Brownian motion in a gas of charged particles under the influence of a magnetic field

The Brownian motion of a particle immersed in a gas of other charged particles is considered when the system is placed in a constant magnetic field. The Zwanzig–Caldeira–Legget particle-bath model is modified so that not only the charged Brownian particle (BP) but also the surrounding it particles respond to the external field. For systems conditioned to be stationary, two non-Markovian equations of motion for the BP across the field are derived. They are of the type of generalized Langevin equations with two different memory functions. As distinct from all the previous theories, the random thermal force is found to depend on the field magnitude. Its time correlation function is connected with one of the found memory functions through the familiar Kubo’s second fluctuation–dissipation theorem. The other memory function disappears when the magnetic force does not affect the bath particles. Analytical expressions can be obtained for the velocity correlation functions and other relevant quantities such as the mean square displacement and the diffusion coefficient of the BP for different distributions of the eigenfrequencies of the bath oscillators. Assuming the Drude distribution of the frequencies, it is found that the motion of the particle at long times is sub-diffusive, with the exponent 1/2. The case of the bath frequencies corresponding to the fractional thermal noise is also analyzed.