  We develop a general framework in the renormalization-group (RG) method for
extracting a mesoscopic dynamics from an evolution equation by incorporating
some excited (fast) modes as additional components to the invariant manifold
spanned by zero modes. We call this framework the doublet scheme. The validity
of the doublet scheme is first tested and demonstrated by taking the Lorenz
model as a simple three-dimensional dynamical system; it is shown that the
two-dimensional reduced dynamics on the attractive manifold composed of the
would-be zero and a fast modes are successfully obtained in a natural way. We
then apply the doublet scheme to construct causal hydrodynamics as a mesoscopic
dynamics of kinetic theory, i.e., the Boltzmann equation, in a systematic
manner with no ad-hoc assumption. It is found that our equation has the same
form as thirteen-moment causal hydrodynamic equation, but the microscopic
formulae of the transport coefficients and relaxation times are different. In
fact, in contrast to the Grad equation, our equation leads to the same
expressions for the transport coefficients as given by the Chapman-Enskog
expansion method and suggests novel formulae of the relaxation times expressed
in terms of relaxation functions which allow a natural physical interpretation
of the relaxation times. Furthermore, our theory nicely gives the explicit
forms of the distribution function and the thirteen hydrodynamic variables in
terms of the linearized collision operator, which in turn clearly suggest the
proper ansatz forms of them to be adopted in the method of moments.
