Hydrodynamic Memory in the Motion of Charged Brownian Particles across the Magnetic Field

An exact solution of the Langevin equation is given for a charged Brownian particle driven in an incompressible fluid by the magnetic field, taking into account the hydrodynamic aftereffect. The stochastic integro-differential Langevin equation is converted to a deterministic equation for the particle mean square displacement. We have found the mean square displacement and other time correlation functions describing the particle motion. For the motion along the field the known results from the theory of the hydrodynamic motion of a free Brownian particle are recovered. The correlation functions across the field contain at long times the familiar Einstein terms and additional algebraic tails. The longest-lived tail in the mean square displacement is proportional to $t^{1//2}$. At short times the motion is ballistic and independent of the magnetic field.