Rotational equilibrated systems are widespread, but relatively little attention has been devoted to studying them from the first principles of statistical mechanics. Here we bridge this gap, as we look at a Brownian particle coupled with a rotational thermal bath modeled via Caldeira-Leggett oscillators. We show that the Langevin equation that describes the dynamics of the Brownian particle contains (due to rotation) long-range correlated noise. In contrast to the usual situation of non-rotational equilibration, the rotational Gibbs distribution is recovered only for a weak coupling with the bath. However, the presence of a uniform magnetic field disrupts equilibrium, even under weak coupling conditions. In this context, we clarify the applicability of the Bohr-van Leeuwen theorem to classical systems in rotational equilibrium, as well as the concept of work done by a changing magnetic field. Additionally, we show that the Brownian particle under a rotationally symmetric potential reaches a stationary state that behaves as an effective equilibrium, characterized by a free energy. As a result, no work can be extracted via cyclic processes that respect the rotation symmetry. However, if the external potential exhibits asymmetry, then work extraction via slow cyclic processes is possible. This is illustrated by a general scenario, involving a slow rotation of a non-rotation-symmetric potential.

The Bohr-Van Leeuwen theorem states that classical systems in equilibrium are unaffected by static magnetic fields. This work explores how a charged Brownian particle in a thermal bath still induces a long-lasting angular momentum in the bath. Our finding shows that while the particle itself remains unaffected in equilibrium, the surrounding bath exhibits a persistent response, challenging traditional views on classical magnetism and opening up new research possibilities in fields like plasma physics and biophysics.

In the study of magnetobiology, understanding how weak magnetic fields interact with biological systems has long been a challenging question. We propose a mechanisms by which weak, static magnetic fields can influence biological matter, despite the high thermal noise and friction present in cellular environments. By examining the nonequilibrium steady states of cellular ions in a confining potential, it is shown that a weak magnetic field can induce cyclotron motion, providing insights into how such fields may impact biological processes at the molecular level. This research sheds light on the fundamental interactions between magnetic fields and biological matter, contributing to the broader understanding of magnetobiology.