In this thesis, we investigate whether an electrical current in a metal can drive the phonon system out of thermal equilibrium, and what the non-equilibrium characteristics of the phonon distribution might be. In the context of flash sintering experiments, it had been proposed that the electron-phonon coupling leads to a proliferation of short wave-length lattice vibrations in an electric field. In this thesis we analyze this by solving two coupled Boltzmann equations, describing a free electron gas in an electric field scattering from a vibrating crystal lattice, which is coupled to a heat bath. Since not only energy, but also momentum is exchanged between the electrons and the phonons, substantial modifications of the theory are required. The electric field imposes cylindrical symmetry, which leads to various numerical challenges. A rather substantial part of this thesis constitutes in demonstrating our techniques in dealing with these numerical intricacies, that come from solving the electron-phonon scattering integrals in the absence of spherical symmetry. The implementation is tested rigorously for the isolated electron-phonon system in various known physical scenarios. 

We find that the shape of the electron distribution is rather robust, due to the Pauli principle and the comparably low phonon energies, however it gets shifted in reciprocal space by a wave vector parallel to the electric field. On the other hand, the phonon distribution deviates strongly from a thermal distribution. More accurately, we find that the phonon distribution shows a strong excess population at the Brillouin zone edge in the direction of the electric field. We argue analytically, that this can be traced back to the shifted Fermi sphere for the electrons. The non-equilibrium distributions remain present in the steady state. Furthermore, since not only energy but also momentum is exchanged in the electron-phonon system, the dynamics cannot be described by a simple two-temperature model.