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.

This chapter describes multidimensional hydrodynamics, and it contains four sections. The first section discusses classical multidimensional shock-capturing hydrodynamics and the application of modern high-order Godunov-type methods that reduce the problem into a class of almost engineering tasks.

Scientific input includes equations of state, reaction rates, and kinetic coefficients, described in the second section. The main difficulties of the kinetic Boltzmann approach discussed in the previous chapter are not only the multidimensionality of the phase space but also the calculation of the reaction rates. These reaction rates usually require using implicit schemes in the case of nontransparent regions, and multidimensional problems become very hard to study. The key point of this section is the proposal to move from the kinetic Boltzmann treatment in 7D phase space (r, p, t) to a hydrodynamic one with diffusion and flux limiters in 5D phase space (r, ϵ, t). The diffusion with flux limiter approach uses some free parameters for the interpolation of spectral energy fluxes in the intermediate case between the transparent (free flow) and the nontransparent (diffusion or heat conduction) regions. The first calculations within such an approach were performed in [131] for the gravitational collapse with the neutrino transport in spherically symmetric case. In such a problem one has to carry out advection by explicit code. At the same time, diffusion of spectral energies is carried out by an implicit scheme, such as the Crank-Nicolson method [100]. Finally, reaction rates are computed using an implicit scheme for the system of ODEs. In the multidimensional 2D or 3D cases the splitting on the time integration along separate directions is made. Given the limited space resolution in the 3D case in comparison with the 1D one, high-order Godunov-type methods are required.

The third section discusses the Riemann problem solver within special relativity. The numerical relativity does not offer universal receipts. Moreover, the current state of numerical relativity in strong gravitational fields is more similar to art than to science. Therefore, in this section, only special relativistic high-order shock-capturing methods are discussed.

The fourth section briefly describes particle-based simulations, in particular, the smoothed-particle hydrodynamics method. Such mesh-free methods are an attractive alternative to traditional grid-based hydrodynamics.

The endeavor of writing this book started from a series of lectures given by the first author for students of the International Relativistic Astrophysics PhD program (IRAP PhD) supported by the Erasmus Mundus program of the European Commission. For this book the material has been expanded and more topics incorporated. It soon became clear that an updated and systematic presentation of relativistic kinetic theory and its numerous applications in astrophysics and cosmology is lacking in the literature. Some existing monographs, presenting fundamental aspects of kinetic theory, are focused on selected applications. Others, which contain applications of kinetic theory in relativistic astrophysics and cosmology, lack the presentation of fundamental concepts of relativistic kinetic theory. Moreover, none of the existing monographs discussed in depth various numerical methods developed and successfully applied in kinetic theory in the recent decades. This last observation urged us to bridge this gap in the literature. This effort eventually resulted in the current monograph, divided in three parts. Parts I and III, with the sole exception of the last chapter, were written by the first author. Part II and the last chapter of Part III were written by the second author.