Invariance of steady-state thermodynamics between different scales of description.

By considering general Markov stochastic dynamics and its coarse graining, we study the framework of stochastic thermodynamics for the original and reduced descriptions corresponding to different scales. We are especially concerned with the case where the irreversible entropy production has a finite difference between the scales. We find that the sum of the increment of the nonequilibrium entropy and the excess part of the entropy production, which are key quantities in the construction of steady-state thermodynamics, is essentially kept invariant with respect to the change in the scales of description. This general result justifies experimental approaches toward steady-state thermodynamics based on coarse-grained variables. We demonstrate our result in a mesoscopic heat engine system.