Kinetic Theory Solutions Research Articles

Local clustering of relic neutrinos: comparison of kinetic field theory and the Vlasov equation

Gravitational clustering in our cosmic vicinity is expected to lead to an enhancement of the local density of relic neutrinos. We derive expressions for the neutrino density, using a perturbative approach to kinetic field theory and perturbative solutions of the Vlasov equation up to second order. Our work reveals that both formalisms give exactly the same results and can thus be considered equivalent. Numerical evaluation of the local relic neutrino density at first and second order provides some fundamental insights into the frequently applied approach of linear response to neutrino clustering (also known as the Gilbert equation). Against the naive expectation, including the second-order contribution does not lead to an improvement of the prediction for the local relic neutrino density but to a dramatic overestimation. This is because perturbation theory breaks down in a momentum-dependent fashion and in particular for densities well below unity.

On the Holway-Weiss debate: Convergence of the Grad-moment-expansion in kinetic gas theory

Moment expansions are used as a model reduction technique in kinetic gas theory to approximate the Boltzmann equation. Rarefied gas models based on so-called moment equations have become increasingly popular recently. However, in a seminal paper by Holway [“Existence of kinetic theory solutions to the shock structure problem,” Phys. Fluids 7(6), 911–913 (1965)], a fundamental restriction on the existence of the expansion was used to explain subshock behavior of shock profile solutions obtained by moment equations. Later, Weiss [“Comments on ‘Existence of kinetic theory solutions to the shock structure problem’ [Phys. Fluids 7, 911 (1964)],” Phys. Fluids 8(6), 1689–1690 (1996)] argued that this restriction does not exist. We will revisit and discuss their findings and explain that both arguments have a correct and an incorrect part. While a general convergence restriction for moment expansions does exist, it cannot be attributed to subshock solutions. We will also discuss the implications of the restriction and give some numerical evidence for our considerations.

Anisotropic nonequilibrium hydrodynamic attractor

We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a 0+1d conformal system using kinetic theory in the relaxation time approximation. We compare our results to the non-equilibrium attractor obtained from exact solution of the 0+1d conformal Boltzmann equation, Navier-Stokes theory, and second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal to transverse pressure ratio and is virtually indistinguishable from the exact attractor. This suggests that kinetic theory involves not only a resummation in gradients (Knudsen number) but also a novel resummation in inverse Reynolds number. We also demonstrate that the DNMR result provides a better approximation to the exact kinetic theory attractor than Mueller-Israel-Stewart theory. Finally, we introduce a new method for obtaining approximate aHydro equations which relies solely on an expansion in inverse Reynolds number, carry out this expansion to third order, and compare these third-order results to the exact kinetic theory solution.

On the Parallel and Oblique Firehose Instability in Fluid Models

A brief analysis of the proton parallel and oblique firehose instability is presented from a fluid perspective and the results are compared to kinetic theory solutions obtained by the WHAMP code. It is shown that the classical CGL model very accurately describes the growth rate of these instabilities at sufficiently long spatial scales (small wavenumbers). The required stabilization of these instabilities at small spatial scales (high wavenumbers) naturally requires dispersive effects and the stabilization is due to the Hall term and finite Larmor radius (FLR) corrections to the pressure tensor. Even though the stabilization is not completely accurate since at small spatial scales a relatively strong collisionless damping comes into effect, we find that the main concepts of the maximum growth rate and the stabilization of these instabilities is indeed present in the fluid description. However, there are differences that are quite pronounced when close to the firehose threshold and that clarify the different profiles for marginally stable states with a prescribed maximum growth rate $\gamma_{max}$ in the simple fluid models considered here and the kinetic description.

Smoothed solutions in the kinetic theory of e+e- vacuum pair creation in strong laser fields. Linear polarization

In this paper, the dynamical Schwinger effect of vacuum creation of electron–positron pairs driven by an intense laser pulse is studied on the basis of correct quantum kinetic theory. In the general case, the numerical solutions of corresponding system of kinetic equations exhibit complex time dependence which makes the analysis of the physical processes complicated. In particular, the question of secondary effects, such as creation of annihilation photons from the focus spot of the colliding laser beams, remains an important open problem. In our previous work [S. A. Smolyansky, M. Bonitz and A. V. Prozorkevich, Contrib. Plasma Phys.53 (2013) 788], we presented a perturbation theory which is able to capture the dominant time dependence of the produced electron–positron pair distribution during the pulse (quasiparticle excitations). In the present work, we develop appreciably this approximation scheme. We demonstrate effectiveness of the proposed method for solution of such kind nonstationary problems in the simplest models of the laser field. However, this approach opens perspective for search of the relevant approximate solutions in kinetic theory of the e+e- quasiparticle plasma for the more realistic field models (arbitrary polarization, space inhomogeneous, etc).

Thermodynamic properties and entropy scaling law for diffusivity in soft spheres.

The purely repulsive soft-sphere system, where the interaction potential is inversely proportional to the pair separation raised to the power n, is considered. The Laplace transform technique is used to derive its thermodynamic properties in terms of the potential energy and its density derivative obtained from molecular dynamics simulations. The derived expressions provide an analytic framework with which to explore soft-sphere thermodynamics across the whole softness-density fluid domain. The trends in the isochoric and isobaric heat capacity, thermal expansion coefficient, isothermal and adiabatic bulk moduli, Grüneisen parameter, isothermal pressure, and the Joule-Thomson coefficient as a function of fluid density and potential softness are described using these formulas supplemented by the simulation-derived equation of state. At low densities a minimum in the isobaric heat capacity with density is found, which is a new feature for a purely repulsive pair interaction. The hard-sphere and n = 3 limits are obtained, and the low density limit specified analytically for any n is discussed. The softness dependence of calculated quantities indicates freezing criteria based on features of the radial distribution function or derived functions of it are not expected to be universal. A new and accurate formula linking the self-diffusion coefficient to the excess entropy for the entire fluid softness-density domain is proposed, which incorporates the kinetic theory solution for the low density limit and an entropy-dependent function in an exponential form. The thermodynamic properties (or their derivatives), structural quantities, and diffusion coefficient indicate that three regions specified by a convex, concave, and intermediate density dependence can be expected as a function of n, with a narrow transition region within the range 5 < n < 8.

On the current-voltage relationship in fluid theory

Abstract. The kinetic theory of precipitating electrons with Maxwellian source plasma yields the well-known current-voltage relationship (CV-relationship; Knight formula), which can in most cases be accurately approximated by a reduced linear formula. Our question is whether it is possible to obtain this CV-relationship from fluid theory, and if so, to what extent it is physically equivalent with the more accurate kinetic counterpart. An answer to this question is necessary before trying to understand how one could combine time-dependent and transient phenomena such as Alfvénic waves with a slowly evolving background described by the CV-relationship. We first compute the fluid quantity profiles (density, pressure etc.) along a flux tube based on kinetic theory solution. A parallel potential drop accumulates plasma (and pressure) below it, which explains why the current is linearly proportional to the potential drop in the kinetic theory even though the velocity of the accelerated particles is only proportional to the square root of the accelerating voltage. Electron fluid theory reveals that the kinetic theory results can be reproduced, except for different numerical constants, if and only if the polytropic index γ is equal to three, corresponding to one-dimensional motion. The convective derivative term v·∇v provides the equivalent of the "mirror force" and is therefore important to include in a fluid theory trying to describe a CV-relationship. In one-fluid equations the parallel electric field, at least in its functional form, emerges self-consistently. We find that the electron density enhancement below the potential drop disappears because the magnetospheric ions would be unable to neutralize it, and a square root CV-relationship results, in disagreement with kinetic theory and observations. Also, the potential drop concentrates just above the ionosphere, which is at odds with observations as well. To resolve this puzzle, we show that considering outflowing ionospheric ions restores the possibility of having the acceleration region well above the ionosphere, and thus the electron kinetic (and fluid, if γ=3) theory results are reproduced in a self-consistent manner. Thus the inclusion of ionospheric ions is crucial for a feasible CV-relationship in fluid theory. Constructing a quantitative fluid model (possibly one-fluid) which reproduces this property would be an interesting task for a future study.Key words. Ionosphere (ionosphere-magnetosphere interactions; particle precipitation) · Magnetospheric physics (magnetosphere-ionosphere interactions)

On the current-voltage relationship in fluid theory

<strong class="journal-contentHeaderColor">Abstract.</strong> The kinetic theory of precipitating electrons with Maxwellian source plasma yields the well-known current-voltage relationship (CV-relationship; Knight formula), which can in most cases be accurately approximated by a reduced linear formula. Our question is whether it is possible to obtain this CV-relationship from fluid theory, and if so, to what extent it is physically equivalent with the more accurate kinetic counterpart. An answer to this question is necessary before trying to understand how one could combine time-dependent and transient phenomena such as Alfvénic waves with a slowly evolving background described by the CV-relationship. We first compute the fluid quantity profiles (density, pressure etc.) along a flux tube based on kinetic theory solution. A parallel potential drop accumulates plasma (and pressure) below it, which explains why the current is linearly proportional to the potential drop in the kinetic theory even though the velocity of the accelerated particles is only proportional to the square root of the accelerating voltage. Electron fluid theory reveals that the kinetic theory results can be reproduced, except for different numerical constants, if and only if the polytropic index &#x03B3; is equal to three, corresponding to one-dimensional motion. The convective derivative term <b>v</b>·&#x2207;<b>v</b> provides the equivalent of the "mirror force" and is therefore important to include in a fluid theory trying to describe a CV-relationship. In one-fluid equations the parallel electric field, at least in its functional form, emerges self-consistently. We find that the electron density enhancement below the potential drop disappears because the magnetospheric ions would be unable to neutralize it, and a square root CV-relationship results, in disagreement with kinetic theory and observations. Also, the potential drop concentrates just above the ionosphere, which is at odds with observations as well. To resolve this puzzle, we show that considering outflowing ionospheric ions restores the possibility of having the acceleration region well above the ionosphere, and thus the electron kinetic (and fluid, if &#x03B3;=3) theory results are reproduced in a self-consistent manner. Thus the inclusion of ionospheric ions is crucial for a feasible CV-relationship in fluid theory. Constructing a quantitative fluid model (possibly one-fluid) which reproduces this property would be an interesting task for a future study.<br><br><b>Key words. </b>Ionosphere (ionosphere-magnetosphere interactions; particle precipitation) · Magnetospheric physics (magnetosphere-ionosphere interactions)

Solar wind in the distant heliosphere

This paper presents a quantitative MHD model to study the effects of pickup protons on the solar wind. The model includes the kinetic theory solution of interstellar neutral hydrogen, the ionization process for the production of pickup protons, the spiral interplanetary magnetic field, and the full MHD equations. In order to study the effects of pickup protons, we first obtain the kinetic theory solutions of interstellar neutral hydrogen. This solution is then used to calculate the flow of interstellar hydrogen and the production of pickup protons. We study a one‐dimensional, steady state model of the pickup proton solar wind in the upwind direction. The solar wind proton parameters predicted by this solution agree reasonably well with observational data from Voyager 2. Throughout the heliosphere the temperature of pickup protons is higher than the temperature of the solar wind protons by two orders of magnitude. Outside the ionization radius, the solar wind temperature and the fast speed begin to increase and the bulk speed and the Mach number of the wind begin to decrease. The component of the equation of motion along the streamline direction can be written in the form of a Bernoulli equation; it can clearly explain the deceleration of the solar wind. In the distant heliosphere the sum of the kinetic energy and the thermal energy decreases with the heliocentric distance due to the charge exchange process; at the same time, the thermal energy of the solar wind plasma increases due to heating by the pickup process. The combined action causes a significant radial decrease in the kinetic energy of the wind.

Transverse energy evolution as a test of parton cascade models

We propose a test of Monte Carlo Parton Cascade models based on analytic solutions of covariant kinetic theory for longitudinally boost and transverse translation invariant boundary conditions. We compute the evolution of the transverse energy per unit rapidity for typical mini-jet initial conditions expected in ultra-relativistic nuclear collisions. The kinetic theory solutions under these conditions test the models severely because they deviate strongly from free-streaming and also from ideal Euler and dissipative Navier-Stokes hydrodynamical approximations. We show that the newly formulated ZPC model passes this test. In addition, we show that the initial mini-jet density would need to be approximately four times higher than estimated with the HIJING generator in central Au+Au collisions at c.m. energies 200 A GeV in order that parton cascade dynamics can be approximated by Navier-Stokes hydrodynamics.

Interstellar neutral hydrogen in the heliosphere

A fluid model is introduced to study the flow of interstellar neutral hydrogen in the heliosphere. We calculate the normalized density N H N H∞ bulk velocity V H V H∞ , and temperature T H T H∞ of interstellar hydrogen as functions of r λ and z λ where λ is the ionization scale length. The fluid solution is much less time-consuming than the kinetic theory solutions.

Comments on ‘‘Existence of kinetic theory solutions to the shock structure problem’’ [Phys. Fluids 7, 911 (1964)

H. Holway presented a proof, based on the Boltzmann equation, in which he shows that for the moment method there exists a critical Mach number beyond which no continuous shock solution is possible. In this paper, the results of Holway’s proof are reinterpreted. From this reinterpretation, it follows that no upper bound for a critical Mach number exists.

Rarefaction in Ultra-Low Flying Sliders Focusing on Analytic and Kinetic Theory Solutions

Rarefaction in Ultra-Low Flying Sliders Focusing on Analytic and Kinetic Theory Solutions

An Approach to Convergency of Kinetic Theory to Fourier Theory in Relation to Laser Heating Process

Fourier theory may be derived from analytical considerations of the heat transfer process in a substance and neglecting of higher order terms in Fourier analysis becomes important at high input power fluxes. It is apparent from the previous work that the kinetic theory equations governing the heat conduction process are more general than the Fourier equation. The special case may exist where the kinetic theory solutions converge to solutions extracted from the Fourier analysis. Consequently, research into the convergence of kinetic theory approach to Fourier analysis becomes necessary. To achieve this, the special case in which local thermal equilibrium is assumed to exist between the electrons and atoms at any section on the surface region of the solid substance is studied in the present work, providing that the study does not allow the existence of phase changes. For this purpose, an analytical approach is introduced and a computer solution of the reduced integral equations is obtained. It is found that the solutions of electron kinetic theory equations based on the assumption of existence of thermal equilibrium between electrons and atoms are identical with those obtained from Fourier theory.

A kinetic theory solution method for the Navier–Stokes equations

AbstractThe kinetic‐theory‐based solution methods for the Euler equations proposed by Pullin and Reitz are here extended to provide new finite volume numerical methods for the solution of the unsteady Navier–Stokes equations. Two approaches have been taken. In the first, the equilibrium interface method (EIM), the forward‐ and backward‐flowing molecular fluxes between two cells are assumed to come into kinetic equilibrium at the interface between the cells. Once the resulting equilibrium states at all cell interfaces are known, the evaluation of the Navier–Stokes fluxes is straightforward. In the second method, standard kinetic theory is used to evaluate the artificial dissipation terms which appear in Pullin's Euler solver. These terms are subtracted from the fluxes and the Navier–Stokes dissipative fluxes are added in. The new methods have been tested in a 1D steady flow to yield a solution for the interior structure of a shock wave and in a 2D unsteady boundary layer flow. The 1D solutions are shown to be remarkably accurate for cell sizes large compared to the length scale of the gradients in the flow and to converge to the exact solutions as the cell size is decreased. The steady‐state solutions obtained with EIM agree with those of other methods, yet require a considerably reduced computational effort.

Shock interference prediction using direct simulation Monte Carlo

The shock interaction produced when an incident shock impinges on an inlet cowl lip is investigated for a cowl lip radius of 0.1 in. and flight conditions of Mach 15 and 35-km altitude. This problem is of interest in the design of hypersonic flight vehicles because the interference heating at the cowl lip surface from such a shock interaction is expected to be very high and the design of adequate thermal protection in this region will be challenging. Noncontinuum effects may be significant at this combination of altitude and small dimensions. If this is so, a computational solution based on the Navier-Stokes equations could significantly overpredict the surface heating on the cowl lip. Therefore, a kinetic theory solution technique, direct simulation Monte Carlo (DSMC), is used to predict the flowfield and surface heating. The details of modeling this problem and some preliminary results are presented. The same qualitative flow patterns are seen with DSMC and a continuum solution of the shock interference case, but the peak heating predicted with DSMC is somewhat lower.

Convergence analysis of asymptotic solutions in kinetic theory by means of a model kinetic equation possessing an exact solution

Convergence analysis of asymptotic solutions in kinetic theory by means of a model kinetic equation possessing an exact solution

Similarity solutions in space dependent extended kinetic theory

An exact analytical solution to a simple problem of extended kinetic theory for a spatially inhomogeneous binary gas mixture is determined in this paper by resorting to Lie group theoretical methods. It is shown that there are cases in which the nonlinear evolution equations for the densities allow transformation groups of infinite order, where the corresponding similarity solution actually represents the general solution to the problem. Results are then briefly commented on.

Class of Exact Solutions in Nonlinear Kinetic Theory

A class of exact solutions of the one-dimensional steady Krooked equation is obtained. The collision frequency of the latter is left arbitrary. The solutions found depict flows widely removed from equilibrium.

Expansion Solutions and Wave Propagation in Partly Ionized Gases

Kinetic theory solutions of the Boltzmann equations for a partly ionized gas are obtained for the study of wave propagation in a three-fluid plasma. The following simplifying assumptions are made: The interactions between charged particles are divided into long-range Coulomb forces which effectively act as an external electric field, and a short-range Coulomb force which is cut off at the Debye length; the interactions between other pairs of particles are assumed to obey an inverse fifth-power law. Further, the gas is assumed to be sufficiently close to equilibrium for the usual expansions of the velocity distribution functions around the equilibrium distribution functions to be used. Under these assumptions, the general sets of coefficient equations are obtained, and a 13 and 20 coefficient approximation (which include the effects of self-collisions or viscosity and thermal conductivity) are given explicitly. For a five-coefficient approximation, corresponding to employment of the conservation equations, a dispersion relation is obtained and solved for the phase velocities and attenuation of longitudinal waves as functions of wave and collision frequencies. It is found that the solutions which correspond to waves of high phase velocity are exactly those obtained when the adiabatic law is assumed to hold. Solutions corresponding to low phase velocity waves, however, are found to be more highly attenuated than their counterparts in the adiabatic theory.