Application of the method of nonequilibrium thermodynamic potentials to systems of fermions interacting with bosons

The method of nonequilibrium thermodynamic potentials is used to construct the statistics of fermion-boson systems ha the case of a fermion-boson coupling that is weak with respect to the fermions but strong with respect to the bosons. It is shown that the temperature shift of the fermion energy is equal to the functional derivative of the boson free energy with respect to the population numbers of the fermion states. The general relations are illustrated by the statistics of electrons in ferromagnetic semiconductors.