Fractional hyper-ballistic transport under external oscillating electric fields

The generalized Langevin equation (GLE) for a tagged particle in a liquid of charged particles under the influence of external AC electric fields is studied. For the fractional memory kernel in the GLE, the mean square displacement (MSD) of the particle is studied analytically in both the underdamped and overdamped regimes. The MSD consists of a part corresponding to the absence of the external field and a part affected by the external field, which is expressed through the mean velocity of the particle. We have identified the time windows when the particle shows unusual behaviors in the oscillating fields including hyper-ballistic diffusion, thus generalizing the results for the memoryless Brownian motion. The theory of Brownian motion, since the time of Einstein and Langevin, has overcome a stormy development and the methods of the description of the irregular movement of small particles in solutions have found use in several areas of science. The time dependence of the key quantities in this theory, such as the particle's MSD in condensed matter physics, has been shown to be anomalous, that is, different from linear, in many experimental observations. The movement of the observed particle shows correlation properties of the thermal noise of the surrounding environment, which can be very different in different systems and are associated with memory effects in the dynamics of the particle. One option, effective in describing complex systems by the method of the GLE, is the use of the fractional kernel of its frictional memory integral that replaces the Stokes friction force in the original Langevin equation of motion. In our work, for the first time, we solve such a GLE with a fractional memory for a particle-in-bath system (the particle can be identical with the surrounding particles) in an external oscillating electric field. All particles are charged, as is the case, for example, in plasma or liquid electrolytes, so both the monitored particle and its surroundings are affected by the external field. The GLE is solved analytically for the entire time scale. The results include solutions to the classical memoryless Langevin equation and new features in the time dependence of the MSD, including unusual near-ballistic or hyper-ballistic particle transport, depending on the way the external AC field is applied.