RG/E Derivation of Dissipative Fluid Dynamics from Classical Non-relativistic Boltzmann Equation

On the basis of the identification of the fluid dynamics with the slow dynamics of the kinetic equation in the asymptotic regime after a long time, we apply the RG method to derive the fluid dynamics with dissipative effects for the non-relativistic system, i.e., the Navier-Stokes equation and the energy equation with the heat current, from the classical Boltzmann equation: The small inhomogeneity in the asymptotic regime is identified as a small parameter for the perturbative expansion. The derivation consists of a two-fold construction, i.e., the explicit construction of deformed distribution function as the attractive/invariant manifold and the the reduced differential equation for the slow variables to be identified with the fluid dynamic equation where the microscopic expressions of the transport coefficients are explicitly given in terms of the distribution function in a form of the Kubo formula.