Most energy generated in the gravitational collapse to a supernova is radiated in neutrinos, hence the role of these particles in a supernova explosion is crucial. Current models of core collapse supernovae focus on multidimensional hydrodynamics and nuclear burning and treat neutrino transport in a simplified manner. In this last chapter an example of accurate neutrino treatment in a spherically symmetric collapse is given. The role of multidimensional effects is discussed. These results are of interest for the multidimensional models with large-scale convection as well as for the ongoing experimental search for neutrinos from supernovae.

Supernova Models and Neutrinos

Supernovae (SNe) are produced by stars that end their late evolution in a catastrophic explosive process. The name supernova was introduced and the difference between SNe and novae in terms of their estimated explosive powers was described in [515]. The luminosity of a SN at its maximum, which lasts for several days, is comparable to the total luminosity of its host galaxy. It was first hypothesized in [516] that SN explosions should be accompanied by the formation of neutron stars; the neutron had been discovered just two years earlier. The total energy involved into explosion is approximately 1053 erg, mostly released in the form of neutrinos. Approximately 1% of this energy, namely, 1051 erg, is released in the form of kinetic energy of the SN ejecta. Only approximately 1% of that kinetic energy, i.e., 1049 erg, is emitted in the form of photons, which are detected as the SN event [517, 518]. The relevant timescales are as follows: collapse of the core occurs on a ≤ 0.1 s timescale, the SW propagation inside the collapsing core takes ~10 ms, and the neutrino cooling time of the hot neutron star is 10 s.

There are two general types of a SN, classified according to the presence of hydrogen absorption lines in observed spectra. Absorption lines are present in the spectra of Type II SNe and absent in the spectra of Type I SNe. In addition, Type II SNe normally contain compact remnants, although such a remnant was not found in the nearby SN 1987A [519]. In this chapter,mainly Type II SNe, or core-collapse SNe, are discussed.

In this chapter, our experience of the integration of Boltzmann equations by the finite difference method is presented. In spherically symmetric geometry the approach operates with finite differences on the fixed grid in 4D phase space for particles (r, μ, ϵ, t), and it is based on the method of lines. The ODE system is solved by implicit Gear's method suitable for integration of stiff equations. The method takes into account all reactions and is applicable to both optically thick and optically thin regions. This chapter also includes description of Monte Carlo (MC) methods for integration of the Boltzmann equation. For small and moderate optical depth, MC is a universal approach able to describe different reactions and is especially suitable for multidimensional geometry. It is shown how the linear MC method is extended for large optical depths with simple reaction rates such as in Compton scattering.

Finite Differences and the Method of Lines

This section is limited to 1D spatial geometry (spherically symmetric case) with full physical processes and implicit methods.

A large number of physical problems requires solution of Boltzmann equations. Analytical solutions are available in exceptional cases, and in the general case one should rely on effective numerical integration. As discussed in the previous chapter, the finite difference technique [117] represents one such effective method.

Finite differences are widely employed in astrophysical problems. Specific examples, considered in Part III, include thermalization of nonequilibrium optically thick electron-positron plasma and neutrino transport in the core collapse supernovae.

Finite differences for the kinetic equation in astrophysical applications were introduced in [118]. In particular, in the gravitational collapse of the iron core of a massive star to a neutron star, neutrinos play a crucial role. Neutrino transport should be described by Boltzmann equations. The core on the verge of collapse has the radius 108 cm, and it is transparent for neutrinos. The collapsed hot proto-neutron star has a radius of 106–107 cm, and neutrinos are trapped with the optical depth. It is impossible to treat such problems by explicit methods due to the presence of different timescales characterizing different processes.

In this chapter the evolution of basic concepts of KT, such as phase space and distribution functions, from nonrelativistic to special and general relativistic frameworks is outlined. The relation between mechanical and kinetic pictures is presented. The physical meaning of the one-particle distribution function is given, and its Lorentz invariance is demonstrated. Then the most useful macroscopic quantities, such as four-current, entropy four-flux, energy-momentum tensor, and hydrodynamic velocity, are obtained. These concepts are essential to proceed with the formulation of kinetic equations as well as to understand the relation between kinetic theory and hydrodynamics discussed in following chapters.

Nonrelativistic Kinetic Theory

In classical (nonrelativistic) mechanics a complete description of a system composed of N interacting particles is given by their N equations of motion. In non-relativistic KT one deals with a space of positions and velocities of these particles, called configuration space or the space of canonical variables: positions and momenta of particles, called the phase space M. Often this mechanical description can be formulated in the language of Hamilton equations, and then an equivalent description of the system is given by a function of time and 6N independent variables, defined on M. An equation can be formulated for this function, called the Liouville equation, that can be written in an apparently very simple form:

where the derivative is over time. Its complexity, however, is equivalent to the complexity of the original N-body problem, and in the majority of cases, it cannot be addressed directly.

A tremendous simplification occurs for such systems, where N is very large. One may define the s-particle distribution function (DF) of states depending on 6s variables with 1 ≤ s ≤ N and time by integrating out the remaining 6(N – s) degrees of freedom in. The hierarchy of integro-differential equations, which connects the s-particle DFs with the s + 1-particle ones, called the Bogolyubov-Born- Green-Kirkwood-Yvon (BBGKY) hierarchy, is obtained in this way. Among these s-particle DFs, the one-particle DF plays a central role in KT, as it describes the probability of finding the particle in a state with momentum in the range (p, p + d3p) and position in the range (x, x + d3x) at the moment t. The s-particle DFs describe joint probabilities, i.e., particle correlations.

This book presents the subject of relativistic kinetic theory (KT). It starts from fundamental concepts and ideas and arrives at a vast spectrum of applications through the bridge of various numerical methods. It is not by chance that we adopted such an approach. KT of gases is perhaps the most fundamental theory in the classical (nonquantum) domain. It has been developed with the goal to derive the properties of matter at the macroscopic level, which is accessible to direct experiments and observations, based on the study of properties of microscopic particles and their mutual interactions, to which one has no direct access. The atomic picture of the world has emerged in this way. Indeed, KT was born in the nineteenth century, the golden age of classical physics. Based on the atomic picture [1], such properties as heat and electrical conductivity as well as viscosity and diffusion found natural explanations. The term originates from the Greek, where κινησις means “motion.” In fact, all the properties of the medium may be understood from the analysis of its microscopical structure and motions.

Nowadays KT has to be considered in a wider context of statistical mechanics, which appeared at the end of the nineteenth century, essentially in the works of Maxwell, Boltzmann, and Gibbs. It should be emphasized that the ideas and principles of KT influenced the development of many other branches of science, including mathematics (probability theory, ergodic theory), biology (evolutionary biology, population genetics), and economics (financial markets, econophysics). Within physics, KT is closely related to statistical physics, thermodynamics, and hydro- and gasdynamics. Today one can say that the main task of kinetic theory is the explanation of various macroscopic properties of a medium based on known microscopic properties and interactions. In a general context, KT is a theory of nonequilibrium systems. Indeed, all the above-mentioned fields of physics assume that the medium is in its most probable microphysical state, called equilibrium. Clearly, any macroscopic manifestation of deviations from this microscopic equilibrium should be considered within KT. Because basic phenomena in the microworld are described in a quantum language, KT uses extensively quantum theory. In fact, the basic principles and equations of KT may be derived from quantum field theory [2].

Gravity is one of the four fundamental interactions, and it is the weakest one. Nevertheless, on largest scales, the structure of the universe is governed by gravity alone. This is essentially due to the long range and unscreened nature of gravitational interaction. These properties make self-gravitating systems with large numbers of particles very different from other systems with short-range interactions, such as gases, or systems with long-range screened interactions, such as neutral plasmas. In this chapter, kinetic properties of self-gravitating systems are discussed and contrasted with kinetic properties of gases and plasmas discussed in previous chapters. While before the focus has been always on relativistic aspects of KT, most essential properties of self-gravitating systems can be understood within a nonrelativistic context.

From the statistical physics point of view, there are fundamental differences between systems like gases and plasmas and systems with unscreened long-range interactions, in particular, self-gravitating systems.

First, extensivity of energy does not hold for self-gravitating systems. Extensivity is a property of thermodynamic variables such as energy entropy and free energy, that is, their proportionality to the size of the system. In contrast, intensive variables, such as entropy or temperature, do not depend on system size. Generally speaking, extensivity does not hold for any system with long-range interactions. In principle, the extensivity property can be restored if the interaction energy scales with the inverse of particle number [407]. However, systems with long-range interactions also lack additivity of energy, namely, such a property that the system can be divided into parts with total energy being the sum of energies of these parts, with the energy of interaction between parts vanishing in the thermodynamic limit. These unusual properties have a consequence, which is ensemble inequivalence [408], namely, that statistical descriptions based on canonical and microcanonical distributions give different results. Recall that a microcanonical ensemble is a statistical ensemble with specified and constant energy. In contrast, a canonical ensemble is a statistical ensemble of particles in thermal equilibrium with a heat bath at a given temperature.