Complementarity relation for irreversible processes near steady states

A relation giving a minimum for the irreversible work in quasi-equilibrium processes was derived by Sekimoto et al. (K. Sekimoto and S. Sasa, J. Phys. Soc. Jpn. {\bf 66} (1997), 3326) in the framework of stochastic energetics. This relation can also be written as a type of "uncertainty principle" in such a way that the precise determination of the Helmholtz free energy through the observation of the work $<W>$ requires an indefinitely large experimental time $\Delta t$. In the present article, we extend this relation to the case of quasi-steady processes by using the concept of non-equilibrium Helmholtz free energy. We give a formulation of the second law for these processes that extends that presented by Sekimoto (K. Sekimoto, Prog. Theo. Phys. Suppl. No. {\bf 130} (1998), 17) by a term of the first order in the inverse of the experimental time. As application of our results, two possible experimental situations are considered: stretching of a RNA molecule and the drag of a dipolar particle in the presence of a gradient of electric force.