Lasting effects of static magnetic field on classical Brownian motion

TL;DR

This paper revisits the Bohr-Van Leeuwen theorem, showing that while static magnetic fields do not affect the equilibrium state of a classical Brownian particle, they do influence the long-time state of the thermal bath, inducing angular momentum.

Contribution

It demonstrates that static magnetic fields induce long-term angular momentum in the thermal bath, challenging the traditional understanding of classical magnetism in equilibrium systems.

Findings

Magnetic field induces angular momentum in the bath.

The long-time state of the particle remains unaffected.

Bath energy retains a small magnetic correction.

Abstract

The Bohr-Van Leeuwen theorem states that an external static magnetic field does not influence the state of a classical equilibrium system: there is no equilibrium classical magnetism, since the magnetic field does not do work. We revisit this famous no-go result and consider a classical charged Brownian particle interacting with an equilibrium bath. We confirm that the Bohr-Van Leeuwen theorem holds for the long-time (equilibrium) state of the particle. But the external static, homogeneous magnetic field does influence the long-time state of the thermal bath, which is described via the Caldeira-Leggett model. In particular, the magnetic field induces an average angular momentum for the (uncharged!) bath, which separates into two sets rotating in opposite directions. The effect relates to the bath going slightly out of equilibrium under the influence of the Brownian particle and persists for arbitrarily long times. In this context we studied the behavior of the two other additive integrals of motion, energy and linear momentum. The situation with linear momentum is different, because it is dissipated away by (and from) the bath modes. The average energy of the bath mode retains the magnetic field as a small correction. Thus, only the bath angular momentum really feels the magnetic field for long times.