On the need for and the possibility of a unified description of kinetic and hydrodynamic processes

Yu. L. Klimontovich

Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 2, pp. 312-330

Résumé

The aim of the paper is to demonstrate the possibility of a nified description of kinetic and hydrodynamic processes on the basis of a eneralized kinetic equation without the use of perturbation theory with respect to the Knudsen number. The derivation of the generalized kinetic equation is based on a oncrete definition of a ontinous medium in the kinetic and hydrodynamic description of nonequilibrium processes in a Boltzmann gas and in a fully ionized plasma. The concept of a “point” of a continuous medium is introduced through the definition of corresponding physically infinitesimally small volumes. On this basis we also give a definition of a Gibbs ensemble to the description of nonequilibrium processes in statistical theory. Besides the usual “collision integral”, which takes into account the dissipation through the redistribution of the particles with respect to the velocities, the generalized kinetic equation contains in the case of the physical definition of the “continuous medium” an additional term of diffusion type. For this reason, it becomes possible to describe kinetic and hydrodynamic processes at all admissible Knudsen numbers. Boltzmann's $H$ theorem is proved for the generalized kinetic equation. The entropy production is determined by the sum of two positive contributions, which are due, respectively, to the redistribution of the particles in the velocity space and in ordinary space. The entropy flux also consists of two terms, one proportional to the entropy and one proportional to the entropy gradient. The presence of the second term makes it possible to give a general definition of a heat flux for arbitrary Knudsen numbers. For small Knudsen numbers and slow processes, it reduces to Fourier's law. The equations of gas dynamics follow from the generalized kinetic equation without the use of perturbation theory with respect to the Knudsen number. They take into account not only processes of viscosity and heat conduction but also self-diffusion. The region of applicability of the equations of gas dynamics is discussed. Generalized kinetic equations are obtained for the distribution functions of the states of the electrons and ions of a partly ionized plasma. Kinetic equations for active media, and also in the theory of Brownian motion, are discussed.

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