Duality principle from rarefied to dense gas and extended thermodynamics with six fields

We present an extended thermodynamics (ET) theory of dissipative dense gases. In particular, we study the ET theory with six fields, where we neglect shear viscosity and heat conductivity. We postulate a simple principle of duality between rarefied and dense gases. This principle is based on the microscopic analysis of the energy exchange between different modes of the molecular motion. The basic system of equations satisfies all principles of ET, that is, Galilean invariance, entropy principle, and thermodynamic stability (entropy convexity), and, as in the ET theory of rarefied gases, the constitutive equations are completely determined by the thermal and caloric equations of state. The system is simplest after the Euler system, but, in contrast to the Euler system, we may have a global smooth solution due to the fact that the system is dissipative symmetric hyperbolic and satisfies the so-called K condition. There emerge two nonequilibrium temperatures; one is due to the translational modes, and the other is due to the internal modes such as rotation and vibration of a molecule. This viewpoint allows us to understand the origin of the dynamic pressure in a more clear way. Furthermore we evaluate the characteristic velocities associated with the hyperbolic system and address the fluctuation-dissipation relation of the bulk viscosity. As a typical example, we analyze van der Waals fluids based on the present theory.