Derivation of the hydrodynamic equation from the quantum transport equation

The Kadanoff-Baym equation is the quantum transport equation which possesses the microscopic trans- port properties. We derive the hydrodynamic equation as an infrared effective dynamics of the microscopic theory from Kadanoff-Baym equation with the dynamical renormalization group method [1]. As a preparation, we check the validity of the dynamical renormalization group method developed in [2] by comparing with the Chapman-Enskog method discussed in [3]. We consider the unitary Fermi gas system as an example. Next, we derive the hydrodynamic equation including the quantum effect from the Kadanoff-Baym equation. Furthermore, we extend this method to the multi-component system. Finally, we apply the hydrodynamic equation derived here to the unitary Fermi gas system and analyze the transport coefficients of the second order hydrodynamics. \\[4pt] [1] D. Boyanovsky and H. J. de Vega, Annals Phys. 307 (2003) 335.\\[0pt] [2] K. Tsumura and T. Kunihiro, Eur. Phys. J. A 48, 162 (2012).\\[0pt] [3] Thomas Schaefer, arXiv:1404.6843.