I am reading this paper, which uses (chiral) kinetic theory. The authors write:
The Boltzmann method is valid only in the semiclassical limit, where $\omega_B \tau \ll 1$, $\omega_B \ll \mu$ and $\mu\tau\gg 1$. Here, $\omega_B$ is the magnetic frequency, $v_F$ is the Fermi velocity, and $\tau$ is the quasiparticle lifetime. (In this paper, we set $\hbar = 1$ and the energy of WP as 0.) In the semiclassical limit, the level smearing caused by the finite quasiparticle lifetime is much larger than the Landau-level splitting, but it is much smaller than the chemical potential; hence, the Landau-level quantization can be ignored, and the Fermi surface remains well defined. Therefore, in the semiclassical limit, the collective dynamics of a Fermi-liquid system can be described by the quasiparticle distribution function $n({\bf k},{\bf r},t)$ through the following Boltzmann equation
$$ \big[\partial_t + {\bf \dot{r}}\cdot\nabla_{\bf r} - {\bf \dot{k}}\cdot\nabla_{\bf k}\big]\delta n ({\bf k}, {\bf r},t) = S[\delta n({\bf k}, {\bf r},t)], $$ where $S[\delta n({\bf k}, {\bf r},t)]$ is a collision integral and $\delta n({\bf k},{\bf r},t)$ the variation in the distribution function.
The issue of validity of kinetic theory has been on my mind for a while. According to the authors of the text above, the chemical potential needs to be the biggest scale in the problem in order to use the (semiclassical) Boltzmann equation. I do not completely understand the explanation they give. Could somebody elaborate on the range of validity of the (semiclassical) Boltzmann equation?
Furthermore, what happens when we consider the limit $\mu \to 0$? In the case of a gapless system like a Dirac or Weyl semimetal, can we still use chiral kinetic theory? Does it break down and how?
