Living cells exhibit spontaneous mechanical fluctuations far exceeding thermal predictions, but the microscopic origin of this “active noise” has remained elusive. Here, we derive it from first principles. We model an active gel as a crosslinked filament network where molecular motors break detailed balance in their binding kinetics. From the stochastic dynamics of individual crosslinkers, we obtain the full noise statistics in the gel’s stress tensor: it is white, decomposes into thermal, driven, active, and cross-coupling contributions, and its active part is directly set by the degree of detailed-balance violation. Applied to a tracer particle, our theory predicts enhanced, anisotropic fluctuations, a testable departure from the fluctuation-dissipation theorem that establishes a new fluctuation-activity relation for living matter.

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 guarantees that a static magnetic field cannot alter the equilibrium state of classical charged particles. We show that while this is true for the particle itself, the thermal environment tells a different story. Using the Caldeira–Leggett model of classical Brownian motion, we demonstrate that uncharged bath oscillators acquire a lasting, nonzero angular momentum that depends on the applied magnetic field, even after the Brownian particle has fully thermalized. Bath modes split into two counter-rotating populations divided at the particle’s natural frequency. This behavior is unique to angular momentum: linear momentum dissipates from all observable bath modes, and the magnetic correction to bath energy is negligible. Our results expose a blind spot in the classical understanding of diamagnetism and point toward a possible mechanism for the experimentally observed effects of weak static magnetic fields on biological systems.

Hundreds of experiments report that weak magnetic fields, as feeble as the Earth’s own, can influence biological processes. Yet no physical mechanism has survived scrutiny: thermal noise is overwhelming and friction on cellular ions is enormous. We show that the answer lies in the cell’s departure from thermodynamic equilibrium. Modeling a confined ion subject to both thermal and non-thermal noise from fast protein motions, we find that even a tiny violation of the fluctuation-dissipation relation bypasses the Bohr-van Leeuwen no-go theorem and lets a static magnetic field drive sustained cyclotron rotation. The resulting angular velocity matches the cyclotron frequency and operates on millisecond timescales, precisely where many metalloproteins carry out their function. This provides a concrete, physically grounded link between weak magnetism and biology.