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Brownian motion of charged particles driven by correlated noise

Stochastic motion of charged particles in the magnetic field was first studied almost half a century ago in the classical works by Taylor and Kursunoglu in connection with the diffusion of electrons and ions in plasma. In their works the long-time limits of the mean square displacement (MSD) of the particles have been found. Later Furuse on the basis of standard Langevin theory generalized their results for arbitrary times. The currently observed revival of these problems is mainly related to memory effects in the diffusion of particles, which appear when colored random forces act on the particles from their surroundings. In the present work an exact analytical solution of the generalized Langevin equation has been found for the motion of the particle in an external magnetic field when the random force is exponentially correlated in the time. The obtained MSD of the particle motion across the field contains a term proportional to the time, a constant term, and contributions exponentially decaying in the time. The results are more general than the previous results from the literature and are obtained in a considerably simpler way applicable to many other problems of the Brownian motion with memory.