Semiclassical model of electron dynamics

Question

I'm now reading Solid State Physics by Neil W. Ashcroft and N. David Mermin (section 12.1).

In cross $\vec{E}$ and $\vec{H}$ fields, when all occupied (or all unoccupied) orbits of electrons in a high magnetic field are closed, for sufficient large $\tau$ ($\omega_c\tau\gg1$), the drift velocity gives the dominant contribution, and we have the current density for occupied (or all unoccupied) orbits.
$$ \lim_{\tau/T\rightarrow\infty}\vec{\jmath}_{\perp}^{\textrm{occupied}}=-\frac{nec}{H}(\vec{E}\times\hat{H}) $$
$$ \lim_{\tau/T\rightarrow\infty}\vec{\jmath}_{\perp}^{\textrm{unoccupied}}=+\frac{n_hec}{H}(\vec{E}\times\hat{H}) $$

However, as they mentioned in chapter 11, from the periodicity of the band structure and semiclassical equations, I think these currents must be identical, and electron and hole perspectives must be consistent (even if they are not instantaneous but averaged perpendicular components of currents given the limit). Therefore, I suspect the assumption that all occupied (or unoccupied) orbits are closed are invalid.

Answer 1

See the footnote given by AM on that very same page. There is no band wherein all orbits are closed; thus, there is no band in which all occupied states and all unoccupied states have closed orbits. There is therefore no contradiction, as only one or the other equation will apply (if at all).