False Onsager relations

Abstract: Recent research suggests that when a system has a "false time reversal violation" the Onsager reciprocity relations hold despite the presence of a magnetic field. The purpose of this work is to clarify that the Onsager relations may well be violated in presence of a "false time reversal violation": that rather guarantees the validity of distinct relations, which we dub "false Onsager relations". We also point out that for quantum systems "false time reversal violation" is omnipresent and comment that, per se, this has in general no consequence in regard to the validity of Onsager relations, or the more general non-equilibrium fluctuation relations, in presence of a magnetic field. Our arguments are illustrated with the Heisenberg model of a magnet in an external magnetic field.