A novel nature in nonequilibrium entropy production with odd-parity variables

We present our recent finding about a novel nature in nonequilibrium entropy production for systems with odd-parity variables under time reversal. In the presence of irreversible forces the entropy production $\Delta S_{env}$ transferred from system to environment is not equal to $Q/T$ where $Q$ is the heat transfer and $T$ the temperature of heat bath. We consider a dissipative force applied by external agent in addition to that given by heat bath. Then $\Delta S_{env}$ has extra contribution to $Q/T$ for which an appropriate physical explanation is still open. Another example for irreversible force is a form of $-A\cdot\vec{p}/m$ for antisymmetric matrix $A$ which is realized by a Lorentz force in a uniform magnetic field. In spite of no heat dissipation $\Delta S_{env}$ has a nonvanishing positive contribution. We find that it is due to a nonzero phase space current remaining through stochastic average, which is in fact a nonzero average force. Basically it plays the same role as a nonzero position space current observed in system with even variables only. We suppose interesting situations for different types of irreversible forces.