Memory effects of a static magnetic field on Brownian motion and the question of the absence of classical magnetism

The Bohr-van Leeuwen (BvL) theorem, stating the absence of classical magnetization in equilibrium, a fundamental result in the field of magnetic phenomena, was originally proved for an electron gas. In the present work, we deal with the problem of whether this theorem applies to particles undergoing a non-Markovian Brownian motion in a static magnetic field. We consider a charged Brownian particle (BP) immersed in a bath of neutral particles. Generalizing the Zwanzig-Caldeira-Legget theory to the presence of a static external magnetic field, we come to the equation of motion for the BP in the form of a generalized Langevin equation that accounts for memory effects in the dynamics of the system. By using its solutions for the displacement and velocity of the BP, we calculate the angular momentum for the Ornstein-Uhlenbeck thermal noise. At long times, when the system should reach equilibrium, this momentum and, consequently, the classical magnetic moment of the BP are nonzero, in contrast to the BvL theorem. With the help of analytical and precise numerical calculations for different sets of system parameters, a simple formula for the angular momentum has been deduced.