Hysteretic thermodynamic uncertainty relation for systems with broken time-reversal symmetry

The thermodynamic uncertainty relation gives a lower bound on the amount of dissipation in a mesoscopic system. By considering the fluctuations in the hysteresis of the current – the sum of the currents in the time-forward and time-reversed processes – we extend this relation to systems with broken time-reversal symmetry, either due to the presence of odd state variables, odd driving field or due to explicit time-dependent driving that is time-reversal asymmetric. We illustrate our predictions on a dilute, weakly-interacting gas driven out of equilibrium by the slow compression of a piston and on a ballistic multi-terminal conductor with an external magnetic field.