The correlated motion of electrons in a one-dimensional system with an externally applied longitudinal electric field is discussed. Within the tight-binding model we show that in addition to the well-known Bloch oscillations, the electron-electron interaction induces time-dependent oscillations of the mobility whose period depends on the strength and range of the coupling only. The oscillations involve transitions between bands of bound and unbound states. The case of two electrons is solved in detail and an extension of the results to more particles is discussed.

A Hamiltonian mean field model, where the potential is inspired by dipole-dipole interactions, is proposed to characterize the behavior of systems with long-range interactions. The dynamics of the system remains in quasistationary states before arriving at equilibrium. The equilibrium is analytically derived from the canonical ensemble and coincides with that obtained from molecular dynamics simulations (microcanonical ensemble) at only long time scales. The dynamics of the system is characterized by the behavior of the mean value of the kinetic energy. The significance of the results, compared to others in the recent literature, is that two plateaus sequentially emerge in the evolution of the model under the special considerations of the initial conditions and systems of finite size. The first plateau decays to a different second one before the system reaches equilibrium, but the dynamics of the system is expected to have only one plateau when the thermodynamics limit is reached because the difference between them tends to disappear as $N$ tends to infinity. Hence, the first plateau is a type of quasistationary state the lifetime of which depends on a power law of $N$ and the second seems to be a true quasistationary state as reported in the literature. We characterize the general behavior of the model according to its dynamics and thermodynamics.