Introduction and Overview

In this chapter, before going into details, we give an overview of the present book, starting with a short history of nonequilibrium thermodynamics and the upbringing of Rational Extended Thermodynamics (RET) of rarefied monatomic gases. The new version of RET includes the 14-field theory of rarefied polyatomic gases that reduces to the classical Navier–Stokes and Fourier theory in the parabolic limit (Maxwellian iteration), to the 13-field RET theory of monatomic gases in a singular monatomic-gas limit, and to the RET theory with 6 fields as a subsystem. The 6-field theory is the minimal dissipative system, where the dissipation is only due to the dynamic pressure, after the Euler system of perfect fluids. For rarefied polyatomic gases, we discuss a theory of molecular ET with an arbitrary number of field variables by using the method of closure based on either the maximum entropy principle or the entropy principle. It can be proved that the two methods are equivalent to each other. Several applications of the RET theory of polyatomic gases are reviewed as well. In the case of high temperature where both molecular rotational and vibrational modes exist, these modes should be taken into account individually to make the RET theory more realistic. A relativistic theory of polyatomic gas is presented, and its classical and ultra-relativistic limits are also discussed. In particular, the classical limit gives a precise structure of hierarchies of moments. We discuss also some recent tentatives for constructing a phenomenological RET theory of dense gases. Moreover, in both classical and relativistic frameworks, we discuss the theory of a mixture of gases with multi-temperature, i.e., a mixture in which each component has its own temperature. We also show an analogy between the behavior of mixture of gases and the collective behavior of many-body system, and extend the Cucker-Smale flocking model to the model with the temperature field and also to the model in the relativistic framework. The qualitative analysis of the differential system is also done by taking into account the fact that, due to the convexity of the entropy, there exists a privileged field (main field) such that the system becomes symmetric hyperbolic. The existence of global smooth solutions and the convergence to equilibrium are also discussed.