As it is well known, electrons at equilibrium (no external field) do not conduct electric current, i.e.

$\int_{BZ} dk\,v_{k}\,f(\epsilon_k)=0$

where $f(\epsilon_k)$ is the Fermi-Dirac distribution

$f(\epsilon)=\frac{1}{e^{\beta(\epsilon-\mu)}+1}$

$v_k$ is the velocity

$v_k=\frac{\partial \epsilon_k}{\partial k}$

and $BZ$ is the Brillouin zone.

Is it possible to demonstrate that just using the explicit expression of the Fermi-Dirac distribution and the fact that we are considering functions periodic on the reciprocal lattice? I mean, is it possible to say that the integral is zero just from the general properties of the quantities involved? (for example, in the free-electron case it comes from the fact that it is the integral of an odd function in $k$).