Unconventional entropy production in the presence of momentum-dependent forces

We investigate the unconventional nature of entropy production (EP) in nonequilibrium systems with odd-parity variables that change signs under time reversal. We consider the Brownian motion of a particle in contact with a heat reservoir, where the particle's momentum is an odd-parity variable. In the presence of an external momentum-dependent force, the EP transferred to the environment is found to be not equivalent to the usual reservoir entropy change due to heat transfer. An additional unconventional contribution to the EP, which is crucial for maintaining the non-negativity of the (average) total EP enforced by the second law of thermodynamics, appears. A few examples are considered to elucidate the novel nature of the EP. We also discuss detailed balance conditions with a momentum-dependent force.