Collisionless Gas Research Articles

Kinetic gases in static spherically symmetric modified dispersion relations

We study the dynamics of a collisionless kinetic gas in the most general static, spherically symmetric dispersion relation. For a static, spherically symmetric kinetic gas, we derive the most general solution to these dynamics, and find that any solution is given by a one-particle distribution function which depends on three variables. For two particular solutions, describing a shell of monoenergetic orbiting particles and a purely radial inflow, we calculate the particle density as a function of the radial coordinate. As a particular example, we study a κ-Poincaré modification of the Schwarzschild metric dispersion relation and derive its influence on the particle density. Our results provide a possible route towards quantum gravity phenomenology via the observation of matter dynamics in the vicinity of massive compact objects.

Axisymmetric, stationary collisionless gas clouds trapped in a Newtonian potential

The properties of an axisymmetric, stationary gas cloud surrounding a massive central object are discussed. It is assumed that the gravitational field is dominated by the central object which is modeled by a nonrelativistic rotationally-symmetric potential. Further, we assume that the gas consists of collisionless, identical massive particles that follow bound orbits in this potential. Several models for the one-particle distribution function are considered and the essential formulae that describe the relevant macroscopical observables, such as the particle and energy densities, pressure tensor, and the kinetic temperature are derived. The asymptotic decay of the solutions at infinity is discussed and we specify configurations with finite total mass, energy and (zero or non-zero) angular momentum. Finally, our configurations are compared to their hydrodynamic analogs. In an accompanying paper, the equivalent general relativistic problem is discussed, where the central object consists of a Schwarzschild black hole.

Axisymmetric, stationary collisionless gas configurations surrounding Schwarzschild black holes

The properties of a stationary gas cloud surrounding a black hole are discussed, assuming that the gas consists of collisionless, identical massive particles that follow spatially bound geodesic orbits in the Schwarzschild spacetime. Several models for the one-particle distribution function are considered, and the essential formulae that describe the relevant macroscopic observables, like the current density four-vector and the stress–energy–momentum tensor are derived. This is achieved by rewriting these observables as integrals over the constants of motion and by a careful analysis of the range of integration. In particular, we provide configurations with finite total mass and angular momentum. Differences between these configurations and their nonrelativistic counterparts in a Newtonian potential are analyzed. Finally, our configurations are compared to their hydrodynamic analogues, the ‘polish doughnuts’.

A coupling approach for the convergence to equilibrium for a collisionless gas

A coupling approach for the convergence to equilibrium for a collisionless gas

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

Convergence toward the steady state of a collisionless gas with Cercignani–Lampis boundary condition

We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani–Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence toward it, in the L 1 norm. The strategy is an application of a deterministic version of Harris’ subgeometric theorem, in the spirit of the recent results of Cañizo-Mischler and of the previous study of Bernou. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani–Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani–Lampis boundary condition in this problem.

Accretion of a Vlasov gas onto a black hole from a sphere of finite radius and the role of angular momentum

The accretion of a spherically symmetric, collisionless kinetic gas cloud on to a Schwarzschild black hole is analysed. Whereas previous studies have treated this problem by specifying boundary conditions at infinity, here the properties of the gas are given at a sphere of finite radius. The corresponding steady-state solutions are computed using four different models with an increasing level of sophistication, starting with the purely radial infall of Newtonian particles and culminating with a fully general relativistic calculation in which individual particles have angular momentum. The resulting mass accretion rates are analysed and compared with previous models, including the standard Bondi model for a hydrodynamic flow. We apply our models to the supermassive black holes Sgr A* and M87*, and we discuss how their low luminosity could be partially explained by a kinetic description involving angular momentum. Furthermore, we get results consistent with previous model-dependent bounds for the accretion rate imposed by rotation measures of the polarised light coming from Sgr A* and with estimations of the accretion rate of M87* from the Event Horizon Telescope collaboration. Our methods and results could serve as a first approximation for more realistic black hole accretion models in various astrophysical scenarios in which the accreted material is expected to be nearly collisionless.

Accretion of the relativistic Vlasov gas onto a moving Schwarzschild black hole: Exact solutions

We derive an exact, axially symmetric solution representing stationary accretion of the relativistic, collisionless Vlasov gas onto a moving Schwarzschild black hole. The gas is assumed to be in thermal equilibrium at infinity, where it obeys the Maxwell-J\"{u}ttner distribution. The Vlasov equation is solved analytically in terms of suitable action-angle variables. We provide explicit expressions for the particle current density and accretion rates. In the limit of infinite asymptotic temperature of the gas, we recover the qualitative picture known form the relativistic Bondi-Hoyle-Lyttleton accretion of the perfect gas with the ultra-hard equation of state, in which the mass accretion is proportional to the Lorentz factor associated with the black-hole velocity. For a finite asymptotic temperature, the mass accretion rate is not in general a monotonic function of the velocity of the black hole.

Phase space mixing in an external gravitational central potential

This article is devoted to the study of the dynamical behavior of a collisionless kinetic gas in d = 1, 2, 3 space dimensions which is trapped in a rotationally symmetric potential well. Although at the microscopic level the trajectories of individual gas particles are quasi-periodic and characterized by their d fundamental frequencies, at the macroscopic level the gas relaxes in time to a stationary state, provided the potential satisfies a certain non-degeneracy condition. In this article, we provide a mathematically precise formulation for this relaxation process which is due to phase space mixing. In particular, we prove that a physically relevant class of macroscopic observables computed from the one-particle distribution function, such as particle and energy densities, pressure and stress tensors, converge in time to the corresponding observables associated with an averaged distribution function. The latter can be determined from the initial datum and depends only on integrals of motion. Thus, the final state of the gas is described by an effective distribution function depending only on integrals of motion, which considerably reduces the degrees of freedom of the gas configuration. We discuss some applications to gravitational physics, including the propagation of a collisionless gas in typical potentials arising in stellar dynamics and the modeling of dark matter halos, and we also generalize our results to a relativistic gas whose individual particles follow bound timelike trajectories in the exterior region of a static, spherically symmetric black hole spacetime.

Boltzmann's H-theorem, entropy and the strength of gravity in theories with a nonminimal coupling between matter and geometry

In this paper we demonstrate that Boltzmann's H-theorem does not necessarily hold in the context of theories of gravity with nonminimally coupled matter fields. We find sufficient conditions for the violation of Boltzmann's H-theorem and derive an expression for the evolution of Boltzmann's H in terms of the nonminimal coupling function, valid in the case of a collisionless gas in a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker universe. We highlight the implications of this result for the evolution of the entropy of the matter fields, briefly discussing the role played by collisions between particles whenever they are relevant. We also suggest a possible link between the high entropy of the Universe and the weakness of gravity in the context of these theories.

A semigroup approach to the convergence rate of a collisionless gas

We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension $n \in \{2,3\}$. By semigroup arguments, we prove that in the $L^1$ norm, the polynomial rate of convergence $\frac{1}{(t+1)^{n-}}$ given by Tsuji et al [2010] and Kuo et al [2013,2014,2015] can be extended to any $C^2$ domain, with standard assumptions on the initial data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependency of the rate with respect to the initial distribution is detailed. Our study includes the case where the temperature at the boundary varies. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Ca\~nizo and Mischler. We also compare our model with a free-transport equation with absorbing boundary.

Неизотермическое течение газа в эллиптическом канале с внутренним круговым цилиндрическим элементом в свободномолекулярном режиме

AbstractThe linearized problem of free-molecular gas flow in a long elliptic channel with a circular cylindrical element inside has been solved in the kinetic approximation. The flow in the channel is caused by temperature and pressure drops between its ends. The Boltzmann kinetic equation for collisionless gas has been used as a basic equation, and the boundary condition has been set in terms of the diffuse reflection model. The distribution of the mass velocity of the gas over the cross section of the channel has been obtained. The mass flow rate of the gas in the channel versus the temperature and pressure drops between its ends has been calculated. It has been found that the mass flow of the gas substantially depends on the radius of the inner cylinder.

Calculation of Gas Flow Between Two Coaxial Cylinders in a Free-Molecular Regime Under Specular-Diffuse Boundary Conditions

Using the method of characteristics, within the framework of the kinetic approach, an analytical solution of the problem on gas heat and mass transfer in a long channel formed by two coaxial cylinders in a free-molecular regime has been constructed. As the basic equation describing the kinetics of the process, use is made of the Boltzmann kinetic equation for a collisionless gas and as a boundary condition of the channel walls — a model of specular-diffuse reflection. The profile of the heat flux vector in the channel has been constructed, and the gas heat and mass flows through the channel cross section have been calculated as functions of the accommodation coefficients of the gas molecule pulses tangent to the surfaces of the cylinders.

Nonisothermal Free-Molecular Flow of Gas in an Elliptic Channel with a Circular Cylindrical Element Inside

The linearized problem of free-molecular gas flow in a long elliptic channel with a circular cylindrical element inside has been solved in the kinetic approximation. The flow in the channel is caused by temperature and pressure drops between its ends. The Boltzmann kinetic equation for collisionless gas has been used as a basic equation, and the boundary condition has been set in terms of the diffuse reflection model. The distribution of the mass velocity of the gas over the cross section of the channel has been obtained. The mass flow rate of the gas in the channel versus the temperature and pressure drops between its ends has been calculated. It has been found that the mass flow of the gas substantially depends on the radius of the inner cylinder.

Collisionless dynamics in two-dimensional bosonic gases

We study the dynamics of dilute and ultracold bosonic gases in a quasi two-dimensional (2D) configuration and in the collisionless regime. We adopt the 2D Landau-Vlasov equation to describe a three-dimensional gas under very strong harmonic confinement along one direction. We use this effective equation to investigate the speed of sound in quasi 2D bosonic gases, i.e. the sound propagation around a Bose-Einstein distribution in collisionless 2D gases. We derive coupled algebraic equations for the real and imaginary parts of the sound velocity, which are then solved taking also into account the equation of state of the 2D bosonic system. Above the Berezinskii-Kosterlitz-Thouless critical temperature we find that there is rapid growth of the imaginary component of the sound velocity which implies a strong Landau damping. Quite remarkably, our theoretical results are in good agreement with very recent experimental data obtained with a uniform 2D Bose gas of $^{87}$Rb atoms.

Collective modes of an imbalanced unitary Fermi gas

We study theoretically the collective mode spectrum of a strongly imbalanced two-component unitary Fermi gas in a cigar-shaped trap, where the minority species forms a gas of polarons. We describe the collective breathing mode of the gas in terms of the Fermi liquid kinetic equation taking collisions into account using the method of moments. Our results for the frequency and damping of the longitudinal in-phase breathing mode are in good quantitative agreement with an experiment by Nascimb\`ene et al. [Phys. Rev. Lett. 103, 170402 (2009)] and interpolate between a hydrodynamic and a collisionless regime as the polarization is increased. A separate out-of phase breathing mode, which for a collisionless gas is sensitive to the effective mass of the polaron, however, is strongly damped at finite temperature, whereas the experiment observes a well-defined oscillation.

Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole

We provide a systematic study for the accretion of a collisionless, relativistic kinetic gas into a nonrotating black hole. To this end, we first solve the relativistic Liouville equation on a Schwarzschild background spacetime. The most general solution for the distribution function is given in terms of appropriate symplectic coordinates on the cotangent bundle, and the associated observables, including the particle current density and stress energy-momentum tensor, are determined. Next, we explore the case where the flow is steady-state and spherically symmetric. Assuming that in the asymptotic region the gas is described by an equilibrium distribution function, we determine the relevant parameters of the accretion flow as a function of the particle density and the temperature of the gas at infinity. In particular, we find that in the low temperature limit the tangential pressure at the horizon is about an order of magnitude larger than the radial one, showing explicitly that a collisionless gas, despite exerting kinetic pressure, behaves very differently than an isotropic perfect fluid, and providing a partial explanation for the known fact that the accretion rate is much lower than in the hydrodynamic case of Bondi–Michel accretion. Finally, we establish the asymptotic stability of the steady-state spherical flows by proving pointwise convergence results which show that a large class of (possibly nonstationary and nonspherical) initial conditions for the distribution function lead to solutions of the Liouville equation which relax in time to a steady-state, spherically symmetric configuration.

Spherical steady-state accretion of a relativistic collisionless gas into a Schwarzschild black hole

In previous work, we derived the most general solution of the collisionless Boltzmann equation describing the accretion of a kinetic gas into a Schwarzschild black hole background, and we gave explicit expressions for the corresponding observables (the current density and stress energy-momentum tensor) in terms of certain integrals over the distribution function. In this article, we numerically compute these integrals for the particular case of the steady-state, spherical symmetric accretion flows which, at infinity, are described by an equilibrium distribution function of given temperature. We analyze in detail the behavior of the observables as a function of the temperature and the radial coordinate, comparing our results with the perfect fluid model of Bondi-Michel accretion.

The dark matter distribution function and halo thermalization from the Eddington equation in galaxies

We find the distribution function [Formula: see text] for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with warm dark matter (WDM). With these methods we find: (i) Cored density profiles behaving quadratically for small distances [Formula: see text] produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for [Formula: see text] where [Formula: see text] is the halo radius. (iii) Analytic expressions for the dispersion velocity and the pressure are derived yielding at each halo point an ideal DM gas equation of state with local temperature [Formula: see text]. [Formula: see text] turns out to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature [Formula: see text] for [Formula: see text], (b) a space dependent temperature [Formula: see text] for [Formula: see text], which slowly decreases with [Formula: see text]. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for [Formula: see text]. (v) [Formula: see text] outside the halo radius nicely follows the decrease of the circular velocity squared.

Detailed flowfield and surface properties for high Knudsen number planar jet impingement at an inclined flat plate

This paper presents two sets of analytical exact solutions for collisionless gas flows from a planar exit, impinging at an inclined flat plate. These analytical results are obtained by using gaskinetic theories. The first set of solutions are for a diffuse reflective plate surface, and the other set of solutions are for a specular reflective plate surface. A virtual nozzle exit is adopted to aid analyzing the specular reflective plate scenario. New formulas for plate surface properties, including velocity slips, pressure, shear stress, and heat flux distributions, are provided. For both problems, the flowfield exact solutions are investigated as well. Numerical simulations with the direct simulation Monte Carlo method are performed to validate these new analytical results, and good agreement is obtained for flows with high Knudsen numbers. The results consider effects from many factors, such as the plate inclination angle, geometry ratios, and exit gas and plate properties (such as exit gas bulk density, gas speed ratio, and exit gas and plate temperatures). Compared with past work, these new solutions are more comprehensive and practical. The results also illustrate that if the plate is quite close to the nozzle exit, it is improper to adopt the traditional treatments of a point source and a simple cosine function.