  We give a detailed derivation of the second-order (local) hydrodynamics for
Boltzmann equation with an external force by using the renormalization group
method. In this method, we solve the Boltzmann equation faithfully to extract
the hydrodynamics without recourse to any ansatz. Our method leads to
microscopic expressions of not only all the transport coefficients that are of
the same form as those in Chapman-Enskog method but also those of the viscous
relaxation times $\tau_i$ that admit physically natural interpretations. As an
example, we apply our microscopic expressions to calculate the transport
coefficients and the relaxation times of the cold fermionic atoms in a
quantitative way, where the transition probability in the collision term is
given explicitly in terms of the $s$-wave scattering length $a_s$. We thereby
discuss the quantum statistical effects, temperature dependence, and
scattering-length dependence of the first-order transport coefficients and the
viscous relaxation times: It is shown that as the temperature is lowered, the
transport coefficients and the relaxation times increase rapidly because Pauli
principle acts effectively. On the other hand, as $a_s$ is increased, these
quantities decrease and become vanishingly small at unitarity because of the
strong coupling. The numerical calculation shows that the relation
$\tau_\pi=\eta/P$, which is derived in the relaxation-time approximation and
used in most of literature without almost any foundation, turns out to be
satisfied quite well, while the similar relation for the relaxation time
$\tau_J$ of the heat conductivity is satisfied only approximately with a
considerable error.
