4.3.2 Specific of the Equilibrium Distribution

When the electric field tends to zero, the distribution function approaches the equilibrium distribution which is in the case of particles with fractional spin represented by the Fermi-Dirac distribution function (2.30). It has the form

$$f_\mathrm{FD}(\epsilon(\vec{k}))=\frac{1}{\exp\bigl(-\frac{E_{f}-\epsilon(\vec{k})}{k_{B}T_{0}}\bigr)+1},$$ (4.46)

where $E_{f}$ denotes the Fermi energy, $\epsilon$ stands for the electron energy and $T_{0}$ is the equilibrium temperature equal to the lattice temperature. Since the stationary distribution is known, there is no need to solve the zeroth order equation (4.10). As can be seen from (4.47), in equilibrium the distribution function depends only on the carrier energy, and the dependence on the quasi-momentum is only introduced through the dispersion law $\epsilon(\vec{k})$.

S. Smirnov: