We just quote Boltzmann's transport equation, indicating that it can be seen as a direct consequence of Liouville's theorem which holds for Hamiltonian systems plus the additional collision term.
Hamiltonian mechanics plus nonconservative collisions
We have seen that the distribution function $f$ of our Hamiltonian system is conserved. We interpret this such that $f(x,v,t){\rm d}^3x{\rm d}^3v$ is the density of particles at time $t$ in phase space volume ${\rm d}^3x{\rm d}^3v$ at the point $(x,v)$. This statement can be generalized to the case where $f$ has an explicit time dependence, such that we eventually obtain
\[\frac{\rm d}{\mathrm{d}t}f
=
\frac{\partial}{\partial t}f
+\sum_i\left(
\frac{\partial x_i}{\partial t}
\frac{\partial f}{\partial x_i}
+
\frac{\partial v_i}{\partial t}
\frac{\partial f}{\partial v_i}
\right)
=
\partial_t f+(v\cdot\nabla_x)f+(\partial_tv\cdot\nabla_v)f
=0\]
for the distribution function, writing $\partial_tx=v=\partial\epsilon/\partial p$ for the group velocity $v$ of particles with dispersion $\epsilon(p)$.
To be able to discuss dissipative transport phenomena, Ludwig Boltzmann (1844-1906) added a collision term as a perturbation to the right-hand side of the above equation. Writing $\partial_tv=a$ for the accelerations (external forces $f_{\rm ext}=\partial_tmv=-\nabla_x\phi$), we obtain
\[\partial_t f+(v\cdot\nabla_x)f+(a\cdot\nabla_v)f=\left.\partial_tf\right|_{\rm streu}.\]
This is the famous Boltzmann transport equation for a distribution function $f=f(x,v,t)$ in phase space. Many authors include the explicit time derivative of $f$ in the collision term and do not distinguish between the conservative and the dissipative time evolution of the distribution function.
We note that (a) because we are discussing fermions subject to the Pauli principle, we must make sure that $f(x,v,t)\le1$ $\forall x,v,t$. If the fermions carry spin, $f$ relates to a specific spin direction, and $f=f_\uparrow=f_\downarrow$ as long as we don’t break time-reversal invariance, for example by turning on a magnetic field $B=\mu_0H$.
(b) Boltzmann’s equation is valid only for phenomena on length scales large compared to intrinsic scales $\Delta x$ and for particles which have momentum differences $\Delta p$ large enough such that $\Delta x\Delta p\gg\hbar$ is fulfilled at any given time $t$. But that’s exactly what hydrodynamics is all about: We don’t care in detail what happens microscopically, but rather follow a more general ansatz using conservation of total energy, particle density, and momentum. In the 19th century this has proven to be very successful in the description of water (indeed that’s where the name comes from), although we nowadays know that the correlations between the interacting water molecules are anything but easy to describe microscopically.
(c) We need not only “large lengths” and “large momenta”, but also well-defined quasiparticles. That is, the quasiparticle lifetime $\Delta t$ must obey the relation $\Delta E\Delta t\gg\hbar$. For Fermi liquids at sufficiently low temperatures $k_{\rm B}T\ll \epsilon_{\rm F}$, we have $\Delta E\approx k_{\rm B}T$ (broadening at the Fermi egde). Because of the Pauli princple (again), only quasiparticles in the broadened area $\propto k_{\rm B}T$ around the Fermi surface can scatter at each other, and also their final states must be in this area. Thus the scattering probability is proportional to the square of this area: For the lifetime we obtain $\Delta t\propto1/T^2$, and for $T\to0$ the condition $\Delta E\Delta t\gg\hbar$ is easily fulfilled.