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We study thermodynamic operations which bring a nonequilibrium steady state(NESS) to another NESS in physical systems under nonequilibrium conditions. Wemodel the system by a suitable Markov jump process, and treat thermodynamicoperations as protocols according to which the external agent varies parametersof the Markov process. Then we prove, among other relations, a NESS version ofthe Jarzynski equality and the extended Clausius relation. The latter can be astarting point of thermodynamics for NESS. We also find that the correspondingnonequilibrium entropy has a microscopic representation in terms of symmetrizedShannon entropy in systems where the microscopic description of states involves"momenta". All the results in the present paper are mathematically rigorous.

Exact equalities and thermodynamic relations for nonequilibrium steady states