Clausius inequality and optimality of quasistatic transformations for nonequilibrium stationary states

Nonequilibrium stationary states of thermodynamic systems dissipate a positive amount of energy per unit of time. If we consider transformations of such states that are realized by letting the driving depend on time, the amount of energy dissipated in an unbounded time window then becomes infinite. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work performed along any given transformation. We then show that the renormalized work satisfies a Clausius inequality and prove that equality is achieved for very slow transformations, that is, in the quasistatic limit. We finally connect the renormalized work to the quasipotential of the macroscopic fluctuation theory, which gives the probability of fluctuations in the stationary nonequilibrium ensemble.