Brownian Motion of Charged Particles in a Bath Responding to an External Magnetic Field

We consider the Brownian motion in a constant magnetic field. Unlike in the previous theories, both the Brownian particle and the surrounding bath particles, if they are charged, respond to the external field. The bath particles are regarded as harmonic oscillators interacting with the Brownian particle. The derived equations of motion for the Brownian particle have a form of generalized Langevin equations. For the motion in the plane perpendicular to the field, two coupled equations for the Brownian particle velocity projections are obtained. Under the condition of stationarity, the velocity correlation functions are determined by the memory functions of the system. The second fluctuation-dissipation theorem is proved. It has the familiar form, but here the thermal noise depends on the magnetic field.