One-dimensional Boltzmann Equation Research Articles

Evolution of the Secondary Electron Spectrum during Cosmic-Ray Discharge in the Universe

We recently found that streaming cosmic rays (CRs) induce a resistive electric field that can accelerate secondary electrons produced by CR ionization. In this work, we study the evolution of the energy spectrum of secondary electrons by numerically solving the one-dimensional Boltzmann equation and Ohm’s law. We show that the accelerated secondary electrons further ionize a gas, that is, an electron avalanche occurs, resulting in increased ionization and excitation of the gas. Although the resistive electric field becomes weaker than the one before the CR discharge, the weak resistive electric field weakly accelerates the secondary electrons. The quasi-steady state is almost independent of the initial resistive electric field but depends on the electron fraction in the gas. The resistive electric field in the quasi-steady state is larger for the higher electron fraction, which makes the number of secondary electrons that can ionize the gas larger, resulting in a higher ionization rate. The CR discharge could explain the high ionization rate that is observed in some molecular clouds.

Why are hydrodynamic theories applicable beyond the hydrodynamic regime?

We present an alternative approach to deriving second-order nonconformal hydrodynamics from the relativistic Boltzmann equation. We demonstrate how constitutive relations for shear and bulk stresses can be transformed into dynamical evolution equations, resulting in Israel-Stewart-like (ISL) hydrodynamics. To understand the far-from-equilibrium applicability of such ISL theories, we investigate the one-dimensional boost-invariant Boltzmann equation using special moments of the distribution function for a system with finite particle mass. Our analysis reveals that the mathematical structure of the ISL equations is akin to that of moment equations, enabling them to approximately replicate even the collisionless dynamics. We conclude that this particular feature is important in extending the applicability of ISL theories beyond the hydrodynamic regime.

From moments of the distribution function to hydrodynamics: The nonconformal case

We study the one-dimensional boost-invariant Boltzmann equation in the relaxation-time approximation using special moments of the distribution function for a system with a finite particle mass. The infinite hierarchy of moments can be truncated by keeping only the three lowest moments that correspond to the three independent components of the energy-momentum tensor. We show that such a three-moment truncation reproduces accurately the exact solution of the kinetic equation after a simple renormalization that takes into account the effects of the neglected higher moments. We derive second-order Israel-Stewart hydrodynamic equations from the three-moment equations, and show that, for most physically relevant initial conditions, these equations yield results comparable to those of the three-moment truncation, albeit less accurate. We attribute this feature to the fact that the structure of Israel-Stewart equations is similar to that of the three-moment truncation. In particular, the presence of the relaxation term in the Israel-Stewart equations, yields an early-time regime that mimics approximately the collisionless regime. A detailed comparison of the three-moment truncation with second-order non-conformal hydrodynamics reveals ambiguities in the definition of second-order transport coefficients. These ambiguities affect the ability of Israel-Stewart hydrodynamics to reproduce results of kinetic theory.

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

Stability Analysis of Quadrature-Based Moment Methods for Kinetic Equations

In this paper, we give a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent literature, the method has revealed its potential for modeling nonequilibrium flows, while a thorough theoretical analysis is largely missing but desirable. We show that the method can yield nonhyperbolic moment systems if the distribution function is approximated by a linear combination of $\delta$-functions. On the other hand, if the $\delta$-functions are replaced by their Gaussian approximations with a common variance, we prove that the moment systems are strictly hyperbolic and preserve the dissipation property (or $H$-theorem) of the kinetic equation. In the proof, we also determine the equilibrium manifold that lies on the boundary of the state space. The proofs are quite technical and involve detailed analyses of the characteristic polynomials of the coefficient matrices.

Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this work we consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation. For any given solution belonging to weighted Sobolev space, we will show it enjoys at positive time the Gelfand-Shilov smoothing effect for the velocity variable and Gevrey regularizing properties for the spatial variable. This improves the result of Lerner-Morimoto-Pravda-Starov-Xu [J. Funct. Anal. 269 (2015) 459-535] on one-dimensional Boltzmann equation to the physical three-dimensional case. Our proof relies on the elementary \begin{document}$ L^2 $\end{document} weighted estimate.

Group Analysis of the One-Dimensional Boltzmann Equation: IV. Complete Group Classification in the General Case

We consider the one-dimensional Boltzmann equation $$f_t+cf_x+(\mathcal{F}f)_c=0$$ with a function $$\mathcal{F}$$ depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and $$dc=\mathcal{F}dt$$, (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them.

Macroscopic Boundary Conditions on a Solid Surface in Rarefied Gas Flow for a One-Dimensional Nonlinear Nonstationary 12-Moment System of Boltzmann Equations

Boundary conditions for a one-dimensional nonlinear nonstationary system of Boltzmann equations are formulated in the fifth approximation. The Maxwell microscopic boundary conditions are approximated in the case of the one-dimensional Boltzmann equation when some of the molecules reflect specularly from the surface, while the others reflect diffusely with Maxwell’s distribution. An initial-boundary value problem for the 12-moment system of Boltzmann equations with Maxwell–Auzhani boundary conditions is stated. For the 12-moment system of Boltzmann equations, six boundary conditions are set at the left and right endpoints of the interval ( $$ - a$$ , $$a$$ ).

Calculation of nonlocal EDF using a one-dimensional Boltzmann equation solver

A method for the calculation of the nonlocal electron distribution function (EDF) with programs commonly used for solving the one-dimensional Boltzmann kinetic equation (e.g., COMSOL Multiphysics) was proposed and implemented. The capabilities of the proposed method were illustrated using the example of the positive column plasma in argon. Significant differences between the local and nonlocal EDFs were observed, especially at the plasma periphery. This can result in significant differences in the electron transport coefficients and reaction rate constants for electron-induced processes. The proposed approach increases the precision of the numerical modeling of gas discharge devices and equipment.

Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation

In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful

Проведена групповая классификация одномерного уравнения Больцмана относительно функции $\mathcal F=\mathcal F(t,x,c)$, характеризующей внешнее силовое поле, в предположении о наличии следующих вытекающих из физических соображений связей между переменными: $dx=c dt$, $dc=\mathcal F dt$, $dt=0$, $dx=0$. Показано, что для всех функций $\mathcal F$ алгебра конечномерна и ее максимальная размерность равна восьми, что соответствует уравнению с нулевой $\mathcal F$.

Group Analysis of the One-Dimensional Boltzmann Equation: III. Condition for the Moment Quantities to Be Physically Meaningful

We present the group classification of the one-dimensional Boltzmann equation with respect to the function F = F(t, x, c) characterizing an external force field under the assumption that the physically meaningful constraints dx = c dt, dc = F dt, dt = 0, and dx =0 are imposed on the variables. We show that for all functions F, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero F.

One-Dimensional Dissipative Boltzmann Equation: Measure Solutions, Cooling Rate, and Self-Similar Profile

This manuscript investigates the following aspects of the one-dimensional dissipative Boltzmann equation associated to a variable hard-spheres kernel: (1) we show the optimal cooling rate of the model by a careful study of the system satisfied by the solution's moments, (2) we give existence and uniqueness of measure solutions, and (3) we prove the existence of a nontrivial self-similar profile, i.e., homogeneous cooling state, after appropriate scaling of the equation. The latter issue is based on compactness tools in the set of Borel measures. More specifically, we apply a dynamical fixed point theorem on a suitable stable set, for the model dynamics, of Borel measures.

Globally Hyperbolic Moment Model of Arbitrary Order for One-Dimensional Special Relativistic Boltzmann Equation

This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors,hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system can converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.

Exact Stationary and Non-stationary Solutions to Inelastic Maxwell Model with Infinite Energy

The one-dimensional inelastic Boltzmann equation with a constant collision rate (the Maxwell model) is considered. It is shown that for special values of restitution parameter there exists a stationary solution with the characteristic function in the form \(e^{-P(\log (z))z},\) where P is a periodic function. The corresponding distribution function belongs to a one special class of stochastic processes termed as a generalized stable in the probability theory. The Fourier transform of the non-stationary equation has the solution \(\bigl (1+P(\log (z))z\bigr )e^{-Q(\log (z))z}\). It is proved that this solution is a characteristic function if periodic functions P, Q satisfy some not very restrictive conditions. The stationary and non-stationary solutions correspond to a gas with infinite temperature.

Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions

We consider the one-dimensional integro-differential Boltzmann equation for Maxwell particles with inelastic collisions. We show that the equation has a five-dimensional algebra of point symmetries for all dissipation parameter values and obtain an optimal system of one-dimensional subalgebras and classes of invariant solutions.

Stability of Superposition of Two Viscous Shock Waves for the Boltzmann Equation

In this paper we study the time-asymptotic stability of a superposition of two viscous shock waves for one-dimensional Boltzmann equation in Eulerian coordinate under the general initial perturbation without the zero total macroscopic mass condition. Roughly speaking, the general perturbation without the zero mass condition will generate not only the shifts on the two viscous shock waves, but also the linear diffusion wave in the linearly degenerate field. Furthermore, the coupled diffusion waves for the macroscopic part are also crucial in the stability analysis. By using the delicate weighted characteristic energy estimates based on the underlying wave structure, we succeed in proving the time-asymptotic stability of a superposition of two viscous shock waves for Boltzmann equation in Eulerian coordinate without the zero initial mass condition.

Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas.

A kinetic theory describing the motion of an impurity particle in a degenerate Tonks-Girardeau gas is presented. The theory is based on the one-dimensional Boltzmann equation. An iterative procedure for solving this equation is proposed, leading to the exact solution in a number of special cases and to an approximate solution with the explicitly specified precision in a general case. Previously we reported that the impurity reaches a nonthermal steady state, characterized by an impurity momentum p(∞) depending on its initial momentum p(0) [E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy, Phys. Rev. A 89, 041601(R) (2014)]. In the present paper the detailed derivation of p(∞)(p(0)) is provided. We also study the motion of an impurity under the action of a constant force F. It is demonstrated that if the impurity is heavier than the host particles, m(i)>m(h), damped oscillations of the impurity momentum develop, while in the opposite case, m(i)<m(h), oscillations are absent. The steady-state momentum as a function of the applied force is determined. In the limit of weak force it is found to be force independent for a light impurity and proportional to √[F] for a heavy impurity.

Plane Poiseuille flow and thermal transpiration of a highly rarefied gas between the two walls of Maxwell-type boundaries with different accommodation coefficients: effect of a weak external force

Plane Poiseuille flow and the thermal transpiration of a highly rarefied gas between the two walls of Maxwell-type boundaries with different accommodation coefficients are studied based on kinetic theory when the gas is subject to a weak external force perpendicular to the walls. The flow behavior is studied numerically based on the spatially one-dimensional Boltzmann equation for a hard-sphere gas derived by the asymptotic analysis for a slow variation in the longitudinal direction. Due to the effect of a weak external force, there is an appreciable difference in the mass-flow rate between the flows in which the two walls are interchanged when the mean free path is sufficiently large. If both of the accommodation coefficients are close to unity, the mass-flow rate is reduced due to the effect of the external force. In contrast, if the accommodation coefficient of one wall is considerably smaller than unity, the mass-flow rate of the gas can be enhanced when this wall is placed on the side to which the external force points.

Simple kinetic-theory tests of dissipative and anisotropic hydrodynamics

We solve the one-dimensional boost-invariant Boltzmann equation in the re- laxation time approximation and compare the results with the predictions of dissipative and anisotropic hydrodynamics. We observe that recent formulations of second-order vis- cous hydrodynamics agree better with the exact solutions than the standard Israel-Stewart approach. In addition, we find that the anisotropic hydrodynamics gives a very good ap- proximation to the exact results provided the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times is introduced.