Maxwellian Equilibrium State Research Articles

Hilbert expansion of Boltzmann equations for gas mixture

The aim of this paper is to study the hydrodynamic limit of Boltzmann equations for a gas mixture containing two kinds of particles, where one is assumed to be near a vacuum and the other is near a Maxwellian equilibrium state. By the method of Hilbert expansion, we could derive the compressible Euler systems together with the terms of Boltzmann equations according to the different orders of the Knudsen number. Next, we truncate the expansion and establish the uniform estimates of the remainder term by using L 2 − L ∞ estimate.

The Vlasov–Poisson–Landau System with the Specular-Reflection Boundary Condition

We consider the Vlasov–Poisson–Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular reflection boundary condition. We provide a new improved \(L^{2}\rightarrow L^{\infty }\) framework: \(L^{2}\) energy estimate combines only with \(S_{ }^{p}\) estimate for the ultra-parabolic equation.

The Landau Equation with the Specular Reflection Boundary Condition

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.

Boltzmann systems for gas mixtures in 1D torus

We study the 1D Boltzmann equation for a mixture of two gases on a torus with the initial condition of one gas near a vacuum and the other near a Maxwellian equilibrium state. An L x ∞ L ξ , β ∞ L^{\infty }_{x}L^{\infty }_{\xi ,\beta } analysis is developed to study this mass diffusion problem, which is based on the Boltzmann equation for the single species hard sphere collision in an earlier work of the author. The decay rate of the solution is algebraic for a small time region and exponential for a large time region. Moreover, the exponential rate depends on the size of the domain.

Global well-posedness for the Fokker–Planck–Boltzmann equation in Besov–Chemin–Lerner type spaces

In this paper, motivated by [16], we use the Littlewood–Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker–Planck–Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov–Chemin–Lerner type spaces C([0,∞);L˜ξ2(B2,rs)) with 1≤r≤2 and s>3/2 or s=3/2 and r=1. Besides, we also obtain the uniform stability of the dependence on the initial data.

Exponential Runge–Kutta for the inhomogeneous Boltzmann equations with high order of accuracy

We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homogeneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi [7] a major difficulty is that the local Maxwellian equilibrium state change with respect to time and thus needs a proper numerical treatment. We show how to derive asymptotic-preserving (AP) schemes of arbitrary order, and in particular by using the Shu–Osher representation of Runge–Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.

Global classical solutions of the Boltzmann equation without angular cut-off

This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials,r−(p−1)r^{-(p-1)}withp>2p>2, for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameterss∈(0,1)s\in (0,1)andγ\gammasatisfyingγ>−n\gamma > -nin arbitrary dimensionsTn×Rn\mathbb {T}^n \times \mathbb {R}^nwithn≥2n\ge 2. Moreover, we prove rapid convergence as predicted by the celebrated BoltzmannHH-theorem. Whenγ≥−2s\gamma \ge -2s, we have exponential time decay to the Maxwellian equilibrium states. Whenγ>−2s\gamma >-2s, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only whenγ≥−2s\gamma \ge -2s, as conjectured by Mouhot and Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.

Global classical solutions of the Boltzmann equation with long-range interactions

This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r(-(p-1)) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.

The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f 0≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L 1 to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f 0 to identify the equilibrium state, both in L 1 and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L 1.

On the Solution of a Boltzmann System for Gas Mixtures

We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green’s function of the Boltzmann equation for the single species hard sphere collision model in Liu andYu (Commun Pure Appl Math 57:1543–1608, 2004). The cross-species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.

Core‐Halo Distribution Functions: A Natural Equilibrium State in Generalized Thermostatistics

Space observations provide a clear signature of non-Maxwellian core-halo electron and ion-velocity distribution functions as an ubiquitous and persistent feature of astrophysical plasma environments. In particular, the detected suprathermal halo populations are accurately represented by the family of κ-distributions, a power law in particle speed. Contrary to observations, the physical relevance of κ-distributions has frequently been criticized because of a lack of theoretical justification. We show that these distributions turn out as consequence of an entropy generalization based on nonextensive thermostatistics, thus providing the missing link for power-law models from fundamental physics. Moreover, upon clarifying that the full nonextensive formalism is also compatible with negative values of the spectral index κ, we demonstrate that core-halo structures are a natural element within pseudoadditive entropy. As a consequence, pronounced core-halo patterns can be generated adiabatically, following density conservation out of a Maxwellian equilibrium state. The physical significance of the proposed nonextensive double-κ distribution family is analyzed with regard to nonthermal entropy evolution and tested on typical core-halo characteristics of observed interplanetary electron and ion velocity space structures, originating as a natural equilibrium state within the generalized entropy concept. The peak separation scale of interplanetary double-humped proton distributions is found to obey a maximum entropy condition.

Numerical Solution of an Ionic Fokker--Planck Equation with Electronic Temperature

We describe a numerical scheme for dealing with an ion/electron collision operator of the Fokker--Planck type; for that purpose, we introduce the notion of the entropic average of two positive quantities. This scheme has the property to be entropic in the sense of Boltzmann's H-theorem under a CFL criteria. Moreover, we prove that the solution of the semidiscrete scheme converges towards a unique Maxwellian equilibrium state when the time grows. Numerical applications are given and show that our scheme is more precise than the classical Chang--Cooper one.

Résolution numérique d'une équation de Fokker-Planck ionique avec température électronique

We describe a numerical scheme for dealing with an ion electron/collision operator of the Fokker-Planck type; for that purpose, we introduce the notion of entropic average of two positive quantities. This scheme has the property to be entropic in the sense of Boltzmann 's H theorem under a CFL criteria. Moreover, we prove that the solution converges when the time grows, towards a unique Maxwellian equilibrium state.

The linear pressure dependence of the viscosity at high densities

Over the years, experimental data of the viscosity coefficient of simple dense fluids at high pressures have been obtained both in our laboratory and elsewhere. These data show that, at high densities, i.e. roughly in the last third of the density range from vacuum to the melting transition, for constant temperature the viscosity coefficient is a linear function of the pressure in a few different but neighbouring ranges, each with different slope and intersection with the pressure axis. The slope appears to be a time of picoseconds. The intersection with the pressure axis can be considered to be a value for the internal pressure. Thus, the viscosity coefficient is directly proportional to the thermal pressure, i.e. the combination of the experimental pressure and the internal pressure, in these ranges. It is shown that such a linear pressure dependence of the viscosity occurs in large ranges of the liquid state and at high densities in the fluid state of simple molecules. We subsequently call attention to old ideas of Maxwell on the equivalence of elasticity and viscosity and the modern basis of it given by Zwanzig and Mountain. This concept leads to Maxwell’s relaxation time. On account thereof, the slope of the linear viscosity–pressure relation is interpreted as the relaxation time, i.e. the time in which a disturbance to the Maxwellian equilibrium state decreases to 1/e of its original value, and as the ideal collision interval. The similarity of the present solution and the case of electrical conductivity in an n-type semiconductor is also pointed out. This leads to a simple view on the transport mechanism and gives reason to consider the temperature dependence of the relaxation times.

THE SEMICONTINUOUS BOLTZMANN EQUATION FOR GAS MIXTURES

This paper proposes a semicontinuous model of the Boltzmann equation to describe the behavior of a mixture of P gases. The semicontinuous Boltzmann equation is a model in which gas particles can move in the plane in all possible directions, but with a finite number of velocity moduli. The model consists of a system of integro-differential equations with one-fold integrals over a suitable angular variable. Thermodynamic equilibrium is studied in details showing that the Maxwellian equilibrium states are uniquely defined in terms of the masses of the components of the mixtures and of the overall momentum and energy.

Exact (2+1)-dimensional solutions for two discrete velocity Boltzmann models with four independent densities

For the square discrete velocity Boltzmann model the authors obtain exact (2+1)-dimensional (space x, y, time t) solutions. These are rational solutions, with three exponential variables, relaxing towards non-uniform Maxwellian equilibrium states with two exponential variables. These are also solutions of another model with four independent densities and eight velocities oriented towards the eight corners of a cube. The positivity problem for the densities is non-trivial.

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.

Electron and Ion Auroral Velocity Distribution Functions Incoherent Scattering of Radar Waves

The electron and the ion velocity distribution functions are studied for the F region of the auroral ionosphere. For large electric field strengths the electrons reach the local Maxwellian equilibrium state and the ions a stationary state very distorted with respect to the Maxwellian shape. Effects that non-Maxwellian ion velocity distribution functions produce on the ionic part of the spectrum of radar waves incoherently scattered are presented.

General solution of a Boltzmann equation, and the formation of Maxwellian tails

A classical Boltzmann equation is studied. The equation describes the evolution towards the Maxwellian equilibrium state of a homogeneous, isotropic gas where the collision cross section is inversely proportional to the relative velocity of the colliding particles. After Tjon & Wu (1979), the problem is transformed into a mathematically equivalent one, itself a model Boltzmann equation in two dimensions. Working in the context of the latter equation, a formal derivation of the general solution is presented. First a countable ensemble of particular solutions, called pure solutions , is constructed. From these, via a non-linear combination mechanism, the general solution is obtained in a form appropriate for direct numerical computation. The validity of the solution depends upon its containment in a well defined Hilbert space H~ Given that the initial condition lies within H~ it is proved that at least for a small finite time interval it remains in H~.

Structure of tangential discontinuities at the magnetopause: the nose of the magnetopause

Observations show that the magnetopause can sometimes be represented as a tangential discontinuity in the magnetic field. In this paper, we use a theoretical model of steady-state tangential discontinuities to analyze the microscale structure of the nose of the magnetopause. The boundary layer is described in terms of a kinetic theory based on the Vlasov-Maxwell equations for the charged particles and electromagnetic fields. The general model represents the magnetosheath and magnetospheric sides as distinct regions with anisotropic displaced Maxwellian equilibrium states and allows the presence of a multi-component plasma whose ionic species have different concentrations, temperatures, anisotropies, etc. For the nose region, we assume for simplicity an isotropic Maxwellian Hydrogen plasma. Transition profiles for the magnetic field, electric potential, electric field, charge separation and density are illustrated. The variations of the magnetic field direction in the discontinuity plane show a great variety of rotational structures in accordance with recent observations. The electric potential difference is seen to control the thickness of the magnetopause. The classical Ferraro boundary layer where the ion current is neglected is recovered when the ion distribution function is a Maxwellian everywhere in the transition. When there is no electric potential difference across the sheath we find again an ‘electron-dominated’ boundary layer where the electric current is mainly carried by the electrons. However, computations of drift and thermal velocities indicate that a two-stream instability occurs in the centre of these sheaths. The result will be a broadening of the layer by wave-particle interactions leading to an ‘ion-dominated’ transition.