Effect of a magnetic field on the thermodynamic uncertainty relation

The thermodynamic uncertainty relation provides a universal lower bound on the product of entropy production and the fluctuations of any current. While proven for Markov dynamics on a discrete set of states and for overdamped Langevin dynamics, its status for underdamped dynamics is still open. We consider a two-dimensional harmonically confined charged particle in a magnetic field under the action of an external torque. We show analytically that, depending on the sign of the magnetic field, the thermodynamic uncertainty relation does not hold for the currents associated with work and heat. A strong magnetic field can effectively localize the particle with concomitant bounded fluctuations and low dissipation. Numerical results for a three-dimensional variant and for further currents suggest that the existence of such a bound depends crucially on the specific current.