Extended Jarzynski equality in general Langevin system

The Speck–Seifert equality and the Hatano–Sasa equality are known to be bases to construct stationary state thermodynamics. The well known nonequilibrium relations, the Jarzynski equality and the fluctuation theorem, share the mathematical structure with the above equalities. This hidden common property motivates us to extend the Jarzynski equality to a relation applicable in the generalized dynamical system. As a result, we find several equalities which are able to be established in this generalized system described by the Langevin equation. These results easily reproduce the Speck–Seifert equality and the Hatano–Sasa equality. We hope that our formulation gives a new insight on nonequilibrium statistical physics.