Recent progress in microscopic derivation procedure of relativistic hydrodynamic equation

Relativistic hydrodynamic equations are often used in the cosmology and ultra high energy heavy ion collisions. The successful description of the RHIC data by the perfect fluid images inspired interest in the transport equations including the dissipation effects. It is well known that the relativistic hydrodynamics of dissipative fluid goes back to Eckart and Landau-Lifshits, which were derived phenomenologically from the requirements of Lorentz covariance and entropy enhancement. Progress as titled above made in collaboration with K. Tsumura and K. Ohnishi is reported here. Essential problems concerning the relativistic hydrodynamic equations for dissipative system are summarized at first. Then 'solutions' for these problems from the analyses by way of both phenomenology and microscopic analysis are presented. Relativistic hydrodynamic equation is derived as an asymptotic infrared effective theory by regarding the relativistic Boltzmann equation as a dynamic system. The renormalization-group method is used as the powerful contraction theory of the dynamic system. The results of derivation of Israel-Stewart type equation is shown as well. In addition a detailed explanation of the 'Renormalization-Group Method' is presented. (S. Funahashi)