Employing the Zubarev method for the derivation of a nonequilibrium statistical operator, Vlasov–Uehling–Uhlenbeck quantum kinetic equations are derived for the distribution functions of clusters with arbitrary mass number. Considering a partially ionized hydrogen plasma, the electronic distribution function is determined numerically for the homogeneous and stationary case. Restricting to elastic scattering processes between the electrons, ions and atoms, the analytical solution of Schenter and Liboff [Phys. Fluids 30, 1789 (1987)] is reproduced. Furthermore, the influence of inelastic collisions such as ionization is investigated. Using the electronic distribution function, the electrical conductivity is calculated for hydrogen plasma of a given composition at arbitrary field strengths. The nonlinear behavior obtained is dependent on the different expressions applied for the electron–atom and the electron–ion cross sections. The effect of inelastic collisions is, in general, a lowering of the conductivity because such processes lead to a limitation of the energy gain of electrons in an electric field.