Onsager relations between spin currents and charge currents

We consider the macroscopic dynamics of systems with charge and spin currents, using the methods of Onsager's irreversible thermodynamics. Applied to systems with spin-orbit interaction (SOI), we derive Onsager relations showing that, if electrical disequilibrium leads to spin currents, magnetic disequilibrium leads to charge currents. We consider three examples of such SOI. Two of these predicted charge currents have not previously appeared. By measuring these charge currents one can infer the corresponding spin currents.