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Relativistic Kinetic Theory: With Applications in Astrophysics and Cosmology
Abstract
Relativistic kinetic theory has widespread application in astrophysics and cosmology. The interest has grown in recent years as experimentalists are now able to make reliable measurements on physical systems where relativistic effects are no longer negligible. This ambitious monograph is divided into three parts. It presents the basic ideas and concepts of this theory, equations and methods, including derivation of kinetic equations from the relativistic BBGKY hierarchy and discussion of the relation between kinetic and hydrodynamic levels of description. The second part introduces elements of computational physics with special emphasis on numerical integration of Boltzmann equations and related approaches, as well as multi-component hydrodynamics. The third part presents an overview of applications ranging from covariant theory of plasma response, thermalization of relativistic plasma, comptonization in static and moving media to kinetics of self-gravitating systems, cosmological structure formation and neutrino emission during the gravitational collapse.
... First let us discuss the physical conditions for quantum degeneracy of plasma. The degree of plasma degeneracy is characterized by the parameter [46,47] ...
... In this paper, we assume the plasma is homogeneous in space, thus transport terms are dropped from the kinetic equations. We will adopt the radiative transport form of kinetic equations of electrons e − , positrons e + and photons γ; see [47]. ...
... These integrals contain the matrix elements of the corresponding QED processes. For binary interactions, the matrix elements are given in most textbooks; see [47,52]. For triple interactions, these matrix elements can be found in the literature: Ref. [53] for double Compton scattering, Ref. [54] for relativistic bremsstrahlung. ...
Relativistic plasma can be formed in strong electromagnetic or gravitational fields. Such conditions exist in compact astrophysical objects, such as white dwarfs and neutron stars, as well as in accretion discs around neutron stars and black holes. Relativistic plasma may also be produced in the laboratory during interactions of ultra-intense lasers with solid targets or laser beams between themselves. The process of thermalization in relativistic plasma can be affected by quantum degeneracy, as reaction rates are either suppressed by Pauli blocking or intensified by Bose enhancement. In addition, specific quantum phenomena, such as Bose–Einstein condensation, may occur in such a plasma. In this review, the process of plasma thermalization is discussed and illustrated with several examples. The conditions for quantum condensation of photons are formulated. Similarly, the conditions for thermalization delay due to the quantum degeneracy of fermions are analyzed. Finally, the process of formation of such relativistic plasma originating from an overcritical electric field is discussed. All these results are relevant for relativistic astrophysics as well as for laboratory experiments with ultra-intense lasers.
... Multicomponent hydrodynamic code [44][45][46] operates with independent variables: concentration densities n i = ρ/m p , internal energies i of components i, momentum density ρv (the velocities of all components of matter are assumed to be the same). One has a system of Euler equations on a fixed coordinate system. ...
... The computational method uses splitting with respect to the physical processes. The hydrodynamic transport is based on an explicit conservative scheme and an original Riemann problem solver for a multicomponent gas mixture with a tabular EOS [44,45] (see also Appendix A). The processes of energy exchange between the components (matter and neutrinos) are considered at a separate time step using the implicit Gear's method [50]. ...
... The processes of energy exchange between the components (matter and neutrinos) are considered at a separate time step using the implicit Gear's method [50]. Moreover, the neutrino diffusion term is considered by lines method [45] and in implicit Gear's method. Key point of the method is joint consideration of matter and neutrinos in conservative hydrodynamic equations. ...
The self-consistent problem of gravitational collapse is solved using 2D gas dynamics with taking into account the neutrino transfer in the flux-limited diffusion approximation. Neutrino are described by spectral energy density, and weak interaction includes a simplified physical model of neutrino interactions with nucleons. I investigate convection on the stage of the collapse and then in the center of the core, where the unstable entropy profile was probably formed. It is shown that convection has large scale. Convection appears only in the semitransparent region near the neutrinosphere due to non-equilibrium nonreversible neutronization. Convection increases the energy of emitted neutrino up to 15÷18 MeV. The obtained neutrino spectrum is important for the registration of low-energy neutrinos from a supernova.
... At linear level, the property of the fluid system is totally determined by a set of phenomenological transport coefficients which can be measured by experiment and can be determined by calculations in the underlying microscopic theory. Although there are multiple approaches with different domain of validity to calculate the transport coefficients, in a curved spacetime background the relativistic kinetic theory turns out to be the most efficient and convenient option with many important applications ranging from stability theory [1][2][3], astrophysics [4,5] to cosmology [6][7][8]. ...
Under relaxation time approximation, we obtain an iterative solution to the relativistic Boltzmann equation in generic stationary spacetime. This solution provides a scheme to study non-equilibrium system order by order. As a specific example, we analytically calculated the covariant expressions of the particle flow and the energy momentum tensor up to the first order in relaxation time. Finally and most importantly, we present all 14 kinetic coefficients for a neutral system, which are verified to satisfy the Onsager reciprocal relation and guarantee a non-negative entropy production.
... Astrophysical plasmas have been widely studied through different theories, simulations, and observations based on plasma physics knowledge. In this case, we have the chance but also face the major challenge of studying plasma dynamics in various contexts, from kinetic processes conditioned by the energy (velocity) distributions of plasma particles to the macro-physics of a hydrodynamic plasma; or from high energy fully relativistic manifestations of plasmas in quasars and AGN jets to non-relativistic solar outflows filling the heliosphere and planetary environments [1][2][3][4]. However, precisely, these complex manifestations of the plasma lead to multiple problems, which must be addressed and solved according to the phenomena of interest in our analysis. ...
Multi-scale modeling of expanding plasmas is crucial for understanding the dynamics and evolution of various astrophysical plasma systems such as the solar and stellar winds. In this context, the Expanding Box Model (EBM) provides a valuable framework to mimic plasma expansion in a non-inertial reference frame, co-moving with the expansion but in a box with a fixed volume, which is especially useful for numerical simulations. Here, fundamentally based on the Vlasov equation for magnetized plasmas and the EBM formalism for coordinates transformations, for the first time, we develop a first principles description of radially expanding plasmas in the EB frame. From this approach, we aim to fill the gap between simulations and theory at microscopic scales to model plasma expansion at the kinetic level. Our results show that expansion introduces non-trivial changes in the Vlasov equation (in the EB frame), especially affecting its conservative form through non-inertial forces purely related to the expansion. In order to test the consistency of the equations, we also provide integral moments of the modified Vlasov equation, obtaining the related expanding moments (i.e., continuity, momentum, and energy equations). Comparing our results with the literature, we obtain the same fluids equations (ideal-MHD), but starting from a first principles approach. We also obtained the tensorial form of the energy/pressure equation in the EB frame. These results show the consistency between the kinetic and MHD descriptions. Thus, the expanding Vlasov kinetic theory provides a novel framework to explore plasma physics at both micro and macroscopic scales in complex astrophysical scenarios.
... The study of kinetic theory started from Boltz-mann's works, and its relativistic version also has a long history (which can be traced back to Jüttner's works in 1911 [1]). Currently, the framework of relativistic kinetic theory looks fairly complete [2][3][4][5]. In contrast, the study of relativistic stochastic dynamics is still far from being accomplished. ...
Two different versions of relativistic Langevin equation in generic curved spacetime background are constructed, both are manifestly general covariant. It is argued that, from the observer's point of view, the version which takes the proper time of the Brownian particle as evolution parameter contains some conceptual issues, while the one which makes use of the proper time of the observer is more physically sound. The two versions of the relativistic Langevin equation are connected by a reparametrization scheme. In spite of the issues contained in the first version of the relativistic Langevin equation, it still permits to extract the physical probability distributions of the Brownian particles, as is shown by Monte Carlo simulation in the example case of Brownian motion in $(1+1)$-dimensional Minkowski spacetime.
The problem of the gravitational collapse of the core of a massive star is considered, taking into account the neutrino transport in the flux-limited diffusion approximation. To reduce the computational domain of a multidimensional problem on a fixed computational grid, the core of a star, which is already at the stage of collapse, is considered. Since the collapse stage is delayed in time compared to the gas-dynamic time scale for an emerging proto-neutron star, we consider the mathematical problem for the initial configuration in equilibrium and neglected the initial radial velocity. Pressure for a long time at the collapse stage is provided by relativistic degenerate electrons, so the relationship between pressure and density in the initial configuration is described by a polytropic equation with the polytropic index n=3. The purpose of this paper is to test the hypothesis that large-scale convection is independent of the 2D and 3D geometry of the mathematical problem and computational grid parameters, as well as the choice of the initial stage of gravitational collapse. The scale of convection is determined by the size of the region of decreasing entropy with neutrino losses, i.e., nonequilibrium neutronization, and the presence of a weak initial rotation.
Two different versions of relativistic Langevin equation in curved spacetime background are constructed, both are manifestly general covariant. It is argued that, from the observer’s point of view, the version which takes the proper time of the Brownian particle as evolution parameter contains some conceptual issues, while the one which makes use of the proper time of the observer is more physically sound. The two versions of the relativistic Langevin equation are connected by a reparametrization scheme. In spite of the issues contained in the first version of the relativistic Langevin equation, it still permits to extract the physical probability distributions of the Brownian particles, as is shown by Monte Carlo simulation in the example case of Brownian motion in (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+1)$$\end{document}-dimensional Minkowski spacetime.
The quasi amplitude of probability, a concept associated with the Wigner function, is used to introduce the notion of symplectic density matrix, \(\rho\). The Liouville-Von Neumann equation is derived in phase space by using symmetry. The physical content of \(\rho\) is analyzed in detail, in particular with the study of the classical limit. The first application addresses the problem of quark-antiquark interaction, described by the so-called Cornell potential. The matrix \(\rho\) is calculated, providing a phase space description of the quark-antiquark state. For the model considered here, the quark confinement is derived, such that the distance of confinement is about 4 GeV\(^{-1}\). This result improves theoretical calculations in the literature, approaching to the order of experimental results. Another application is the calculation of \(\rho\) considering mixed state systems. Since \(\rho\) is a density of probability, the entropy is defined without ambiguity, and as such the Gibbs ensemble is introduced for a quantum system in the phase space representation. Then a harmonic oscillator system at finite temperature is studied. The second mixed state system is a maximally entangled bipartite state. In these applications, when it is possible, the nature of \(\rho\) is compared with the Wigner function.
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