a rigidly rotating bar, allowing for diffusion of the particles’ slow actions. We solve this equation for various values of the dimensionless diffusion strength Δ, and then apply our theory to the calculation of bar–halo dynamical friction. For Δ = 0, we recover the classic result of Tremaine and Weinberg that friction ultimately vanishes, owing to the phase mixing of resonant orbits. However, for Δ > 0, we find that diffusion suppresses phase mixing, leading to a finite torque. Our results suggest that stochasticity—be it physical or numerical—tends to increase bar–halo friction, and that bars in cosmological simulations might experience significant artificial slowdown, even if the numerical two-body relaxation time is much longer than a Hubble time.

Sharp depletion of radial distribution function of particles due to collision and coagulation inside turbulent flow: A systematic study

Sharp depletion of radial distribution function of particles due to collision and coagulation inside turbulent flow: A systematic study

Formation of the Injection Function of Solar Energetic Particles in Gradual Events

We present a model for solar energetic particle injection into interplanetary space in gradual events, in which particle acceleration occurs in a limited region of the solar atmosphere. The distribution function of particles accelerated by the diffusion mechanism is calculated. The flux of injected solar energetic particles is determined as a function of time and energy. We provide an explanation of the characteristic properties of the injection function and their dependence on the particle energy. Comparing the calculation results with ground-based measurements in the 2001 April 15 event shows a rough agreement with the particle density as a function of time and good agreement with the spectrum of maximum intensity values.

Acceleration of cosmic rays in presence of magnetohydrodynamic fluctuations at small scales

ABSTRACT This work investigates the evolution of the distribution of charged particles (cosmic rays) due to the mechanism of stochastic turbulent acceleration (STA) in presence of small-scale turbulence with a mean magnetic field. STA is usually modelled as a biased random walk process in the momentum space of the non-thermal particles. This results in an advection-diffusion type transport equation for the non-thermal particle distribution function. Under quasi-linear approximation, and by assuming turbulent spectra with power being available only in the sub-gyroscale range, we find that the Fokker–Planck diffusion coefficients Dγγ and Dμμ scale with the Lorentz factor γ as Dγγ ∝ γ−2/3 and Dμμ ∝ γ−8/3. We consider Alfvèn and fast waves in our calculations, and find a universal trend for the momentum diffusion coefficient irrespective of the properties of the small-scale turbulence. Such universality has already been reported regarding the spatial diffusion of the cosmic rays, and, here too, we observe a universality in the momentum diffusion coefficient. Furthermore, with the calculated transport coefficients, we numerically solve the advection-diffusion-type transport equation for the non-thermal particles. We demonstrate the interplay of various mircophysical processes such as STA, synchrotron loss, and particle escape on the particle distribution by systematically varying the parameters of the problem. We observe that the effect of the small-scale turbulence is more impactful for the high-energy protons as compared to the electrons and such turbulence is capable of sustaining the energy of the protons from catastrophic radiative loss processes. Such a finding is novel and helps us to enhance our understanding about the hadronic emission processes that are typically considered as a competitor for the leptonic emission for certain astrophysical systems.